Gromov-Witten Invariants: Crepant Resolutions and Simple Flops Citation Cheong, Wan Keng (2010) Gromov-Witten Invariants: Crepant Resolutions and Simple Flops. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6KZZ-MT72. https://resolver.caltech.edu/CaltechTHESIS:05212010-060144793 Abstract Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Sym n (S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilb n (S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Sym n (S). Moreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Sym n (A r )], where A r is the minimal resolution of the cyclic quotient singularity C 2 /Z r+1 . Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to Bryan-Graber's Crepant Resolution Conjecture on [Sym n (A r )] and Hilb n (A r ). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym n (A r )]/Hilb n (A r ) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P 1 . Finally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space P r for positive integers r and d. The result generalizes the multiple cover formula for P r and reveals that any simple P r flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: symmetric product; Hilbert scheme; orbifold; crepant resolution; simple flop; Gromov-Witten invariant Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2010 Thesis Availability: Public (worldwide access) Research Advisor(s): Graber, Thomas B. Thesis Committee: Graber, Thomas B. (chair) Aschbacher, Michael Ramakrishnan, Dinakar Wales, David B. Defense Date: 29 April 2010 Record Number: CaltechTHESIS:05212010-060144793 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05212010-060144793 DOI: 10.7907/6KZZ-MT72 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5820 Collection: CaltechTHESIS Deposited By: Wan Keng Cheong Deposited On: 25 May 2010 16:11 Last Modified: 08 Nov 2019 18:09 Thesis Files Preview PDF - Final Version See Usage Policy. 873kB Repository Staff Only: item control page

Gromov-Witten Invariants: Crepant Resolutions and Simple Flops Thesis by Wan Keng Cheong In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2010 (Defended April 29, 2010) ii c 2010 Wan Keng Cheong All Rights Reserved iii To the memory of my mom iv Acknowledgements My deepest gratitude goes to my adviser Tom Graber for his guidance, patience, and support in the past four years. His suggestions are extremely helpful during the preparation of this thesis. Things I have learned from him are invaluable. I am thankful to my previous adviser Hee Oh though I did not follow her to Brown. While being my adviser, she was always available to help me. Her care for students was impressive. I also thank Dongping Zhuang, who helped me learn algebraic topology in my first year at Caltech when I basically knew nothing about the area. It is great to have such a nice study partner and friend. It is a privilege to acknowledge Caltech Mathematics Department for providing a nurturing environment for graduate students. Moreover, I thank Michael Aschbacher, Dinakar Ramakrishnan, and David Wales for serving on my thesis committee. I am indebted to Kaen Koh and Chun-Kai Li for their encouragement and support, which have played an important role in my academic life at Caltech. Many thanks are also due to my lovely friends Baolian Huang, Chia-Yun Kang, Cheng-Yeaw Ku, Chern-Yang Lee, Kian-Yi Lee, Chiu-Ju Lin, Yu-Ju Wei, Kathy Wu, Kai-Yee Tee, and Tzu-Yi Yang, just to name a few, for filling my life with joy, love, and surprise. This thesis is dedicated to my family. They are behind me all the way... v Abstract Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Symn (S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilbn (S) of n points in S. This proves a generalization of Ruan’s Cohomological Crepant Resolution Conjecture for the case of Symn (S). We determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Symn (Ar )], where Ar is the minimal resolution of the cyclic quotient singularity C2 /Zr+1 . Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to BryanGraber’s Crepant Resolution Conjecture on [Symn (Ar )] and Hilbn (Ar ). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symn (Ar )]/Hilbn (Ar ) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar ×P1 . Finally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space Pr for positive integers r and d. The result generalizes the multiple cover formula for P1 and reveals that any simple Pr flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan’s Minimal Model Conjecture holds. vi vii Contents Acknowledgements iv Abstract v 1 Introduction 1 1.1 1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Crepant Resolution Conjecture . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Tetrahedron of equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Minimal Model Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Setting for Chapters 2–5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Smooth toric surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Notation and convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Hilbert scheme of points 9 2.1 Fixed-point basis and Nakajima basis . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Comparison to symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Three-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 The product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Symmetric Product Stack 19 3.1 Inertia stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.1 A description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.2 Fixed-point classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Extended Gromov-Witten theory of orbifolds . . . . . . . . . . . . . . . . . . . . 25 3.3.1 The space of twisted stable maps . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Gromov-Witten invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 viii 3.4 Extended Chen-Ruan product and the product formula . . . . . . . . . . . . . . 29 3.5 Ruan conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5.1 SYM-HILB correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5.2 The cup product structure on the Hilbert scheme . . . . . . . . . . . . . . 35 4 Orbifold quantum cohomology of the symmetric product of Ar 36 4.1 Resolutions of cyclic quotient surface singularities . . . . . . . . . . . . . . . . . . 36 4.2 Connected invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Orbifold quantum product and divisor operators . . . . . . . . . . . . . . . . . . 39 4.4 Fixed loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.5 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.6 Counting branched covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.7 Localization contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.7.1 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.7.2 The setup for localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.7.3 Virtual normal bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7.4 Vanishing and relation to connected invariants . . . . . . . . . . . . . . . 54 Combinatorial descriptions of two-point extended invariants . . . . . . . . . . . . 59 4.8 5 Comparison to other theories 5.1 5.2 5.3 Relative Gromov-Witten theory of threefolds . . . . . . . . . . . . . . . . . . . . 68 5.1.1 Degree (0, n) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.2 Ar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Quantum cohomology of Hilbert schemes of points . . . . . . . . . . . . . . . . . 71 5.2.1 Quantum cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.2 SYM-HILB correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 71 The Crepant Resolution Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3.2 Nonderogatory Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3.3 Multipoint functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Ordinary flops 6.1 68 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 ix 6.2 The main formula 6.3 Minimal Model Conjecture Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 87 91 x 1 Chapter 1 Introduction 1.1 Main Results 1.1.1 The Crepant Resolution Conjecture A principle in physics states that string theory on an orbifold is equivalent to string theory on any crepant resolution of singularities. Over the years, it has been put into various mathematical frameworks. Among them, we are particularly interested in the formulations pioneered by Ruan in the context of Gromov-Witten theory (see, e.g., [R, BG, CoIT, CoR]). Let n be a positive integer and T = (C× )2 a torus. The T-equivariant cohomology of a point is simply Q[t1 , t2 ]. Given a smooth toric surface S, which comes with a T-action, the symmetric group Sn acts on the n-fold Cartesian product S n by permuting coordinates. Thus, we obtain a quotient scheme Symn (S) := S n /Sn , the n-fold symmetric product of S, and a quotient stack [Symn (S)], the n-fold symmetric product stack of S. The stack [Symn (S)] is a smooth orbifold, whose coarse moduli space is none other than the symmetric product Symn (S). Further, it has a unique crepant resolution given by the Hilbert scheme Hilbn (S) of n points in S. The string theories of [Symn (S)] and Hilbn (S) are expected to be equivalent. In order to make this precise in mathematics, Ruan [R] proposes the Cohomological Crepant Resolution Conjecture (CCRC), which asserts that the Chen-Ruan cohomology ring of [Symn (S)] is isomorphic to the so-called quantum corrected cohomology ring of the crepant resolution Hilbn (S). The first goal of this thesis, which covers Chapters 2 and 3, is to justify CCRC for symmetric products of smooth toric surfaces. In fact, we obtain a slightly stronger result, which says that there is a SYM-HILB correspondence between the degree zero Gromov-Witten theory of [Symn (S)] and the extremal Gromov-Witten theory of Hilbn (S): 2 Theorem 1.1.1. After making an appropriate change of variables and extending scalars to a suitable field F , there is a ring isomorphism ∗ L : HT,orb ([Symn (S)]; F ) → HT∗ (Hilbn (S); F ), which preserves gradings and is also an isometry with respect to Poincaré pairings. Here the left side is the equivariant extended Chen-Ruan cohomology whose structure constants are defined by three-point extended invariants of degree zero while the right side is the equivariant extremal quantum cohomology whose structure constants are defined by three-point extremal invariants. The above result holds, in particular, for surfaces such as P2 , P1 × P1 , Hirzeburch surfaces, and the total spaces of line bundles over P1 . The quantum corrected cohomology of the Hilbert scheme Hilbn (S), in the sense of [R], is defined as the extremal quantum cohomology with the unique extremal quantum parameter being specialized to −1. The following result, being an immediate consequence of Theorem 1.1.1, justifies CCRC for [Symn (S)] and Hilbn (S). Corollary 1.1.2. The equivariant Chen-Ruan cohomology of [Symn (S)] is isomorphic to the equivariant quantum corrected cohomology of Hilbn (S). We may encode three-point extended [Symn (S)]-invariants of degree zero in a three-point function (cf. Section 3.4), which depends only on the equivariant parameters t1 , t2 and the quantum parameter u corresponding to the twisted divisor. Theorem 1.1.1 allows us to reconstruct the cup product of the Hilbert scheme from the Gromov-Witten invariants of the symmetric product stack as in the following statement, which is not covered by Ruan Conjecture, however. Corollary 1.1.3. The equivariant ordinary cohomology of Hilbn (S) is isomorphic to a certain correction to the equivariant extended Chen-Ruan cohomology of [Symn (S)]. Precisely, the cup product of Hilbn (S) can be recovered from the three-point functions of [Symn (S)] by taking the limit u → i∞, where i is a square root of −1. If we have closed-form formulas for these symmetric product orbifold invariants, the cup product of the Hilbert scheme can be written down explicitly. We also study a little bit about the relative Gromov-Witten theory of S ×P1 as it seems quite close to the orbifold theory and may yield an alternative way to compute the cup product of the Hilbert scheme of points. Indeed, the degree (0, n) relative theory turns out to be equivalent to 3 the extended Chen-Ruan cohomology of [Symn (S)], and thus it is also equivalent to the extremal quantum cohomology of Hilbn (S) (see Section 5.1.1). Given a positive integer r, we let Ar be the minimal resolution of the type Ar surface singularity. The second goal of the thesis, covering Chapters 4 and 5, is to compare the equivariant orbifold Gromov-Witten theories of the symmetric products of Ar with the equivariant GromovWitten theories of the crepant resolutions in the spirit of Bryan-Graber’s Crepant Resolution Conjecture [BG]. The correspondence we obtain is stronger than Theorem 1.1.1 for the case of Ar . To formulate explicitly the correspondence, we consider the three-point functions n hhα1 , α2 , α3 ii[Sym (Ar )] ∈ Q(t1 , t2 )[[u, s1 , . . . , sr ]], which encode three-point extended Gromov-Witten invariants of [Symn (Ar )] (see (4.3.1)). These generating functions add a multiplicative structure to the equivariant Chen-Ruan cohomology ∗ HT,orb ([Symn (Ar )]; Q). The multiplication so obtained is called the small orbifold quantum product. The T-equivariant quantum cohomology of Hilbn (Ar ) has been explored by Maulik and Oblomkov in [MO1], so we need only deal with the quantum ring of the orbifold [Symn (Ar )]. We fully cover two-point extended Gromov-Witten invariants of [Symn (Ar )] and find that the calculation of these invariants is tantamount to the question of counting certain branched covers of rational curves. Our discovery can be summarized in the following statement. Theorem 1.1.4. Two-point extended equivariant Gromov-Witten invariants of [Symn (Ar )] are expressible in terms of equivariant orbifold Poincaré pairings and one-part double Hurwitz numbers. One-part double Hurwitz numbers, as shown by Goulden, Jackson, and Vakil [GJV], admit explicit closed formulas (cf. (4.8.4)), and therefore Theorem 1.1.4 provides a complete solution to the divisor operators, i.e., the operators of quantum multiplication by divisor classes. These operators correspond naturally to the divisor operators on the Hilbert scheme Hilbn (Ar ): Theorem 1.1.5. Let L be as in Theorem 1.1.1. For any Chen-Ruan cohomology classes α1 , α2 and divisor D, we have the following identity for three-point functions: n n hhα1 , D, α2 ii[Sym (Ar )] = hL(α1 ), L(D), L(α2 )iHilb (Ar ) . 4 n Here h−, −, −iHilb (Ar ) are the three-point functions of Hilbn (Ar ) in variables t1 , t2 , q, s1 , . . . , sr (see (5.2.1)), and we make the substitution q = −eiu , where i2 = −1. In addition to the relation to the Hilbert schemes, the orbifold theory is in connection with the relative Gromov-Witten theory of threefolds. Theorem 1.1.6. Given cohomology-weighted partitions λ1 (~η1 ), λ2 (~η2 ) of n and α = 1(1)n , (2) or Dk , k = 1, . . . , r (see Sections 3.2.1 and 5.1 for these classes), we have n hhλ1 (~η1 ), α, λ2 (~η2 )ii[Sym (Ar )] = GW(Ar × P1 )λ1 (~η1 ),α,λ2 (~η2 ) , where the right hand side is a shifted partition function (cf. (5.1.1)). 1.1.2 Tetrahedron of equivalences The above theorems form a triangle of equivalences. We can include the Donaldson-Thomas theory to make up a tetrahedron. In fact, Theorems 1.1.5 and 1.1.6, in conjunction with the results of [M, MO1, MO2], establish the following equivalences for divisor operators. Orbifold quantum cohomology of [Sym(Ar )] B @ B@ B @ B @ Relative Relative B @ Donaldson-Thomas Gromov-Witten XXX  XXX B  theory of Ar × P1 theory of Ar × P1 X XB  Quantum cohomology of Hilb(Ar ) Figure 1.1. A tetrahedron of equivalences. Before the study of the Gromov-Witten theory of [Symn (Ar )], the case of the affine plane C2 was the only known example for the above tetrahedron to hold for all operators (cf. [BG, BP, OP1, OP2]). If the nonderogatory conjecture (Conjecture 5.3.1) of Maulik and Oblomkov is assumed, these four theories will be equivalent in our case of Ar as well. The base triangle of “equivalences” is the work of Maulik and Oblomkov. And the triangle facing the rightmost corner is worked out in this thesis: Proposition 1.1.7. Let L be as in Theorem 1.1.1 and λ1 (~η1 ), λ2 (~η2 ), λ3 (~η3 ) any cohomology- 5 weighted partitions of n. Assuming the nonderogatory conjecture, the identities n hhλ1 (~η1 ), λ2 (~η2 ), λ3 (~η3 )ii[Sym (Ar )] n = hL(λ1 (~η1 )), L(λ2 (~η2 )), L(λ3 (~η3 ))iHilb (Ar ) = GW(Ar × P1 )λ1 (~η1 ),λ2 (~η2 ),λ3 (~η3 ) hold under the substitution q = −eiu . Once the WDVV equations are used, we can make a more general statement on [Symn (Ar )] and Hilbn (Ar ). Proposition 1.1.8. Let q = −eiu . Assuming the nonderogatory conjecture, the map L, in Theorem 1.1.1, equates the extended multipoint functions of [Symn (Ar )] to the multipoint functions of Hilbn (Ar ). Moreover, these functions are rational functions in t1 , t2 , q, s1 , . . . , sr . (Multipoint functions are those with at least three insertions). This answers positively the Crepant Resolution Conjecture, proposed by Bryan and Graber, on the symmetric product case. We will see that Propositions 1.1.7 and 1.1.8 are valid in the case of n = 2, r = 1 even without presuming the nonderogatory conjecture (see Section 5.3.1). 1.1.3 Minimal Model Conjecture The Hilbert scheme of points in a smooth surface S gives the unique crepant resolution of the symmetric product of S. In general, a singular variety may admit no crepant resolutions or more than one crepant resolution. The Minimal Model Conjecture, proposed by Ruan, is closely related to the Crepant Resolution Conjecture. It asserts that if a variety X admits two different crepant resolutions Y1 and Y2 , the Gromov-Witten theories of Y1 and Y2 are equivalent. A simple ordinary Pn flop is a K-equivalence with exceptional locus Z = Pn . In [LLW], Lee, Lin, and Wang justify the genus zero Minimal Model Conjecture for the simple flop case. We will consider extremal Gromov Witten theories for simple flops in all genera and obtain the following result. Theorem 1.1.9. Let f : X 99K X 0 be a simple ordinary Pn flop. There is a correspondence which identifies extremal Gromov-Witten theories of X and X 0 for any genus. 6 1.2 Setting for Chapters 2–5 1.2.1 Smooth toric surfaces Throughout Chapters 2–5, we let S be a smooth toric surface, which is acted on by the torus T = (C× )2 . The surface S is determined by a fan Σ that is a finite collection of strongly convex rational polyhedral cones σ contained in N = Hom(M, Z), where M ∼ = Z2 . That is, S is obtained by gluing together affine toric varieties Sσ and Sτ along Sσ∩τ for σ, τ ∈ Σ. Here, for example, Sσ has coordinate ring C[σ ∨ ∩ M ], which is the C-algebra with generators χm for m ∈ σ ∨ ∩ M and 0 0 multiplication χm χm = χm+m . Note that σ ∨ ∩ M is by definition the set of elements m ∈ M satisfying v(m) ≥ 0 for all v ∈ σ. In addition, S has finitely many T-invariant subvarieties, and so it has a finite number of T-fixed points x1 , . . . , x s . (We do not study smooth toric surfaces without T-fixed points as they are not interesting in equivariant theory.) For each i, xi is contained in Ui := Sσi for some σi ∈ Σ. As S is smooth and Ui possesses a unique T-fixed point xi , we see that Ui must be isomorphic to the affine plane with xi corresponding to the origin. However, S is not Ss necessarily the union i=1 Ui . We denote by Li and Ri the tangent weights at xi . 1.2.2 Notation and convention The following notations will be used in Chapters 2–5 without further comment. Some other notations will be introduced along the way. 1. To avoid doubling indices, we identify Ai (X) = H 2i (X; Q), Ai (X) = H2i (X; Q), and Ai (X; Z) = H2i (X; Z), 7 just to name a few, for any complex variety X to appear in this article (note that we drop Q but not Z). They will be referred to as cohomology or homology groups rather than Chow groups. 2. An orbifold X is a smooth Deligne-Mumford stack of finite type over C. Denote by c : X → X the canonical map to the coarse moduli space. 3. For every positive integer s, µs is the cyclic subgroup of C× of order s. For any finite group G, BG is the classifying stack of G, i.e., [Spec C/G]. 4. (a) T = (C× )2 is always a two-dimensional torus, and t1 , t2 are the generators of the T-equivariant cohomology A∗T (point) of a point, that is, A∗T (point) = Q[t1 , t2 ]. (b) Vm = V ⊗Q[t1 ,t2 ] Q(t1 , t2 ) for each Q[t1 , t2 ]-module V . 5. Given any object O, On means that O repeats itself n times. 6. For i = 1, 2, i is a function on the set of nonnegative integers such that i (m) =    0 if m < i;   1 if m ≥ i. 7. Let σ be a partition of a nonnegative integer. (a) `(σ) is the length of σ. (b) Unless otherwise stated, σ is presumed to be written as σ = (σ1 , . . . , σ`(σ) ) with σ1 ≥ · · · ≥ σ`(σ) . To make a emphasis, if σk is another partition, it is simply (σk1 , . . . , σk`(σk ) ). (c) Let α ~ := (α1 , . . . , α`(σ) ) be an `(σ)-tuple of cohomology classes associated to σ so that we may form a cohomology-weighted partition σ(~ α) := σ1 (α1 ) · · · σ`(σ) (α`(σ) ). The group Aut(σ(~ α)) is defined to be the group of permutations on {1, 2, . . . , `(σ)} fixing ( (σ1 , α1 ), . . . , (σ`(σ) , α`(σ) ) ). Let Aut(σ) be the group Aut(σ(~ α)) when all entries of α ~ are identical. Its order is Qn simply i=1 mi ! if σ = (1m1 , . . . , nmn ). 8 (d) |σ| = n if σ1 + · · · + σ`(σ) = n, and o(σ) = lcm(|σ1 |, . . . , |σ`(σ) |) is the order of any permutation of cycle type σ. (e) (2) := (1n−2 , 2) and 1 := (1n ) are partitions of length n − 1 and length n respectively. 9 Chapter 2 Hilbert scheme of points The Hilbert scheme of n points in S, denote by Hilbn (S) or S [n] , parametrizes zero-dimensional closed subscheme Z of S satisfying dimC H 0 (Z, OZ ) = n. The Hilbert-Chow morphism ρ : Hilbn (S) → Symn (S) is defined by sending [Z] to P p∈S `(OZ,p )[p], where the length `(OZ,p ) is simply the multiplicity of p in Z. The map ρ is not only a resolution of singularities, but it is also crepant, i.e., KHilbn (S) = ρ∗ KSymn (S) . We remark that ρ actually gives a unique crepant resolution. (This is, however, no longer true for higher-dimensional varieties). The action of T on S lifts to Hilbn (S). Each element of the fixed locus Hilbn (S)T has support in (S n )T = {x1 , . . . , xs }, and Hilbn (S)T is isolated. 2.1 Fixed-point basis and Nakajima basis Fixed-point basis. As seen in [ES], there is a one-to-one correspondence between partitions of n and T-fixed point of Hilbn (C2 ). Indeed, for every partition λ, the corresponding T-fixed points λ(xi ) ∈ Hilb|λ| (Ui ) is defined as follows: let C[u, v] be the coordinate ring for Ui ∼ = C2 , λ(xi ) is then the subscheme of Ui with ideal Iλ(xi ) being (uλ1 , vuλ2 , . . . , v `(λ)−1 uλ`(λ) , v `(λ) ). 10 The point λ(xi ) is supported at xi and is mapped to |λ| · [xi ] ∈ Sym|λ| (Ui ) by the Hilbert-Chow morphism. Note that each T-fixed point of Hilbn (S) is the sum λ1 (x1 ) + · · · + λs (xs ) for some partitions λi ’s satisfying Ps i=1 |λi | = n (by “the sum” we mean the disjoint union of e := (λ1 , . . . , λs ), write λi (xi )’s). For λ Iλe = [λ1 (x1 ) + · · · + λs (xs )], which we call T-fixed point classes of Hilbn (S), and which form a basis for the localized cohomology A∗T (Hilbn (S))m . Nakajima basis. Another important basis for A∗T (Hilbn (S)) is the Nakajima basis, which we now describe. For further details, see [Gro, N1, N2, V, LQW]. Given a partition λ of n and an `(λ)-tuple ~η := (η1 , . . . , η`(λ) ) with entries in A∗T (S). Let |0i = 1 ∈ A0T (S [0] ), we define `(λ) Y 1 1 p−λi (ηi )|0i, aλ (~η ) = |Aut(λ(~η ))| i=1 λi ∗+λi −1+deg(γ) where p−λi (ηi ) : A∗T (S [k] ) → AT (S [k+λi ] ) are Heisenberg creation operators. (Some- times we denote the class aλ (~η ) by aλ1 (η1 ) · · · aλ`(λ) (η`(λ) )). Choose a basis B for A∗T (S). The classes aλ (~η )’s, running through all partitions λ of n and all ηi ∈ B, give a basis for A∗T (Hilbn (S)). They are referred to as the Nakajima basis associated to B. We may also work with the Nakajima basis associated to the T-fixed point classes [x1 ], . . . , [xs ]. e = Ps `(λi ) and For partitions λ1 , . . . , λs of n1 , . . . , ns respectively, we define `(λ) i=1 aλe := aλ11 ([x1 ]) · · · aλ1`(λ1 ) ([x1 ]) · · · aλs1 ([xs ]) · · · aλs`(λs ) ([xs ]). 11 The Chow degree of aλe is s X e = (n − `(λ)) e + 2`(λ) e = n + `(λ). e (|λi | − `(λi )) + 2`(λ) i=1 In the case of Ui , we have exactly one T-fixed point xi . Hence, aλei , denoting the classes aλi1 ([xi ]) · · · aλi`(λi ) ([xi ])|0i (with |λi | = ni ) on Hilbni (Ui ), form a basis for A∗T (Hilbni (Ui ))m . The equivariant Poincaré pairing h•|•i of A∗T (Hilbni (Ui ))m is determined by the formula haλei |aµfi i = δλi ,µi (−1)|λi |−`(λi ) (Li Ri )`(λi ) 2.2 1 , |λi | = |µi | = ni . zλi (2.1.1) Comparison to symmetric functions Let pi (z) = P∞ i th power sum. Given a partition µ, write k=1 zk be the i `(µ) pµ (z) = Y 1 1 pµ (z). |Aut(µ)| i=1 µi i This family of symmetric functions forms a basis for the ring RSym of symmetric functions over Q(t1 , t2 ). Let αi = Ri /Li . We denote the integral Jack symmetric functions corresponding to αi and the partitions µ by Jµαi (z), which actually provide an orthogonal basis for RSym (cf. [S]). The relationship between T-fixed point basis and Nakajima basis is indeed the relationship between Jack symmetric functions and the power sums. More precisely, the Nakajima basis element a(λ1 ,...,λs ) is identified with `(λi ) ⊗si=1 Li pλi (z(i)), while the T-fixed point class I(µ1 ,...,µs ) is identified with |µ | ⊗si=1 Li i Jµαii (z(i)). 12 For more details, see [N1], [V] or [LQW]. For i = 1, . . . , s, let λi be a partition of ni . As the fixed-point classes [µi (xi )]’s (with |µi | = ni ) P span A∗T (Hilbni (Ui ))m , we can write aλei = |µi |=ni cλi ,µi [µi (xi )] for some cλi ,µi ∈ Q(t1 , t2 ). By the above identifications, a(λ1 ,...,λs ) = X cλ1 ,µ1 · · · cλs ,µs I(µ1 ,...,µs ) . (2.2.1) |µi |=ni ;i=1,...,s 2.3 Three-point functions Extremal Gromov-Witten invariants. The kernel of the morphism ρ∗ : A1 (Hilbn (S); Z) → A1 (Symn (S); Z) is one-dimensional and is generated by an effective rational curve class βn that is dual to −a2 (1)a1 (1)n−2 . For every positive integer k, we let M 0,k (Hilbn (S), d) be the moduli space parametrizing stable maps from genus zero, k-pointed, nodal curves to Hilbn (S) of degree dβn . Let ei : M 0,k (Hilbn (S), d) → Hilbn (S) be the evaluation map at the ith marked point. Although Hilbn (S) is not necessarily compact, the k-point, T-equivariant, extremal GromovWitten invariant Hilbn (S) hα1 , . . . , αk id := Z [M 0,k (Hilbn (S),d)]vir T e∗1 (α1 ) · · · e∗k (αk ) (2.3.1) is well-defined when all αi ’s are T-fixed point classes because the space of stable maps meeting these classes is compact. Since T-fixed point basis spans the cohomology A∗T (Hilbn (S))m , the Hilbn (S) invariant hα1 , . . . , αk id with insertions being any classes on Hilbn (S) can be defined by writing each αi in terms of fixed-point classes and by linearity. Another interpretation of (2.3.1) is to treat the integral as a sum of residue integrals over T-fixed connected components of M 0,k (Hilbn (S), d) via virtual localization formula. The invariants in both treatments take values in Q(t1 , t2 ). 13 We will explore the following three-point function of Hilbn (S): Extremal quantum product. Hilbn (S) hα1 , α2 , α3 i (q) := ∞ X Hilbn (S) d hα1 , α2 , α3 id q . d=0 Let {} be a basis for A∗T (Hilbn (S))m and {∨ } its dual basis. Define the extremal quantum product ∪crep for A∗T (Hilbn (S))m as follows: Given any classes a, b ∈ A∗T (Hilbn (S))m , a ∪crep b := X Hilbn (S) ha, b, i (q) ∨ .  Equivalently, a ∪crep b is defined to be the unique element satisfying ha ∪crep b|ci = ha, b, ci Hilbn (S) (q), ∀c ∈ A∗T (Hilbn (S))m . By extending scalars, the vector space A∗T (Hilbn (S)) ⊗Q[t1 ,t2 ] Q(t1 , t2 )((q)) endowed with the multiplication ∪crep is referred to as the extremal quantum cohomology ring of Hilbn (S). 2.4 The product formula For each T-fixed connected component Γ, we denote by γ : Γ → M 0,3 (Hilbn (S), d) the natural inclusion and by NΓvir the virtual normal bundle to Γ. In this section, we are going to express our three-point invariant in aλe , aµe , aνe in terms of Gromov-Witten invariants of Hilbert schemes of points in the affine plane. First of all, let us see a vanishing statement. Hilbn (S) Proposition 2.4.1. aλe , aµe , aνe d does not vanish only if |λi | = |µi | = |νi | for each i = 1, . . . , s. (2.4.1) Proof. Suppose (2.4.1) fails, we would like to show γ ∗ (e∗1 (aλe ) · e∗2 (aµe ) · e∗3 (aνe)) = 0 for every connected component Γ. By (2.2.1), we merely have to verify γ ∗ (e∗1 (Iσe ) · e∗2 (Iτe) · e∗3 (Iθe)) = 0 (2.4.2) 14 for |σi | = |λi |, |τi | = |µi |, |θi | = |νi |, ∀i = 1, 2, 3. We note that the images of all T-fixed stable maps in Γ go to the same point after composition with the Hilbert-Chow morphism. That is, ρ ◦ ei (Γ)’s are the same for each i, which means that at least one of e−1 1 (σ1 (x1 ) + · · · + −1 σs (xs )), e−1 2 (τ1 (x1 ) + · · · + τs (xs )), e3 (θ1 (x1 ) + · · · + θs (xs )) does not meet Γ. Thus, (2.4.2) follows. It remains to study the three-point invariants Hilbn (S) aλe , aµe , aνe d under condition (2.4.1): ni := |λi | = |µi | = |νi | for each i = 1, . . . , s and Ps i=1 ni = n. We fix such ni and partitions λi , µi , νi of ni throughout the remainder of this section. Let U = Hilbn1 (U1 ) × · · · × Hilbns (Us ), P = ρ−1 (n1 [x1 ] + · · · + ns [xs ]). ni ni −1 In fact, P ∼ = ρ−1 1 (n1 [x1 ]) × · · · × ρs (ns [xs ]) ⊆ U , where ρi : Hilb (S) → Sym (S) is the Hilbert-Chow morphism, ∀i. (In case ni = 0, ρ−1 i (ni [xi ]) will be missing from the product). Let N = {i = 1, . . . , s : ni ≥ 1}. As each ρ−1 i (ni [xi ]) is irreducible and has complex dimension ni − 1 for i ∈ N , P is then irreducible and has dimension n − |N |. Let ξ = µ1 (x1 ) + · · · + µs (xs ) ∈ S [n] and ξµe = µ1 (x1 ) × · · · × µs (xs ) ∈ U . We have Tξµe U = Tξ S [n] . Indeed, Tξµe U = M HomOUi (Iµi (xi ) , Oµi (xi ) ) = i∈N = M HomOS,xi (Iξ,xi , Oξ,xi ) i∈N HomOS (Iξ , Oξ ) = Tξ S [n] . Denote by ιP , P the inclusion of P into Hilbn (S) and U respectively. We have a simple lemma. Lemma 2.4.2. ι∗P (aλe ) = ∗P (aλf1 ⊗ · · · ⊗ aλfs ). Proof. By (2.2.1), it suffices to show that ι∗P I(µ1 ,...,µs ) = ∗P ([µ1 (x1 )] ⊗ · · · ⊗ [µs (xs )]). (2.4.3) Let ξ and ξµe be the points as in the discussion preceding the lemma. We can see that the left 15 side of (2.4.3) is given by X iη ∗ ( η∈P T (ιP ◦ iη )∗ I(µ1 ,...,µs ) eT (Tξ S [n] ) ) = iξ ∗ ( ), eT (Nη/P ) eT (Nξ/P ) where iη : {η} → P is the natural inclusion. Similarly, the right side of (2.4.3) coincides with iξµe ∗ (eT (Tξµe U )/eT (Nξµe /P )). Thus, the equality (2.4.3) follows from Tξµe U = Tξ S [n] . Hilbn (S) To determine the three-point invariant aλe , aµe , aνe d , we only need to consider those connected components of M 0,3 (Hilbn (S), d)T whose images under the map ρ ◦ ei are the point n1 [x1 ] + · · · + ns [xs ], ∀i = 1, 2, 3. Observe that any T-fixed stable map f , representing an element in these components, factors through P . Hence, we may calculate aλe , aµe , aνe d over connected components lying in a M 0,3 (P, (d1 , . . . , ds ))T . d1 +···+ds =d Write Γd1 ,...,ds for Γ whenever Γ is contained in the moduli space M 0,3 (P, (d1 , . . . , ds )). On the other hand, we also note that Γd1 ,...,ds ’s form a complete set of T-fixed connected components of M 0,3 (U, (d1 , . . . , ds )). The following diagram summarizes their relationships: Γd1 ,...,ds V VVVV jj j VVVV γ j j γU jjj VVVV j γ P j VVVV j j j j VV* j t  j  _  o ? / M 0,3 (U, (d1 , . . . , ds )) M 0,3 (P, (d1 , . . . , ds )) M 0,3 (Hilbn (S), d1 + · · · + ds ) Here γU , γP are the natural inclusion γ with the target replaced with M 0,3 (U, (d1 , . . . , ds )) and M 0,3 (P, (d1 , . . . , ds )) respectively. Proposition 2.4.3. The extremal invariants of Hilb(S) can be expressed in terms of the invariants of Hilb(Ui )’s. Precisely, Hilbn (S) aλe , aµe , aνe d = X s D EHilbni (Ui ) Y aλei , aµfi , aνei . d1 +···+ds =d i=1 di 16 Proof. First of all, we would like to show that Hilbn (S) aλe , aµe , aνe d X = h⊗si=1 aλei , ⊗si=1 aµfi , ⊗si=1 aνei iU (d1 ,...,ds ) . (2.4.4) d1 +···+ds =d Denote by e¯i : M 0,3 (P, (d1 , . . . , ds )) → P the evaluation map at the ith marked point. With Hilbn (S) notation as in the above discussion, the invariant aλe , aµe , aνe d X is γP∗ (ē∗1 ι∗P (aλe ) · ē∗2 ι∗P (aµe ) · ē∗3 ι∗P (aνe)) . eT (NΓvir ) Γd1 ,...,ds d ,...,ds Z X d1 +···+ds =d Γd1 ,...,ds 1 On the other hand, we find that h⊗si=1 aλei , ⊗si=1 aµfi , ⊗si=1 aνei iU (d1 ,...,ds ) is X Γd1 ,...,ds γP∗ (ē∗1 ∗P (⊗si=1 aλei ) · ē∗2 ∗P (⊗si=1 aµfi ) · ē∗3 ∗P (⊗si=1 aνei )) Z eT (NΓvir ) d ,...,ds ,U Γd1 ,...,ds , 1 is the virtual normal bundle to Γd1 ,...,ds in M 0,3 (U, (d1 , . . . , ds )). By Lemma where NΓvir d ,...,ds ,U 1 2.4.2, it is given by γP∗ (ē∗1 ι∗P (aλe ) · ē∗2 ι∗P (aµe ) · ē∗3 ι∗P (aνe)) . eT (NΓvir ) Γd1 ,...,ds d ,...,ds ,U Z X Γd1 ,...,ds 1 Hence (2.4.4) is a consequence of the equality 1 eT (NΓvir ) d1 ,...,ds = 1 eT (NΓvir ) d1 ,...,ds ,U for each Γd1 ,...,ds . (2.4.5) Now we prove (2.4.5). Given any T-fixed stable map [f : (Σ, p1 , p2 , p3 ) → P ] in the T-fixed connected component Γd1 ,...,ds . For X = S [n] or X = U, in order to verify (2.4.5), we need only examine the infinitesimal deformations of f with the source curves fixed. In fact, it suffices to check two things: eT (H 0 (Σv , f ∗ T X)mov ) eT (H 0 (Σe , f ∗ T X)mov ) and eT (H 1 (Σv , f ∗ T X)mov ) eT (H 1 (Σe , f ∗ T X)mov ) are independent of X for every connected contracted component Σv and noncontracted irreducible component Σe . The first independence holds due to the fact that Σv is of genus 0 and 17 Tf (Σv ) S [n] = Tf (Σv ) U . Thus, it remains to justify the second independence. Since Σe ∼ = P1 , L2n X f ∗ T X is a direct sum of line bundles over Σe , i.e., f ∗ T X = i=1 OΣe (`i ) for some integers `X i ’s, and we also have 2n mov eT (H 0 (Σe , f ∗ T X)mov ) Y eT (H 0 (Σe , OΣe (`X ) i )) . = X 1 ∗ mov mov 1 eT (H (Σe , f T X) ) i=1 eT (H (Σe , OΣe (`i )) ) (2.4.6) Note that the T-action on f (Σe ) (independent of X) induces a natural action on Σe and hence actions on OΣe (`X i )’s. Suppose that T acts on OΣe (1)|p0 with weight w, then the T-weight of X [n] OΣe (`)|p0 is `w. This means that Tf (p0 ) X comes with weights `X = 1 w, . . . , `2n w. But Tf (p0 ) S [n] [n] U Tf (p0 ) U , so the numbers `S1 , . . . , `S2n are exactly `U 1 , . . . , `2n after a suitable reordering. By (2.4.6), this shows the independence and ends the proof of (2.4.5). Further, the equality h⊗si=1 aλei , ⊗si=1 aµfi , ⊗si=1 aνei iU (d1 ,...,ds ) = s D EHilbni (Ui ) Y aλei , aµfi , aνei di i=1 holds. This is clear for (d1 , . . . , ds ) = (0, . . . , 0); in general, the result follows from the Product formula [Be] in equivariant context. Combining it with (2.4.4), we obtain the proposition. Proposition 2.4.4. The three-point function aλe , aµe , aνe Hilbn (S) (q) is an element of Q(t1 , t2 , q) and is given by s D Y aλei , aµfi , aνei EHilbni (Ui ) (q). i=1 ni (Ui ) Proof. Each term haλei , aµfi , aνei iHilb (q) is a rational function in t1 , t2 , q (cf. [OP1]). By Propositions 2.4.1 and 2.4.3, aλe , aµe , aνe Hilbn (S) (q) = = = ∞ X X ( s D Y d=0 d1 +...+ds =d i=1 s X ∞ D Y aλei , aµfi , aνei aλei , aµfi , aνei i=1 di =0 s D Y aλei , aµfi , aνei EHilbni (Ui ) EHilbni (Ui ) )q d di q di di EHilbni (Ui ) (q), i=1 as desired. Together with (2.1.1), we have the following result on Poincaré pairings. Proposition 2.4.5. The equivariant Poincaré pairing on Hilbn (S) can be written in terms of 18 the pairings on the Hilbert schemes of points in the affine plane. More precisely, haµe |aνei = s Y δµi ,νi (−1)|µk |−`(µk ) (Li Ri )`(µi ) i=1 In other words, aµe ’s give an orthogonal basis for A∗T (Hilbn (S))m . 1 . zµi 19 Chapter 3 Symmetric Product Stack Let X be a smooth complex variety. Given any finite set N , let SN be the symmetric group on N and X N = {(xi )i∈N : xi ’s are elements of X}, a set of |N |-tuples of elements of X. We denote by Sn the group S{1,...,n} and by X n the set X {1,...,n} . The symmetric group Sn acts on the n-fold product X n by permutation of coordinates. The quotient scheme Symn (X) := X n /Sn is referred to as the n-fold symmetric product of X and is the coarse moduli space of the quotient stack [Symn (X)] defined as follows: • An object over U is a pair (p : P → U, f : P → X n ) where p is a principal Sn -bundle, and f is a Sn -equivariant morphism. • Suppose that (p0 : P 0 → U 0 , f 0 : P 0 → X n ) is another object, a morphism from (p0 , f 0 ) to (p, f ) is a Cartesian diagram α P 0 −−−−→ P    0 p yp y β U 0 −−−−→ U such that f 0 = α ◦ f . Note that the stack [Symn (X)] is an orbifold with atlas X n → [X n /Sn ]. 20 3.1 Inertia stack There is a natural stack associated to the symmetric product, i.e., the inertia stack I[Symn (X)] := a HomRep(Bµs , [Symn (X)]), s∈N where HomRep(Bµs , [Symn (X)]) is the stack of representable morphisms from the classifying stack Bµs to [Symn (X)]. Moreover, I[Symn (X)] is isomorphic to the disjoint union of orbifolds a [Xgn /C(g)], (3.1.1) [g]∈C where C is the set of conjugacy classes, Xgn is the g-fixed locus of X n , and C(g) is the centralizer of g. The component [X n /Sn ] is called the untwisted sector while all other components are called twisted sectors. As there is a one-to-one correspondence between the conjugacy classes of Sn and the partitions of n, these sectors can be labeled with the partitions of n. If [g] is the conjugacy class corresponds to the partition λ and C(g) := C(g)/hgi, we may write X(λ) := Xgn /C(g), and X(λ) := Xgn /C(g) (see below). The Chen-Ruan cohomology A∗orb ([Symn (X)]) is by definition the cohomology A∗ (I[Symn (X)]) of the inertia stack ([ChR1]). By (3.1.1), it is M [g]∈C A∗ (Xgn /C(g)) = M A∗ (Xgn )C(g) . [g]∈C (For any orbifold Y with coarse moduli space Y , we identify A∗ (Y) = A∗ (Y ) by the pushforward c∗ : A∗ (Y) → A∗ (Y ) defined by c∗ ([V]) = 1s [c(V)], where V is a closed integral substack and s is the order of the stabilizer of a generic geometric point of V). The age (or the degree shifting number) of the sector [X(λ)] is given by age(λ) := n − `(λ). Additionally, the Chen-Ruan cohomology is graded by ages. If α ∈ Ai (X(λ)), the orbifold 21 (Chow) degree of α is defined to be i + age(λ). In other words, A∗orb ([Symn (X)]) = M A∗−age(λ) (X(λ)). |λ|=n We may rigidify the inertia stack to remove the actions of µs ’s. Each Bµs acts on the stack HomRep(Bµs , [Symn (X)]), and the quotient by this action is a stack of gerbes banded by µs to [Symn (X)]. The stack I[Symn (X)] := a HomRep(Bµs , [Symn (X)])/Bµs , s∈N is called the rigidified inertia stack of [Symn (X)]. One of the reasons why we mention this is that the rigidified stack is where the evaluation maps land (see (3.3.1)). For more details on the rigidification procedure, consult [ACV, AGV1, AGV2]. In fact, the procedure amounts to removing the action of the permutation g from (3.1.1) (note that g acts trivially on Xgn ). Concretely, the stack I[Symn (X)] is the disjoint union a [Xgn /C(g)] (or 3.2 Bases 3.2.1 A description [X(λ)]). |λ|=n [g]∈C However, its coarse moduli space a n [g]∈C Xg /C(g) is identical to that of the inertia stack. ` Assume that X admits a T-action. We can see easily that there are induced T-actions on the spaces Xgn /C(g) (∀g ∈ Sn ) and I[Symn (X)]. So we may put the above cohomologies into an equivariant context by considering T-equivariant cohomologies. Now we would like to understand the module structure of the equivariant Chen-Ruan cohomology A∗T,orb ([Symn (X)]); particularly, we need a precise description of bases. Given a partition λ of n, we would like to give a basis for the cohomology A∗T (Xgn )C(g) , where g ∈ Sn has cycle type λ. The permutation g has a cycle decomposition, i.e., a product of disjoint cycles (including 1-cycles), g = g1 . . . g`(λ) , 22 with gi being a λi -cycle. For each i, let Ni be the minimal subset of {1, . . . , n} such that ``(λ) gi ∈ SNi . Thus |Ni | = λi and i=1 Ni = {1, . . . , n}. It is clear that `(λ) Xgn = Y XgNi i , and XgNi i ∼ = X. i=1 To the partition λ, we associate an `(λ)-tuple ~η = (η1 . . . η`(λ) ) with entries in A∗T (X). Let us put `(λ) g(~η ) = (|Aut(λ(~η ))| Y λi ) −1 i=1 `(λ) X O gih (ηi ) ∈ A∗T (Xgn )C(g) . (3.2.1) h∈C(g) i=1 This requires some explanations: • gih := h−1 gi h. • Let N be a subset of {1, . . . , n}. For each |N |-cycle α ∈ SN and η a class on X, let α(η) be the pullback of η by the obvious isomorphism XαN ∼ = X. N`(λ) h1 i=1 gi (ηi ) N`(λ) h2 i=1 gi (ηi ) on Xgn coincide for some h1 , h2 ∈ C(g), N`(λ) h and a straightforward verification shows that each term i=1 gi (ηi ) repeats precisely Q`(λ) Q`(λ) |Aut(λ(~η ))| i=1 λi times. Hence, (|Aut(λ(~η ))| i=1 λi )−1 is a normalization factor to • Two classes and ensure that no repetition occurs in (3.2.1). • If g = k1 · · · k`(λ) is another cycle decomposition with λi -cycles ki ’s, then there exists h ∈ C(g) such that `(λ) O ki (ηi ) = i=1 `(λ) O gih (ηi ). i=1 Thus, the expression (3.2.1) is independent of the cycle decomposition. Let B be a basis for A∗T (X). The classes g(~η )’s, with ηi ’s elements of B, form a basis for A∗T (Xgn )C(g) . Suppose that ĝ is another permutation of cycle type λ, i.e., ĝ = g α for some α ∈ Sn . The classes g(~η ) and ĝ(~η ) are identical in A∗T,orb ([Symn (X)]) though they are related by ring isomorphism α∗ : A∗T (Xgn )C(g) → A∗T (Xĝn )C(ĝ) induced by α. In fact, α∗ sends g(~η ) to ĝ(~η ) and is independent of the choice of α due to the fact that ϑ∗ : A∗T (Xgn )C(g) → A∗T (Xgn )C(g) is the identity for every ϑ ∈ C(g). We use the cohomology-weighted partition λ1 (η1 ) · · · λ`(λ) (η`(λ) ) or simply λ(~η ) 23 to denote the class g(~η ) (and hence ĝ(~η )). Now the classes λ(~η )’s, running over all partitions λ of n and all ηi ∈ B, serve as a basis for the Chen-Ruan cohomology A∗T,orb ([Symn (X)]). For classes λ(~η ) ∈ A∗T,orb ([Symn (X)]) and ~ ∈ A∗ ([Symm (X)]), keep in mind that the class ρ(ξ) T,orb λ1 (η1 ) · · · λ`(λ) (η`(λ) )ρ1 (ξ1 ) · · · ρ`(ρ) (ξ`(ρ) ) ∈ A∗T,orb ([Symn+m (X)]) is denoted by ~ λ(~η )ρ(ξ). We use the shorthand (2) for the divisor class 2(1)1(1)n−2 . Also, we define the age of λ(~η ), denoted by age(λ(~η )), to be the age of the sector [X(λ)], i.e., n − `(λ). 3.2.2 Fixed-point classes We can work with λ(~η )’s with ηk ’s in the localized cohomology A∗T (X)m to give a basis for A∗T,orb ([Symn (X)])m . Assume that X has exactly p T-fixed points z1 , . . . , zp . For partitions σ1 , . . . , σp , we denote the class σ11 ([z1 ]) · · · σ1`(σ1 ) ([z1 ]) · · · σp1 ([zp ]) · · · σp`(σp ) ([zp ]) by σ e := (σ1 , . . . , σp ). The classes σ e’s form a basis for A∗T,orb ([Symn (X)])m . Note also that each σ e corresponds to a T-fixed point, which we denote by [e σ ], in the sector indexed by the partition (σ11 , . . . , σ1`(σ1 ) , . . . , σp1 , . . . , σp`(σp ) ). So we refer to σ e’s as T-fixed point classes. 24 Moreover, given δe ∈ A∗T,orb ([Symn (X)])m and σ e ∈ A∗T,orb ([Symm (X)])m (m ≤ n), we say that δe ⊃ σ e if σk is a subpartition of δk , ∀k = 1, . . . , p; in this case, we let δe − σ e := (δ1 − σ1 , . . . , δp − σp ) ∈ A∗T,orb ([Symn−m (X)])m . (e.g., the difference (1, 1, 2, 2, 3) − (1, 2, 3) of two partitions is the partition (1, 2).) T-weights. Given any fixed-point class σ e, let m ¯ (X)]). t(e σ ) = eT (T[eσ] I[Sym A simple analysis shows that t(e σ) = p Y `(σk ) eT (Tzk X) . (3.2.2) k=1 e = t(e Thus, for each δe ⊃ σ e, t(δ) σ )t(δe − σ e). ~ ∈ A∗ ([Symm (X)])m , we have Coefficients with respect to fixed-point basis. For θ(ξ) T,orb ~ = θ(ξ) X hθ(ξ)|e ~ σi σ e he σ |e σi σ e, where h•|•i are T-equivariant orbifold pairings on A∗T,orb ([Symm (X)])m . Now let αθ(ξ) σ ) := ~ (e ~ σi hθ(ξ)|e he σ |e σi ~ relative to σ be the components of θ(ξ) e’s. We have two properties by direct verification: (1) Suppose λ(~η ), ρ(~ε) ∈ A∗T,orb ([Symn (X)])m have explicit forms mi n Y Y i=1 j=1 i(ηij ) and `i n Y Y i=1 j=1 i(εij ) 25 respectively, we have hλ(~η )|ρ(~ε)i =    0 if mi 6= `i for some i;  Qmi Q Q i   ni=1 h m j=1 i(εij )i if mi = `i for each i. j=1 i(ηij ) · (2) Given η1 , . . . , ηn ∈ A∗T (X)m and T-fixed points y1 , . . . , yn of X. For m ≤ n, the coefficient αi(η1 )···i(ηn ) (i([y1 ]) · · · i([yn ])) equals X αi(ξ1 )···i(ξm ) (i([y1 ]) · · · i([ym ]))αi(ξm+1 )···i(ξn ) (i([ym+1 ]) · · · i([yn ])), where the sum is over all possible i(ξ1 ) · · · i(ξm ) and i(ξm+1 ) · · · i(ξn ) such that i(ξ1 ) · · · i(ξn ) = i(η1 ) · · · i(ηn ). We may combine (1) with (2) to get a general statement, which presents an algorithm to calculate e the coefficient αλ(~η) (δ). Proposition 3.2.1. Given λ(~η ), δe ∈ A∗T,orb ([Symn (X)])m and σ e ∈ A∗T,orb ([Symm (X)])m with δe ⊃ σ e, e = αλ(~η) (δ) X αθ(ξ) σ )αµ(~γ ) (δe − σ e), ~ (e (3.2.3) P where the index P under the summation symbol means that the sum is taken over all possible ~ ∈ A∗ ([Symm (X)])m and µ(~γ ) ∈ A∗ ([Symn−m (X)])m satisfying λ(~η ) = θ(ξ)µ(~ ~ γ ). θ(ξ) T,orb T,orb In the proposition, δe is separated into two parts σ e and δe − σ e. In general, we can break it as many parts as possible. The form (3.2.3) is, however, convenient for later use. 3.3 Extended Gromov-Witten theory of orbifolds To make our exposition as self-contained as possible, we review some relevant background on orbifold Gromov-Witten theory. We take the algebro-geometric approach in the sense of Abramovich, Graber and Vistoli’s works [AGV1, AGV2]. The reader may also want to consult the original work [ChR2] of Chen and Ruan in symplectic category. In what follows, we utilize the isomorphism A1 (Symn (X); Z) ∼ = A1 (X n ; Z)Sn ∼ = A1 (X; Z). 26 3.3.1 The space of twisted stable maps For any curve class β ∈ A1 (X; Z), the moduli space M 0,k ([Symn (X)], β)1 parametrizes genus zero, k-pointed, twisted stable map (or orbifold stable map in [ChR2]) f : (C, P1 , . . . , Pk ) → [Symn (X)] with the following conditions: • (C, P1 , . . . , Pk ) is an twisted nodal k-pointed curve. The marking Pi is an étale gerbe banded by µri , where ri is the order of the stabilizer of the twisted point. Moreover, over a node, C has a chart isomorphic to Spec C[u, v]/(uv)/µs where µs acts on Spec C[u, v] by ξ · (u, v) = (ξu, ξ −1 v), and the canonical map c : C → C is given by x = us , y = v s in this chart. • f is a representable morphism and gives rise to a genus zero, k-pointed, stable map fc : (C, c(P1 ), . . . , c(Pk )) → Symn (X) of degree β by passing to coarse moduli spaces. Note that the canonical map c : C → C is an isomorphism away from the nodes and marked gerbes and that whenever we say that f is of degree β, we actually mean fc is. There are evaluation maps on the moduli space M 0,k ([Symn (X)], β), which take values in the rigidified inertia stack. At the level of Spec(C)-points, the ith evaluation map evi : M 0,k ([Symn (X)], β) → I[Symn (X)] (3.3.1) is defined by [f : (C, P1 , . . . , Pk ) → [Symn (X)]] 7−→ [f |Pi : Pi → [Symn (X)]]. The moduli space M 0,k ([Symn (X)], β) can be decomposed into open and closed substacks: M 0,k ([Symn (X)], β) = a M ([Symn (X)], σ1 , . . . , σk ; β). σ1 ,...,σk −1 Here M ([Symn (X)], σ1 , . . . , σk ; β) = ev−1 1 ([X(σ1 )])∩· · ·∩evk ([X(σk )]), which can be empty for monodromy reason (e.g., the component M ([Sym3 (X)], (2), (2), (2); β) is empty), and the union 1 [AGV1] and [AGV2] adopt the notation K instead of M . Also, we just describe the Spec(C)-points of the moduli stack in this article. This is enough because our main purpose is the calculation of Gromov-Witten invariants. 27 is taken over all partitions σ1 , . . . , σk of n. Keep in mind that the substack carries a virtual class [M ([Symn (X)], σ1 , . . . , σk ; β)]vir of dimension −K[Symn (X)] · β + n · dim(X) + k − 3 − k X age(σi ). i=1 The twisted map f that represents an element of M ([Symn (X)], σ1 , . . . , σk ; β) amounts to the following commutative diagram f0 PC −−−−→   y Xn  π y f C −−−−→ [Symn (X)]    c cy y (3.3.2) fc C −−−−→ Symn (X) where π is the natural map, PC := C ×[Symn (X)] X n is a scheme by representability of f , and f 0 is Sn -equivariant. Away from the marked points and nodes, PC is a principal Sn -bundle of C. It is branched over the markings with ramification types σ1 , . . . , σk . Additionally, there is such a diagram C̃   py f˜ −−−−→ X (3.3.3) (C, c(P1 ), . . . , c(Pk )) associated to f that p : C̃ → C is an admissible cover branched over c(P1 ), . . . , c(Pk ) with monodromy given by σ1 , . . . , σk , and f˜ : C̃ → X is a degree β morphism such that if Σ ⊂ C is a rational curve possessing less than 3 special points, then there is a component of p−1 (Σ) which is not f˜-contracted. In fact, (3.3.3) is induced by the diagram (3.3.2) by taking f 0 mod Sn−1 and composing with the nth projection. The diagram (3.3.3) will be particularly helpful later in the descriptions of T-fixed loci for the space of twisted stable maps to [Symn (Ar )]. The reader should look closely at the above notation. We will use (3.3.2) and (3.3.3) and the symbols there mostly without further comment. 28 3.3.2 Gromov-Witten invariants For any cohomology classes αi ∈ A∗T,orb ([Symn (S)]) (i = 1, . . . , k), the k-point equivariant Gromov-Witten invariant is defined by [Symn (S)] hα1 , . . . , αk iβ := Z [M ([Symn (S)],β)]vir T ev∗1 (α1 ) · · · ev∗k (αk ), (3.3.4) where the symbol [ ]vir T stands for the T-equivariant virtual class. The underlying moduli space is not necessarily compact, but the above definition makes sense because of a similar treatment given in Section 2.3. Moreover, it is convenient to express the integral in (3.3.4) as a sum of integrals against the virtual fundamental classes of the components M ([Symn (S)], σ1 , . . . , σk ; β)’s. In the context of orbifolds, it is in reality more natural to study the Gromov-Witten theory in twisted degrees, i.e., in curve classes of Aorb,1 ([Symn (S)]; Z) = A0 ([S((2))]; Z) ⊕ A1 ([Symn (S)]); Z). This makes a lot of sense because the direct sum matches A1 (Hilbn (S); Z) (cf. Section 5.2.1). Let us identify A0 ([S((2))]; Z) with Z. To define the k-point extended Gromov-Witten invari[Symn (S)] ant hα1 , . . . , αk i(a,β) of twisted degree (a, β) ∈ Z⊕A1 (S; Z) with a ≥ 0, we include additional a unordered markings in the twisted stable map of degree β above such that these markings go to the age one sector under the corresponding evaluation maps. To make this precise, we present a formula: [Symn (S)] hα1 , . . . , αk i(a,β) = 1 [Symn (S)] hα1 , . . . , αk , (2)a iβ . a! (3.3.5) Note that in the expression, the last a insertions are all (2) and the invariant is defined to be zero in case a < 0. For later convenience of explanation, we refer to the markings associated to α1 , . . . , αk as distinguished marked points and to the other a markings as simple marked points. Also the markings corresponding to the twisted sectors are called twisted and are otherwise called untwisted. The expression (3.3.5) is almost identical to the nonextended version except for the appear1 due to the fact that we do not order simple markings. Additionally, we say ance of the factor a! [Symn (S)] that hα1 , . . . , αk i(a,β) is of nonzero (resp. zero) degree if it is a Gromov-Witten invariant [Symn (S)] (up to a multiple) of nonzero (resp. zero) degree and that hα1 , . . . , αk i(a,β) k ≥ 3. is multipoint if 29 Like ordinary Gromov-Witten theory, if β 6= 0 or k ≥ 3, we have a forgetful morphism f tk+1 : M ([Symn (S)], σ1 , . . . , σk , 1; β) → M ([Symn (S)], σ1 , . . . , σk ; β) defined by forgetting the last untwisted marked points. The (untwisted) divisor equation holds as well in the orbifold case. Unfortunately, we are not allowed to forget twisted markings in general. 3.4 Extended Chen-Ruan product and the product formula In the remainder of this chapter, we will be primarily interested in three-point extended degree zero invariants. Let us simplify our notation a little. For partitions σ1 , . . . , σm of n, let M ([Symn (S)], σ1 , . . . , σm ; a) = m \ a \ ev−1 i ([S(σi )]) ∩ i=1 ev−1 m+j ([S((2))]) j=1 be an open and closed substack of the moduli space M m+a ([Symn (S)], 0). Given any αi ∈ A∗T ([Symn (S)]) for i = 1, . . . , m, we put [Symn (S)] hα1 , . . . , αm ia [Symn (S)] := hα1 , . . . , αm i(a,0) . We encode the invariants in a three-point function: hα1 , α2 , α3 i [Symn (S)] (u) = X [Symn (S)] hα1 , α2 , α3 ia ua . a Now let {γ} be a basis for A∗T,orb ([Symn (S)]) and {γ ∨ } its dual basis. Define the extended Chen-Ruan product ∪orb for A∗T,orb ([Symn (S)]) in this way: α1 ∪orb α2 = X hα1 , α2 , γi [Symn (S)] (u) γ ∨ . γ We call the vector space A∗T,orb ([Symn (S)]) ⊗Q[t1 ,t2 ] Q(t1 , t2 )((u)) with the multiplication ∪orb the extended Chen-Ruan cohomology ring of [Symn (S)]. (Note that the associativity of ∪orb follows from the WDVV equation.) 30 We use the notation ~g = (g1 , . . . , gs ) (3.4.1) to represent an s-tuple of partitions or an s-tuple of nonnegative integers. In the case of integers, we say that |~g | = ` if the entries of ~g add up to `. Fixed loci. Given s-tuples σ1 , . . . , σm of partitions and an s-tuple ~a of nonnegative integers with |~a| = d. Let M(~σ1 , . . . , ~σm ; ~a) 0 be the union of T-fixed components of M ([Symn (S)], σ10 , . . . , σm ; d) (here σi0 admits a decompo- sition (σi1 , . . . , σis )) with the following configuration: Let [f : C → [Symn (S)]] be any element. As discussed earlier, it comes naturally with diagram (3.3.2). In addition, the twisted stable map f has the following properties: • The associated admissible cover C̃ of C is ramified with monodromy σ1 , . . . , σm , (2)d and has components C̃k , k = 1, . . . , s. Each C̃k , if nonempty, is contracted by f˜ to xk . (C̃k is possibly empty or disconnected. Empty sets are included just for the simplicity of notation.) • The cover C̃k → C is ramified with monodromy σ1k , . . . , σmk , (2)ak , 1d−ak for every k. Suppose that |σik | = nk for some nk , ∀i = 1, . . . , m, we have a natural morphism φ : M(~σ1 , . . . , ~σm ; ~a) → s Y M ([Symnk Uk ], σ1k , . . . , σmk ; ak )T k=1 defined as follows: Let [f : C → [Symn (S)]] be an element of M(~σ1 , . . . , ~σm ; ~a). For each k, we stabilize the target of the covering C̃k → C and the domain accordingly (by forgetting those simple markings of C over which the points of C̃k are unramified). The output is the following setting C̃kstk −−−−→ {xi }   y C stk (3.4.2) 31 with the vertical map being an admissible covering. It gives rise to a T-fixed twisted stable map fk : Ck → [Symnk (Uk )], which represents a T-fixed point of M ([Symnk Uk ], σ1k , . . . , σmk ; ak ). We then take φ([f ]) := ([f1 ], . . . , [fs ]). Now we focus on m = 3, in which case the morphism φ is surjective, and both its source and target have dimension d. Given any T-fixed connected component F (~a) ⊆ M(~σ1 , ~σ2 , ~σ3 ; ~a) and denote by πk the k th projection, we let Fk (~a) = πk ◦ φ(F (~a)). Qs The collection k=1 Fk (~a)’s form a complete set of T-fixed connected components of the product Qs space k=1 M ([Symnk Uk ], ~σ1k , ~σ2k , ~σ3k ; ak ). Determination. [Symn (S)] e µ We want to investigate the invariant hλ, e, νeid . It is clearly zero if the condition |λk | = |µk | = |νk | = nk for each k = 1, . . . , s (3.4.3) fails. In general, we have the following product formula. e µ Proposition 3.4.1. Given any T-fixed point classes λ, e, νe, [Symn (S)] e µ hλ, e, νeid s Y X = nk fk , µ hλ fk , νek i[Sym ak (Uk )] . a1 +···+as =d k=1 Proof. The statement is obvious when (3.4.3) does not hold. So let us assume (3.4.3). The only fixed loci that can make contribution to the three-point extended invariant [Symn (S)] e µ hλ, e, νeid (3.4.4) are M(~λ, µ ~ , ~ν ; ~a)’s with |~a| = d. Precisely, (3.4.4) is given by 1 X X d! |~ a|=d F (~ a) Z ∗ e · ev∗ (e ν )) ι∗F (~a) (ev∗1 (λ) 2 µ) · ev3 (e eT (NFvir(~a) ) F (~ a) , where F (~a) ⊂ M(~λ, µ ~ , ~ν ; ~a) runs over all T-fixed connected components. Given any T-fixed component F (~a) ⊂ M(~λ, µ ~ , ~ν ; ~a) and [f ] ∈ F (~a), we have eT (H i (C, f ∗ T [Symn (S)])) = φ∗ s O k=1 eT (H i (Ck , fk∗ T [Symnk Uk ])) 32 for i = 0, 1, and so s O eT (NFvir(~a) ) = φ∗ eT (NFvir a) ). k (~ k=1 Moreover, by (3.2.2), we check immediately that ∗ e · ev∗ (e ι∗F (~a) (ev∗1 (λ) ν )) = 2 µ) · ev3 (e s Y ∗ fk ) · ev∗ (f ι∗Fk (~a) (ev∗1 (λ 2 µk ) · ev3 (νek )). k=1 Hence the contribution of M(~λ, µ ~ , ~ν ; ~a) to (3.4.4) equals s XY 1 a1 ! · · · as ! F (~ a) k=1 Z ∗ fk ) · ev∗ (f ι∗Fk (~a) (ev∗1 (λ 2 µk ) · ev3 (νek )) eT (NFvir a) ) k (~ Fk (~ a) , where the prefactor accounts for the distribution of simple marked points. The sum is nothing but Z s ∗ fk ) · ev∗ (f Y ι∗Fk (~a) (ev∗1 (λ 1 X 2 µk ) · ev3 (νek )) . ak ! eT (NFvir Fk (~ a) a) ) k (~ k=1 Fk (~ a) Since Fk (~a)’s run through all connected components of M ([Symnk Uk ], ~λ1k , ~λ2k , ~λ3k ; ak )T , (3.4.4) P Qs [Symnk (Uk )] fk , µ and |~a|=d k=1 hλ fk , νek iak coincide. Put another way, we have the following. e µ Corollary 3.4.2. For any T-fixed point classes λ, e, νe, n e µ hλ, e, νei[Sym (S)] (u) = s Y nk fk , µ hλ fk , νek i[Sym (Uk )] (u). k=1 Moreover, any extended three-point function is a rational function in t1 , t2 , eiu , where i2 = −1. Proof. The first statement is immediate from Proposition 3.4.1. The second statement is due to n fk , µ the fact that each hλ fk , νek i[Sym k (Uk )] (u) is an element of Q(t1 , t2 , eiu ). The orbifold Poincaré pairing h•|•i on A∗T,orb ([Symnk (Uk )])m is determined by fk |f hλ µk i = δλk ,µk (Lk Rk )`(λk ) 1 , |λk | = |µk | = nk . zλk (3.4.5) The argument of Proposition 3.4.1 may be applied to show that the orbifold pairing on [Symn (S)] is expressible in terms of those on [Symnk (Uk )]’s. 33 Proposition 3.4.3. The equivariant orbifold Poincaré pairing on [Symn (S)] is determined by the formula: e µi = hλ|e s Y δλk ,µk (Lk Rk )`(λk ) k=1 1 zλk . e provide an orthogonal basis for A∗ ([Symn (S)])m . Thus, λ’s T,orb 3.5 Ruan conjecture 3.5.1 SYM-HILB correspondence In this section, we give a SYM-HILB correspondence that relates the Gromov-Witten theories of Hilbert scheme of points and symmetric products. Let q = −eiu , where i2 = −1, and F = Q(i, t1 , t2 , q). Define L : A∗T,orb ([Symn (S)]) → A∗T (Hilbn (S)) by e = (−i)age(λ) ae . L(λ) λ e As we have a bijection between the bases on both sides, L extends to a Q(i, t1 , t2 )((u))-linear isomorphism. However, extended three-point functions of [Symn (S)] and three-point functions of Hilbn (S) are elements of F . So we may view L as an F -linear isomorphism. e has orbifold Chow degree Note that λ e + age(λ) e = n + `(λ), e 2`(λ) which matches the Chow degree of aλe . Further, L is an isometry: e Proposition 3.5.1. For every fixed-point class λ, e λi e = hL(λ)|L( e e hλ| λ)i. e e λi e = (−1)age(λ) Proof. Propositions 3.4.3 and 2.4.5 say that hλ| haλe |aλe i. 34 Theorem 3.5.2. The map L : A∗T,orb ([Symn (S)]) ⊗Q[t1 ,t2 ] F → A∗T (Hilbn (S)) ⊗Q[t1 ,t2 ] F is an F -algebra isomorphism. In particular, equivariant Ruan conjecture holds for Symn (S). Proof. The proof relies on the affine plane case. Indeed, Okounkov-Pandharipande and BryanGraber determine the structures of T-equivariant quantum cohomology rings of Hilbn (C2 ) and [Symn (C2 )] respectively. Also, these rings are related by the correspondence LC2 : A∗T,orb ([Symn (C2 )]) ⊗Q[t1 ,t2 ] F → A∗T (Hilbn (C2 )) ⊗Q[t1 ,t2 ] F, which sends µ1 ([0]) · · · µ`(µ) ([0]) to (−i)age(µ) aµ1 ([0]) · · · aµ`(µ) ([0]). Bryan and Graber show that LC2 preserves three-point functions. That is, for any Chen-Ruan classes δ1 , δ2 , δ3 on the orbifold [Symn (C2 )], we have n Hilbn (C2 ) 2 hδ1 , δ2 , δ3 i[Sym (C )] (u) = hLC2 (δ1 ), LC2 (δ2 ), LC2 (δ3 )i (q). (3.5.1) As Uk ∼ = C2 , we denote the associated correspondence by LUk : A∗T,orb ([Symn (Uk )]) ⊗Q[t1 ,t2 ] F → A∗T (Hilbn (Uk )) ⊗Q[t1 ,t2 ] F. Given any equivariant Chen-Ruan classes α1 , α2 , α3 on [Symn (S)], let us show the identity n n hα1 , α2 , α3 i[Sym (S)] (u) = hL(α1 ), L(α2 ), L(α3 )iHilb (S) (q). e µ It is, however, enough to establish that for all λ, e, νe satisfying (3.4.3), n n e µ e L(e hλ, e, νei[Sym (S)] (u) = hL(λ), µ), L(e ν )iHilb (S) (q). (3.5.2) n e µ In fact, by (3.5.1) and Corollary 3.4.2, the invariant hλ, e, νei[Sym (S)] (u) is given by s Y nk (Uk )] fk , µ hλ fk , νek i[Sym k=1 Clearly, age(e σ) = (u) = s Y nk fk ), LU (f hLUk (λ µk ), LUk (νek )iHilb (Uk ) (q). k k=1 Ps e. k=1 age(σk ) for any fixed-point class σ By applying Proposition 2.4.4, we 35 obtain (3.5.2). In addition, according to Proposition 3.5.1, L preserves Poincaré pairings, and so we have an equality: hL(α1 ∪orb α2 )|L(α3 )i = hL(α1 ) ∪crep L(α2 )|L(α3 )i. This yields L(α1 ∪orb α2 ) = L(α1 ) ∪crep L(α2 ) and proves the isomorphism of algebras. 3.5.2 The cup product structure on the Hilbert scheme An upshot of Theorem 3.5.2 is that three-point degree zero invariants of Hilbert schemes is expressible in terms of degree zero invariants of symmetric product stacks. Corollary 3.5.3. Given cohomology-weighted partitions λ1 (η~1 ), λ2 (η~2 ), λ3 (η~3 ), the degree zero Hilbn (S) invariant aλ1 (η~1 ) , aλ2 (η~2 ) , aλ3 (η~3 ) 0 P3 i k=1 age(λk ) is given by n lim hλ1 (η~1 ), λ2 (η~2 ), λ3 (η~3 )i[Sym (S)] (u). u→+i∞ (3.5.3) Proof. It follows immediately from Theorem 3.5.2 by taking q → 0. This is the precise statement of Corollary 1.1.3. Since the ordinary cup product of Hilbn (S) is defined by three-point degree zero invariants, the corollary says that the three-point extended [Symn (S)]-invariants of degree zero completely determine the cup product of Hilbn (S). This may shed new light on the explicit calculation of the ordinary cohomology ring of the Hilbert scheme of points. 36 Chapter 4 Orbifold quantum cohomology of the symmetric product of Ar In Chapters 4 and 5,1 we basically restrict our attention to the case where S is the minimal resolution of type Ar surface singularity. 4.1 Resolutions of cyclic quotient surface singularities We fix a positive integer r once and for all. Let the cyclic group µr+1 act on C2 by the diagonal matrices   ζ  0 0  , ζ −1 where ζ ∈ µr+1 . The quotient C2 /µr+1 is a surface singularity. We denote by π : Ar → C2 /µr+1 its minimal resolution. It is actually well-known that π can be obtained via a sequence of b r+1 2 c blow-ups at the unique singularity. The exceptional locus Ex(π) of π is a chain of (−2)-curves, r [ Ei , i=1 1 Some results in these two chapters have also appeared in [CG] but the materials presented here are due solely to myself. 37 with Ei−1 and Ei intersect transversally. The intersection numbers of the exceptional curves are given by Ei · Ej =     −2     if i = j; if |i − j| = 1; 1       0 otherwise. In particular, the intersection matrix is negative definite (as expected from the general theory of complex surfaces). Additionally, E1 , . . . , Er give a basis for A1 (Ar ; Z). We also have two noncompact curves E0 and Er+1 attached to E1 and Er respectively. The curve E0 (resp. Er+1 ) can be arranged to map to the µr+1 -orbit of x-axis (resp. y-axis). The natural action of T on C2 comes with tangent weights t1 and t2 at the origin. It commutes with the µr+1 -action, so we have an induced T-action on the quotient C2 /µr+1 and thus on the resolved surface Ar . We fix these actions of T throughout Chapters 4 and 5. The T-invariant curves on Ar are E1 , . . . , Er , and there are r + 1 T-fixed points x1 , . . . , xr+1 , which are the nodes of the chain ∪r+1 i=0 Ei of curves. We may assume that {xi } = Ei−1 ∩ Ei . Let Li and Ri be respectively the weights of the T-action on the tangent spaces to Ei−1 and Ei at xi . We have L1 = (r + 1)t1 , Rr+1 = (r + 1)t2 , and the following equalities Li + Ri = t1 + t2 , Ri = −Li+1 , for each i = 1, . . . r. x2 • Z   L2   =ZZ · Z   } >R1  (r + 1)t1 = LZ Z 1Z• Z Z x1 Z E0ZZ · · Z Z  xr  Er+1 • Z  R Z  Z ~Zr  Z  Z } Z >  Z Lr+1Z•Rr+1 = (r + 1)t2 Z Z xr+1 Figure 4.1. The middle chain is the exceptional locus Ex(π). The labeled vectors stand for the tangent weights at the fixed points. The above information will be sufficient for our calculation of Gromov-Witten invariants. 38 One can also compute explicitly to obtain (Li , Ri ) = ((r − i + 2)t1 + (1 − i)t2 , (−r + i − 1)t1 + it2 ). Obviously, Lk Rk ≡ −(r + 1)2 t21 mod (t1 + t2 ) for k = 1, . . . , r + 1. It is convenient to take τ = −(r + 1)2 t21 . In this manner, e ≡ τ `(δ) t(δ) mod (t1 + t2 ). e (4.1.1) e is the sum Pr+1 `(δk ). Here `(δ) k=1 4.2 Connected invariants Let ◦ M 0,k ([Symn (Ar )], β) be the component of M 0,k ([Symn (Ar )], β) parametrizing connected covers (i.e., each cover C̃ ◦ associated to [f : C → [Symn (Ar )]] ∈ M 0,k ([Symn (Ar )], β) is connected). We define k-point connected Gromov-Witten invariant as the contribution of the component ◦ M 0,k ([Symn (Ar )], β) to the extended Gromov-Witten invariant. That is, [Symn (Ar )],conn hα1 , . . . , αk iβ Z = ◦ [M 0,k ([Symn (Ar )],β)]vir T ev∗1 (α1 ) · · · ev∗k (αk ). ◦ Note that M 0,k ([Symn (Ar )], β) is compact whenever β 6= 0, in which case the corresponding connected invariant is an element of Q[t1 , t2 ]. Similarly, the connected invariant has an extended version. We define k-point extended connected invariant by [Symn (Ar )],conn hα1 , . . . , αk i(a,β) = 1 [Symn (Ar )],conn hα1 , . . . , αk , (2)a iβ . a! 39 4.3 Orbifold quantum product and divisor operators We may view curve classes E1 , . . . , Er as a basis for A1 (Symn (Ar ); Z). Let {ω1 , . . . , ωr } be the dual basis of {E1 , . . . , Er } with respect to the Poincaré pairing. For any classes α1 , . . . , αk ∈ A∗T,orb ([Symn (Ar )]), we define the extended k-point function of [Symn (Ar )] by n hhα1 , . . . , αk ii[Sym (Ar )] = ∞ X [Symn (Ar )] X hα1 , . . . , αk i(a,β) ua s1β·ω1 · · · srβ·ωr , (4.3.1) a=0 β∈A1 (Ar ;Z) and denote by n hα1 , . . . , αk i[Sym (Ar )] n the usual k-point function hhα1 , . . . , αk ii[Sym (Ar )] |u=0 . Now let {γ} be a basis for the Chen-Ruan cohomology A∗T,orb ([Symn (Ar )]) and {γ ∨ } its dual basis. Define the small orbifold quantum product on A∗T,orb ([Symn (Ar )]) in this way: α1 ∗orb α2 = X n hhα1 , α2 , γii[Sym (Ar )] γ ∨ . γ By extending scalars, we work with QA∗T,orb ([Symn (Ar )]), which is defined as the vector space A∗T,orb ([Symn (Ar )])⊗Q[t1 ,t2 ] Q(t1 , t2 )((u, s1 , . . . , sr )) endowed with quantum multiplication ∗orb . The extended k-point functions were first studied by Bryan and Graber [BG] in the case of [Symn (C2 )] so as to link the Gromov-Witten theory of [Symn (C2 )] to that of Hilbn (C2 ). When it comes to the whole group of multipoint functions, it is clear that the extended and the usual versions share the same information. However, extended three-point functions are a wider group than the usual three-point functions, and the quantum product defined above retains more information than the usual small quantum product. We are going to study the operators D ∗orb − on the (small) quantum cohomology of the orbifold [Symn (Ar )] for divisor classes D. We refer 40 to them as divisor operators. We let Dk = 1(1)n−1 1(ωk ), k = 1, . . . , r. These classes, along with (2), form a basis for divisors on [Symn (Ar )]. Thus, the divisor operators are determined by (2) ∗orb −, D1 ∗orb −, . . . , Dr ∗orb −, which are governed by two-point extended invariants to be calculated in this chapter. 4.4 Fixed loci Fix a nonnegative integer a throughout the rest of this section. We shorten our notation by declaring M ([Symn (Ar )], σ1 , . . . , σk ; (a, β)) = M ([Symn (Ar )], σ1 , . . . , σk , (2)a ; β). Also, as in (3.4.1), we use ~g = (g1 , . . . , gr+1 ) to denote an (r + 1)-tuple, whose entries are either all partitions or all nonnegative integers. Moreover, given a partition σ0 and a multi-partition ~σ , we put σ̂ := (σ0 , ~σ ) = (σ0 , . . . , σr+1 ), which we also realize as a partition of Pr+1 k=0 |σk |. Let us now describe the fixed loci that will play an important role in our virtual localization calculation. Given nonnegative integers i, j, s with 1 ≤ i ≤ j ≤ r and s ≤ a. We consider effective curve classes Eij = Ei + · · · + Ej . L R L (Note that Eii = Ei ). For each bL 0 ∈ {0, . . . , s} and u0 ∈ {0, . . . , a − s}, put b0 = s − b0 and 41 L uR 0 = a − s − u0 . We let bL ,σ0 ,uL 0 {M 00 bL ,σ0 ,uL 0 (1)} (resp. {M 00 (2)}) (4.4.1) be the set consisting of all T-fixed connected components of the moduli space ◦ M ([Sym|λ0 | (Ar )], λ0 , ρ0 , (2)s , 1a−s ; dEij ) bL ,σ0 ,uL 0 such that each point [f0 : C → [Sym|λ0 | (Ar )]] ∈ M 00 bL ,σ0 ,uL 0 (1) (resp. M 00 (2)) has the following properties: (i) f0 has its source curve decomposed as C = CL0 ∪ D0 ∪ CR0 . Here Ck0 ’s are disjoint f0 -contracted components, D0 is a chain of noncontracted components with f0∗ ([D0 ]) = dEij , and Ck0 ∩ D0 = {Pk } is a twisted point, k = L, R. Let D0 , C, Pk be coarse moduli spaces of D0 , C, Pk respectively (k = L, R) and C̃0 the admissible cover associated to C. (ii) C̃0 := C̃L0 ∪ D̃0 ∪ C̃R0 is connected with admissible covers D̃0 → D0 and C̃k0 → Ck0 (k = L, R). Moreover, • each irreducible component of the cover D̃0 → D0 is totally branched over two points (either nodes or markings) and branched nowhere else. • the covering C̃L0 → CL0 is branched with monodromy L L L L λ0 , (2)b0 , 1u0 , σ0 (resp. λ0 , ρ0 , (2)b0 , 1u0 , σ0 ), around markings and PL while the covering C̃R0 → CR0 is branched with monodromy R R R R ρ0 , (2)b0 , 1u0 , σ0 (resp. (2)b0 , 1u0 , σ0 ), around markings and PR . (iii) In the cover D̃0 , there exists a unique chain ε formed by rational curves not contracted by f˜0 . Additionally, 42 • ε possesses j − i + 1 irreducible components which are mapped to Ei , . . . , Ej with the same degree d under the map f˜0 . • the contracted components attached to the two ends of ε collapse to xi and xj+1 respectively. Now we turn our attention to the fixed locus on the moduli space M ([Symn (Ar )], Λ, ℘, (a, dEij )). R ~R We fix ~bL and ~bR , tuples of nonnegative integers, with |~bL | = uL 0 and |b | = u0 . We define L L b ,σ0 ,u0 Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~)[i, j, s] = {M 0 (1)} 0 0 to be the set of T-fixed loci of M ([Symn (Ar )], Λ, ℘, (a, dEij )) (so Λ = λ̂ and ℘ = ρ̂ as partitions) bL ,σ0 ,uL 0 such that any element [f : C → [Symn (Ar )]] ∈ M 0 (1) of these fixed loci satisfies the following properties: (a) The domain curve C of f decomposes into three pieces C = CL ∪ D ∪ CR , (4.4.2) where Ck ’s are disjoint f -contracted components; D is a chain of noncontracted components, which maps to [Symn (Ar )] with degree dEij ; and the intersection Ck ∩ D := {Qk } is a twisted point, k = L, R. As in (3.3.3), there is an associated morphism f˜ : C̃ → Ar . Let D, C, Ck , Qk be coarse moduli spaces of D, C, Ck , Qk respectively (k = L, R). L R R L L (b) CL carries bL 0 + u0 + 1 marked points, and CR carries the other b0 + u0 + 1 = a − b0 − u0 + 1 marked points. (c) The covering C̃ → C has components C̃k := C̃Lk ∪ D̃k ∪ C̃Rk , k = 0, . . . , r + 1. (4.4.3) For k 6= 0, C̃k , if nonempty, is contracted to xk in Ar . (Note that, as in Chapter 3, C̃k is possibly empty or disconnected for k 6= 0, and we include empty sets just for the simplicity 43 of notation). (d) For k = 0, . . . , r + 1, • the covering `r+1 k=0 C̃Lk → CL (resp. L `r+1 k=0 C̃Rk → CR ) is ramified with monodromy L R R λ̂, (2)b0 +u0 , σ̂ (resp. ρ̂, (2)b0 +u0 , σ̂), around markings and QL (resp. QR ); • each irreducible component of the cover D̃k → D is totally branched over two points and branched nowhere else; • each C̃Lk → CL (resp. C̃Rk → CR ) is a covering ramified with monodromy L L L L R R R R λk , (2)bk , 1b0 +u0 −bk , σk (resp. ρk , (2)bk , 1b0 +u0 −bk , σk ), around markings and QL (resp. QR ). (e) The diagram of maps f˜|C̃ C̃0 −−−−0→ Ar   y (4.4.4) C L bL 0 ,σ0 ,u0 corresponds to [f0 ] ∈ M 0 (1) above. Note that Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~)[i, j, s] does not exist for certain parameters. If it does, 0 0 bL ,σ0 ,uL 0 it is indexed by M 00 bL ,σ0 ,uL 0 (1)’s. Each fixed locus M 0 (1) is, however, a union of T-fixed connected components in general. H •H λ̂H • H C (2) · HH L ·· H • (2) HH •H σ̂ D   • ρ̂  CR  • · (2)  · • ·   (2)  •  σ̂ bL ,σ0 ,uL 0 Figure 4.2. This is the configuration of a typical domain curve C for M 0 (1). Each straight line represents a chain of curves. All markings and Qk ’s are labeled with their monodromy and there are bk0 + uk0 copies of (2) on Ck , k = L, R. In case bk0 + uk0 = 0, Ck is simply a twisted point. Details on the covering C̃ associated to C are included in the above properties. 44 Define L L b ,σ0 ,u0 Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~, ~bL | ~bR )[i, j, s] := {M 0 (2)} 0 0 in an analogous manner. The differences occur in properties (b), (d), and (e). Precisely, (b) the L R R curve CL carries bL 0 + u0 + 2 marked points while the curve CR carries the other b0 + u0 marked `r+1 `r+1 points; (d) the covering k=0 C̃Lk → CL (resp. k=0 C̃Rk → CR ) is ramified with monodromy L L R R λ̂, ρ̂, (2)b0 +u0 , σ̂ (resp. (2)b0 +u0 , σ̂) around markings and QL (resp. QR ), and the monodromy L L L L associated to the cover C̃Lk → CL (resp. C̃Rk → CR ) is now λk , ρk , (2)bk , 1b0 +u0 −bk , σk (resp. R R R bL ,σ0 ,uL 0 R (2)bk , 1b0 +u0 −bk , σk ); (e) the diagram (4.4.4) corresponds to [f0 ] ∈ M 00 H • H λ̂ HH • ρ̂ HH CL (2)•· HH ·· H (2)• HH • σ̂H D • σ̂ (2).   •  (2)  C R  · ·   · • (2)  bL ,σ0 ,uL 0 Figure 4.3. This is the configuration of a typical domain curve C for M 0 (2). There are bk0 + uk0 copies of (2) on Ck , k = L, R. CL is always a twisted curve. CR is of dimension R R R 2 (bR 0 + u0 ); in particular, it is a twisted point when b0 + u0 ≤ 1. 4.5 Valuations Given moduli space M ([Symn (Ar )], Λ, ℘, (a, dEij )) as above. For each T-fixed connected component F , the virtual normal bundle to F is denoted by NFvir . Let [f : C → [Symn (Ar )] ∈ F and ` v Cv the union of one-dimensional, contracted, connected components of C. We have a natural morphism φF : F → F c := Y M 0,val(v) , v defined by φF ([f ]) = ([c(Cv )])v . That is, all noncontracted components, zero-dimensional contracted components, stack structures at special points and the map f are forgotten. Also, val(v) denotes the number of special points on Cv . Let Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s] 0 0 45 ~, ~bL | ~bR )[i, j, s]. be the union Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~)[i, j, s] ∪ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ 0 0 0 0 L bL bL ,σ0 ,uL 0 ,σ0 ,u0 0 L The indices bL (k) and M 00 (k) are going to be 0 , σ0 , u0 , (k) (k = L, R) from M ~; ~bL , ~bR )[i, j, s], we let suppressed. We simply write M , M 0 . For each M ∈ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ 0 0 MT be the collection of all T-fixed connected components of M . There are other T-fixed loci on the moduli space M ([Symn (Ar )], Λ, ℘, (a, dEij )). The reason why Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s]’s are singled out will be discussed later. It will turn out 0 0 that these fixed loci are enough for our study of two-point extended invariants as a consequence of (t1 + t2 )-valuation below. Proposition 4.5.1. If M∈ [ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s] and F ∈ M T , 0 (4.5.1) 0 where the union ranges over all possible parameters, the inverse Euler class eT (N1 vir ) has valuation F 1 with respect to (t1 + t2 ). Otherwise, it has valuation at least 2. Proof. Consider any T-fixed connected component F of M ([Symn (Ar )], Λ, ℘; (a, β)). We let f : C → [Symn (Ar )] be a twisted stable map representing a point of F . As discussed earlier, there are a morphism f˜ : C̃ → Ar and an ordinary stable map fc : C → Symn (Ar ) associated to f . Recall that τ = −(r + 1)2 t21 . To establish the assertion, we need to analyze the contribution of following situations (cf. [GP]) to the inverse Euler class eT (N1 vir ) . F 1. Infinitesimal deformations and obstructions of f with C held fixed: (a) Any contracted component contributes zero (t1 + t2 )-valuation. Let C 0 ⊂ C be a contracted component and pick any connected component Z of the cover associated to C 0 . We see that Z contributes eT (H 1 (Z, f˜∗ T Ar )) eT (H 0 (Z, f˜∗ T Ar )) (4.5.2) and is collapsed by f˜ to xk for some k. So the numerator is, by Mumford’s relation, congruent modulo t1 + t2 to Λ∨ (Lk )Λ∨ (Rk ) ≡ τ g , 46 where g = rank(H 0 (Z, ωZ )) and Λ∨ (t) = Pg i=0 ci (H 0 (Z, ωZ )∨ )tg−i . The denominator of (4.5.2) is eT (Txk Ar ). Thus, the contribution of Z is simply τ g−1 mod (t1 + t2 ). In other words, the contribution of C 0 , being the product of the contributions of such Z’s, is not divisible by t1 + t2 . (b) The nodes joining contracted curves to noncontracted curves have zero (t1 + t2 )valuation because each of them gives some positive power of τ modulo (t1 + t2 ). (c) Noncontracted curves: Suppose D is a noncontracted component with D̃ its associated (possibly disconnected) covering. Its contribution is eT (H 1 (D̃, f˜∗ T Ar ))mov eT (H 1 (D, f ∗ T [Symn (Ar )]))mov = . n 0 ∗ mov eT (H (D, f T [Sym (Ar )])) eT (H 0 (D̃, f˜∗ T Ar ))mov Here ( )mov stands for the moving part. It is clear from (a) that each f˜-contracted component of D̃ has zero (t1 + t2 )-valuation. However, any irreducible component Σ of D̃ that is not f˜-contracted contributes t1 + t2 τ mod (t1 + t2 )2 . (4.5.3) This can be seen as follows. Assume that f˜ maps Σ to E := f˜(Σ) with degree ` > 0. Let S1 = {0, . . . , 2` − 2} − {` − 1} and S2 = {0, . . . , 2`} − {`}. The moving part of eT (H 1 (Σ, f˜∗ T Ar ))) arises from ∨ H 1 (Σ, f˜∗ NE/Ar ) = H 0 (Σ, ωΣ ⊗ f˜∗ NE/A )∨ . r The curve E having self-intersection −2 implies NE/Ar ∼ = OP1 (−2), and so the in∨ vertible sheaf ωΣ ⊗ f˜∗ NE/A has degree 2` − 2. Hence, the moving part is r (t1 + t2 ) Y k( `−1 (r + 1)t1 ) + (2` − 2 − k)( 1−` (r + 1)t1 ) ` k∈S1 ` 2` − 2 mod (t1 + t2 )2 47 (which is simply (t1 + t2 ) for ` = 1). We further simplify it to get (t1 + t2 )τ `−1 `−1 Y ( k=1 `−k 2 ) ` mod (t1 + t2 )2 . (4.5.4) On the other hand, eT (H 0 (Σ, f˜∗ T Ar ))mov equals eT (H 0 (Σ, f˜∗ T E))mov , that is congruent modulo (t1 + t2 ) to Y k(−(r + 1)t1 ) + (2` − k)((r + 1)t1 ) 2` k∈S2 ≡ τ` `−1 Y ( k=1 `−k 2 ) . ` (4.5.5) Dividing (4.5.4) by (4.5.5) gives (4.5.3). 2. Infinitesimal automorphisms of C: We need only investigate the nonspecial points p, which lie on noncontracted curves Σ and are mapped to fixed points. In fact, each p gives the weight of the tangent space to Σ at p. This has zero (t1 + t2 )-valuation. 3. Infinitesimal deformations of C: Given any node P joining two curves V1 and V2 . Let P, V1 , V2 be coarse moduli spaces of P, V1 , V2 respectively and Stab(P) the stabilizer of P. In each of the following, we study the contribution arising from smoothing the node P. (a) V1 and V2 are noncontracted: We may assume that the restriction of fc to Vk is a dk -sheeted covering fc |Vk : Vk → Σk := fc (Vk ) ∼ = P1 for some dk > 0, k = 1, 2. The node-smoothing contribution is |Stab(P)| ( w2 −1 w1 + ) , d1 d2 (4.5.6) where wk is the tangent weight of the rational curve Σk at the fixed point fc (P ). Thus, (4.5.6) is proportional to (t1 + t2 )−1 only if d1 = d2 and w1 + w2 is a multiple of t1 + t2 . (b) V1 is noncontracted but V2 is contracted: Let w be the tangent weight of V1 at the node P and L the tautological line bundle formed by the cotangent space TP∗ V2 (see 48 Chapter 6). Denote by ψ the first Chern class of L. The node smoothing contributes |Stab(P)| . w−ψ (4.5.7) So, neither (t1 + t2 ) nor (t1 + t2 )−1 is generated in this case. Thus, only 1(c) and 3(a) may produce any power of (t1 + t2 ). We conclude that F gives positive (t1 +t2 )-valuation because the number of noncontracted curves is more than the number of nodes joining them. Suppose F is a T-fixed component described in (4.5.1), in which case we have a unique chain of noncontracted rational components for the cover associated to C. The discussion in 3(a) shows that each node in the chain gives (t1 +t2 )-valuation −1. In total, the node smoothing contributes i − j in valuation. On the other hand, the chain has j − i + 1 irreducible components. By the result of 1(c), eT (N1 vir ) has valuation 1, which establishes the first assertion. F Assume that F is not as in (4.5.1). If the associated cover has at least two disjoint chains of noncontracted rational curves, a (t1 + t2 )-valuation at least 2 is obtained because each chain gives valuation at least 1. Otherwise, the cover has a unique chain but property (e) (and hence (iii)) in Section 4.4 is not fulfilled for each i, j, s. In this case, we have the same consequence by the discussion in 3(a) and the result of 1(c). This shows the second assertion. 4.6 Counting branched covers Later, we will count certain coverings of (chains of) rational curves. Let us now review some related notions and fix notation. For partitions η1 , . . . , ηs of n, the Hurwitz number H(η1 , . . . , ηs ) is the weighted number of possibly disconnected covers π : X → (P1 , p1 , . . . , ps ) such that π are branched over p1 , . . . , ps with ramification profiles η1 , . . . , ηs and unbranched away from p1 , . . . , ps . (Each cover is counted with weight 1 over the size of its automorphism group). The Hurwitz number H(η1 , . . . , ηs ) is essentially a combinatorial object. It can be described combinatorially by 1 |H(η1 , . . . , ηs )|. n! 49 Here H(η1 , . . . , ηs ) is the set consisting of (g1 , . . . , gs ) ∈ Qs i=1 Sn satisfying (i) for each i = 1, . . . , s, gi has cycle type ηi ; (ii) g1 · · · gs = 1. Let us introduce some other Hurwitz-type numbers. Let Hσ (η1 , . . . , ηs | τ1 , . . . , τt ) be the subset of H(η1 , . . . , ηs , τ1 , . . . , τt ) such that each element (g1 , . . . , gs , h1 , . . . , ht ) has an additional property that g1 · · · gs has cycle type σ (and so h1 · · · ht has the same cycle type as well). Put Hσ (η1 , . . . , ηs | τ1 , . . . , τt ) := |Hσ (η1 , . . . , ηs | τ1 , . . . , τt )| n! (in case σ is a vacuous partition, we set Hσ (η1 , . . . , ηs | τ1 , . . . , τt ) = 1). We readily find the following relations. Lemma 4.6.1. The number Hσ (η1 , . . . , ηs | τ1 , . . . , τt ) is exactly the product |C(σ)| H(η1 , . . . , ηs , σ) H(σ, τ1 , . . . , τt ). Moreover, we have H(η1 , . . . , ηs , τ1 , . . . , τt ) = X Hσ (η1 , . . . , ηs | τ1 , . . . , τt ). |σ|=n 4.7 Localization contributions 4.7.1 Reduction From now on, fix cohomology-weighted partitions µ1 (~η1 ) and µ2 (~η2 ) of n with ηk` ’s 1 or divisors on Ar . We concentrate on the two-point extended invariant [Symn (Ar )] hµ1 (~η1 ), µ2 (~η2 )i(a,β) (4.7.1) of twisted degree (a, β), β 6= 0. We will leave out the superscript [Symn (Ar )] when there is no likelihood of confusion. Let us write µi (~η1 ) = κi (~ηi1 )θi (~ηi2 ) 50 where the entries of ~ηi1 ’s are all 1 and the entries of ~ηi2 ’s are all divisors, i = 1, 2. We may assume that `(κ1 ) ≤ `(κ2 ). Use the identity 1 = Pr+1 1 k=1 Lk Rk [xk ], we see readily that (4.7.1) is a Q(t1 , t2 )-linear combination of the invariants of the form D E κ11 ([xm1 ]) · · · κ1`(κ1 ) ([xm`(κ1 ) ])θ1 (~η12 ), µ2 (~η2 ) . (4.7.2) (a,β) Additionally, (4.7.2) is an element of Q[t1 , t2 ] as the first insertion has compact support. Also, the sum of the degrees of the insertions is at most 1 larger than the virtual dimension. Precisely, the difference is `(κ1 ) − `(κ2 ) + 1. Thus, the invariant (4.7.2) is a linear polynomial if `(κ1 ) = `(κ2 ); otherwise, it is a rational number. Assume that β is not a multiple of Eij for any i, j. Clearly, the fixed loci (4.5.1) make no contribution. By Proposition 4.5.1, the invariant (4.7.2) is zero by divisibility of (t1 + t2 )2 (each of the two insertions is a linear combination of fixed-point classes with coefficients being 0 or having nonnegative (t1 + t2 )-valuation; for details, consult the discussion preceding Lemma 4.7.3). It follows that (4.7.1) is zero as well. So we can now set our mind on the invariant hµ1 (~η1 ), µ2 (~η2 )i(a,dEij ) , d, i, j > 0. (4.7.3) We fix positive integers i, j, d with i ≤ j from here on. Let β = dEij . By virtual localization, (4.7.2) can be expressed as a sum of residue integrals over T-fixed loci. By Proposition 4.5.1, the invariant (4.7.2) is α(t1 + t2 ) for some rational number α, and it suffices to evaluate (4.7.2) `i,j ~σ `i,j over all T-fixed loci lying in the union Fλ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s], where means 0 0 that only i, j are fixed and the other parameters can vary. 4.7.2 The setup for localization Given M ∈ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s] and F ∈ M T , we let 0 0 ιF : F → M ([Symn (Ar )], Λ, ℘, (a, dEij )) 51 be the natural inclusion (as partitions, Λ = λ̂, and ℘ = ρ̂). ~)[i, j, s] (resp. M ∈ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~, ~bL | ~bR )[i, j, s]). Let M ∈ Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ 0 0 0 0 As mentioned earlier, there are natural morphisms R R φF : F → M 0,bL0 +uL0 +2 × M 0,bR (resp. M 0,bL0 +uL0 +3 × M 0,bR ) 0 +u0 +2 0 +u0 +1 for F ∈ M T and R R φM 0 : M 0 → M 0,bL0 +uL0 +2 × M 0,bR (resp. M 0,bL0 +uL0 +3 × M 0,bR ). 0 +u0 +2 0 +u0 +1 c Obviously, F c = M 0 . We intend to calculate our Gromov-Witten invariants by localization, which will be reduced to integrals over F c ’s. So it is necessary to understand the degree deg(φF ) of the morphism φF . For F ∈ M T with M ∈ Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~)[i, j, s], we let [f : CL ∪ D ∪ CR → 0 0 n [Sym (Ar )]] ∈ F (in the notation of Section 4.4) be a typical element. We obtain deg(φF ) = m1 · m2 . Here • m1 = c0 (o(σ̂)−1 Qr+1 k=1 |C(σk )|) ε(F ) is a factor arising from the nodes, which are glued over the rigidified inertia stack. Here c0 is an overall factor coming from nodes of the cover C̃0 → C (we do not have to give a careful description here as c0 will be cancelled by an L R R identical term in deg(φM 0 )), and ε(F ) = 1 (bL 0 + u0 ) + 1 (b0 + u0 ) + j − i (the terms L R R 1 (bL 0 + u0 ) and 1 (b0 + u0 ) record the dimensions of CL and CR respectively). • m2 is given by dj−i+1 m0 r+1 Y L L L L R R R R H(λk , (2)bk , 1b0 +u0 −bk , σk ) H(σk , σk )j−i+1 H(σk , (2)bk , 1b0 +u0 −bk , ρk ), k=1 where dj−i+1 is an automorphism factor that takes care of the restriction f |D forgotten by φF , m0 is the contribution of C̃0 , and the other terms account for the overall contribution `r+1 of the disconnected curve k=1 C̃k . 52 Also, the degree of φM 0 can be calculated in a similar fashion. That is, deg(φM 0 ) = c0 ( 1 ε(F ) j−i+1 ) d m0 . o(σ0 ) By Lemma 4.6.1, we may write deg(φF ) as r+1 deg(φM 0 )( L L L L R R R R o(σ0 ) ε(F ) Y ) Hσk (λk , (2)bk , 1b0 +u0 −bk | (2)bk , 1b0 +u0 −bk , ρk ). o(σ̂) (4.7.4) k=1 ~, ~bL | ~bR )[i, j, s], deg(φF ) is given by Similarly, for F ∈ M T with M ∈ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ 0 0 r+1 deg(φM 0 )( L L L L R R R R o(σ0 ) ε(F ) Y ) Hσk (λk , ρk , (2)bk , 1b0 +u0 −bk | (2)bk , 1b0 +u0 −bk ). o(σ̂) (4.7.5) k=1 R Now ε(F ) is set to be 1 + 2 (bR 0 + u0 ) + j − i. Remark. 0 ) ε(F ) will cancel with a similar term in eT (N1 vir ) (see Lemma 4.7.1 The term ( o(σ o(σ̂) ) F below). Moreover, forgetting the indices involving the partition 1 does not change the value of the Hurwitz-type numbers. We did not do this in the above formulas so as to keep track of the ramification profiles corresponding to the simple marked points. 4.7.3 Virtual normal bundles Let us determine eT (N1 vir ) modulo (t1 + t2 )2 for each connected component F described in (4.5.1). F The following outcome should be within our expectation. Recall again that τ = −(r + 1)2 t21 . ~; ~bL , ~bR )[i, j, s] and any connected component F ∈ Lemma 4.7.1. Given M ∈ Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ 0 0 M T , we have the congruence equation 1 ~ 1 o(σ̂) ε(F ) τ 2 (a−s−`(λ)−`(~ρ)) ≡( ) vir vir ) o(σ0 ) eT (NF ) eT (NM mod (t1 + t2 )2 . 0 Here ε(F )’s are as in (4.7.4), (4.7.5) respectively. Proof. We just investigate the case where M ∈ Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~) and F ∈ M T , the 0 0 other case being similar. Pr+1 Pr+1 R Let p = k=0 bL k and q = k=0 bk , and so p + q = a. Assume that p, q > 0. Pick any point [f ] ∈ F . Again, we follow the notation of Section 4.4. The contribution of the contracted 53 component CL is P eT (H 1 (CL , f ∗ [Symn (Ar )])) k (gk −1) ≡ τ n eT (H 0 (CL , f ∗ [Sym (Ar )])) mod (t1 + t2 ). Here gk ’s are the genera of connected components of the covering associated to CL . We find, by P Riemann-Hurwitz formula, that k (gk − 1) = 21 (p − `(λ̂) − `(σ̂)). Hence CL contributes 1 τ 2 (p−`(λ̂)−`(σ̂)) mod (t1 + t2 ). Similarly, CR contributes 1 τ 2 (q−`(ρ̂)−`(σ̂)) mod (t1 + t2 ). And the contribution of nodes joining contracted components to D is τ 2`(σ̂) mod (t1 + t2 ). These three contributions, taken together, yield 1 τ 2 (a−`(λ̂)−`(ρ̂)+2`(σ̂)) mod (t1 + t2 ). One can check that the same formula holds when p = 0 or q = 0. As for the cover C̃L0 ∪ D̃0 ∪ C̃R0 , by a similar argument, the combined contribution of C̃L0 , C̃R0 , and nodes joining C̃L0 , C̃R0 to D̃0 is given by 1 τ 2 (s−`(λ0 )−`(ρ0 )+2`(σ0 )) mod (t1 + t2 ). Further, the covers D̃1 , . . . , D̃r+1 (including the nodes inside) contribute 1 τ `(~σ) mod (t1 + t2 ). We now study the infinitesimal deformations of C. Let k = L, R. When Ck is a curve, smoothing the node Pk joining Ck to D contributes o(σ̂) , wk − ψk 54 where wk is the T-weight of the tangent space to c(D) at the point c(Pk ), and ψk is the class ∗ associated to Tc(P Ck (cf. (4.5.7)). By property (e) in Section 4.4, f˜ : C̃0 → Ar corresponds to k) the point [f0 : CL0 ∪ D0 ∪ CR0 → [Sym|λ0 | (Ar )]] ∈ M 0 , so o(σ0 ) wk − ψk is the factor smoothing nodes joining Ck0 and D0 and is o(σ0 )/o(σ̂) times the preceding factor. Similarly, the overall contributions of node smoothing inside D and node smoothing inside D0 differ by a factor (o(σ̂)/o(σ0 ))j−i . Hence, deformations of C contribute (o(σ̂)/o(σ0 ))ε(F ) times those of CL0 ∪ D0 ∪ CR0 , and the term ( o(σ̂) ε(F ) 1 ) vir ) o(σ0 ) eT (NM 0 is the combined contribution of the deformations of C and the unique noncontracted connected component C̃0 of the associated cover C̃. Putting all these together, we get 1 eT (NFvir ) 1 ≡ o(σ̂) ε(F ) τ 2 (a−`(λ̂)−`(ρ̂)+2`(σ̂)) 1 1 ( · 1 (s−`(λ )−`(ρ )+2`(σ )) · `(~σ) ) vir 0 0 0 o(σ0 ) τ eT (NM ) τ 2 ≡ o(σ̂) ε(F ) τ 2 (a−s−`(λ)−`(~ρ)) ( ) vir ) o(σ0 ) eT (NM 0 ~ 1 mod (t1 + t2 )2 , 0 as desired. 4.7.4 Vanishing and relation to connected invariants Let us look closely at the invariant (4.7.2) with β = dEij . We will go back to (4.7.3) in the end. For every nonnegative integer s, let I(s) be the contribution of the T-fixed loci `i,j,s Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~; ~bL , ~bR )[i, j, s] (all but i, j, s vary) 0 0 to the invariant (4.7.2) with β = dEij . We claim the following. Proposition 4.7.2. For any s < a, I(s) ≡ 0 mod (t1 + t2 )2 . 55 Now fix a nonnegative integer s < a as well. For simplicity, we drop the index [i, j, s] from the notation of fixed loci. We would like to deduce Proposition 4.7.2 by replacing the first two insertions with T-fixed ~ B. ~ Define point classes. Fix T-fixed point classes A, I := X X Z ι∗ (ev∗ (A)ev ~ ∗ (B)) ~ 1 2 F , vir eT (NF ) F (4.7.6) M F ∈M T where M is taken over all possible T-fixed loci in ~ σ ~ ~; ~bL , ~bR ). L σ ,~ L (λ, ρ σ0 ,bL bL ,~bR Fλ0 ,ρ0 ,σ0 ;bL 0 ,u0 ,~ 0 ,u0 ` The coefficient ~ hµ (~η )|Bi ~ hκ11 ([xm1 ]) · · · κ1`(κ1 ) ([xm`(κ1 ) ])θ1 (~η12 )|Ai 2 2 · , ~ ~ ~ ~ hA|Ai hB|Bi is either zero or has nonnegative valuation with respect to t1 + t2 , so Proposition 4.7.2 follows from the following lemma. Lemma 4.7.3. I≡0 mod (t1 + t2 )2 . Proof of Lemma 4.7.3 Lemma 4.7.3 is clear if the condition λk ⊂ Ak and ρk ⊂ Bk , ∀k = 1, . . . , r + 1 (4.7.7) does not hold, in which case I is identically zero. Now we assume (4.7.7), and the idea of the proof in this case is to relate I to certain connected invariants. We put λ̄k = Ak − λk , ρ̄k = Bk − ρk . ~ = ((λ1 , λ̄1 ), . . . , (λr+1 , λ̄r+1 )) and B ~ = ((ρ1 , ρ̄1 ), . . . , (ρr+1 , ρ̄r+1 )). Let That is, we may write A Ā = (λ̄1 , . . . , λ̄r+1 ) and B̄ = (ρ̄1 , . . . , ρ̄r+1 ) be T-fixed point classes. First of all, it is good to have some observations on hand. 56 Lemma 4.7.4. For every partition σ0 and (r + 1)-tuples ~bL , ~bR , ~σ , deg(φM 0 ) ~ ~ L | ~bR ,~ ρ) L (λ,b λ0 ,σ0 ,ρ0 ;bL 0 ,u0 σ M ∈F ~ is deg(φM 0 ) ~ ~ cL | ~ M ∈F θ cR ,~ ρ) L (λ,~ λ0 ,σ0 ,ρ0 ;bL 0 ,u0 0 vir ) eT (NM σ M ∈F ~ deg(φM 0 ) ~ ρ,~bL | ~bR ) L (λ,~ ∗ (ev∗1 (Ā)ev∗2 (B̄)) ιM 0 vir ) eT (NM c M0 0 λ0 ,ρ0 ,σ0 ;bL 0 ,u0 is ι∗M (ev∗1 (Ā)ev∗2 (B̄)) Z X deg(φM 0 ) ~ ~ ρ,~ M ∈F θ cL | ~ cR ) L (λ,~ λ0 ,ρ0 ,σ0 ;bL 0 ,u0 , 0 Z X 0 0 c M0 and L J2 (σ0 ; bL 0 , u0 ) := vir ) eT (NM c M0 ι∗M (ev∗1 (Ā)ev∗2 (B̄)) Z X ∗ ιM (ev∗1 (Ā)ev∗2 (B̄)) Z X L J1 (σ0 ; bL 0 , u0 ) := 0 c M0 vir ) eT (NM , 0 for any ~cL , ~cR and θ~ satisfying |θk | = |σk | for each k = 1, . . . , r + 1. Here the collections of T-fixed loci under the summation symbols are all nonempty. ~ Proof. The first identity follows as Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~) and Fλθ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~cL | ~cR , ρ ~) 0 0 0 0 have the same number of elements and the same configuration for the unique noncontracted connected component of the associated cover (see the description in Section 4.4). The second identity holds for similar reasons. We apply Proposition 4.5.1 to the connected invariant conn Ā, B̄, (2)s , 1a−s dE (4.7.8) ij and find that (4.7.8) is given by X L L L (J1 (σ0 ; bL 0 , u0 ) + J2 (σ0 ; b0 , u0 )) mod (t1 + t2 )2 . L σ0 ,bL 0 ,u0 As a − s > 0, (4.7.8) is zero. We have X L L L (J1 (σ0 ; bL 0 , u0 ) + J2 (σ0 ; b0 , u0 )) ≡ 0 L σ0 ,bL 0 ,u0 Here is an elementary but helpful combinatorial fact. mod (t1 + t2 )2 . (4.7.9) 57 Lemma 4.7.5. Given nonnegative integers k, p and p1 , . . . , pk with p1 + · · · + pk = p. For any nonnegative integer m ≤ p,  p p1 , . . . , p k Note that ` `1 ,...,`k
  X = m1 ,...,mk m m1 , . . . , mk   p−m . p1 − m1 , . . . , pk − mk is declared to be 0 if ` is smaller than some of `i ’s or if some entries are negative integers. We continue the proof of Lemma 4.7.3. Let θ= 1 (a − s + `(~λ) + `(~ ρ)). 2 For every (r + 1)-tuple ~q with |~q| = a − s, let R Q(~q) = {(~bL , ~bR ) | bL k + bk = qk , ∀k = 1, . . . , r + 1}. L Fix σ0 , bL 0 , u0 , we consider two cases: (1) The contribution of Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~)’s to I with the constraint (~bL , ~bR ) ∈ Q(~q) 0 0 is X X X Z ι∗ (ev∗ (A)ev∗ (B)) 1 2 F eT (NFvir ) F X σ (~bL ,~bR )∈Q(~ q) ~ mod (t1 + t2 )2 , (4.7.10) M F ∈M T where M ∈ Fλ~σ0 ,σ0 ,ρ0 ;bL ,uL (~λ, ~bL | ~bR , ρ ~) runs through all T-fixed loci. By (4.1.1), for each 0 0 F ∈ M T, ~ ι∗F (ev∗1 (A) · ev∗2 (B)) ≡ τ `(λ)+`(~ρ) ι∗M 0 (ev∗1 (Ā) · ev∗2 (B̄)) mod (t1 + t2 ). Applying the pushforward φF ∗ and Lemma 4.7.1, (4.7.10) is given by τθ X X X X deg(φF ) ~ ~ L | ~bR ,~ ρ) F ∈M T L (λ,b σ σ M ∈F ~ (~bL ,~bR )∈Q(~ q) ~ λ0 ,σ0 ,ρ0 ;bL 0 ,u0 ·( o(σ̂) ε(F ) ) o(σ0 ) ∗ (ev∗1 (Ā)ev∗2 (B̄)) ιM Z 0 c M0 vir ) eT (NM 0 mod (t1 + t2 )2 . 58 By (4.7.4), (4.7.10) is congruent modulo (t1 + t2 )2 to τ  X θ (~bL ,~bR )∈Q(~ q) · X r+1 Y uL 0 L bL 1 , . . . br+1 L L  L uR 0 R bR 1 , . . . , br+1 L R  R R R Hσk (λk , (2)bk , 1b0 +u0 −bk | (2)bk , 1b0 +u0 −bk , ρk ) ~ σ k=1 0 deg(φM 0 ) c M0 σ ~ ~ L | ~bR ,~ M ∈F ~ ρ) L (λ,b λ0 ,σ0 ,ρ0 ;bL 0 ,u0 where the product ∗ (ev∗1 (Ā)ev∗2 (B̄)) ιM Z X · vir ) eT (NM , 0

uL uR 0 0 is the number of choices to distribute simple ramL R bL bR 1 ,...br+1 1 ,...,br+1 ification points lying above simple markings. By Lemmas 4.6.1, 4.7.4 and 4.7.5, (4.7.10) is simplified to   r+1 Y a−s L τθ H(λk , (2)qk , 1a−qk , ρk )J1 (σ0 ; bL 0 , u0 ) q1 , . . . , qr+1 mod (t1 + t2 )2 . k=1 (2) By a similar argument, the contribution of Fλ~σ0 ,ρ0 ,σ0 ;bL ,uL (~λ, ρ ~, ~bL | ~bR )’s to I with the 0 0 constraint (~bL , ~bR ) ∈ Q(~q) is   r+1 Y a−s L τθ H(λk , (2)qk , 1a−qk , ρk )J2 (σ0 ; bL 0 , u0 ) q1 , . . . , qr+1 mod (t1 + t2 )2 . k=1 In total, I is given by H· X L L L (J1 (σ0 ; bL 0 , u0 ) + J2 (σ0 ; b0 , u0 )) mod (t1 + t2 )2 , L σ0 ,bL 0 ,u0 where H := a−s |~ q |=a−s q1 ,...,qr+1 P
θ Qr+1 qk a−qk , ρk ). By (4.7.9), τ k=1 H(λk , (2) , 1 I≡0 mod (t1 + t2 )2 . This shows Lemma 4.7.3 and ends the proof of Proposition 4.7.2. (4.7.11) 59 4.8 Combinatorial descriptions of two-point extended invariants Now we return to the invariant (4.7.3). We will interpret it combinatorially in terms of orbifold Poincaré pairings and connected invariants. In general, we have the following fact on two-point extended invariants of nonzero degree. Theorem 4.8.1. Given partitions µ1 , µ2 of n and `(µ1 )-tuple ~η1 , `(µ2 )-tuple ~η2 with entries 1 or divisors on Ar . For any curve class β 6= 0, the invariant hµ1 (~η1 ), µ2 (~η2 )i(a,β) (4.8.1) X conn hθ(ξ~1 )|θ(ξ~2 )i hν1 (~γ1 ), ν2 (~γ2 )i(a,β) . (4.8.2) is given by the sum Here the sum is taken over all possible cohomology-weighted partitions θ(ξ~1 ), θ(ξ~2 ), ν1 (~γ1 ), ν2 (~γ2 ) satisfying µ1 (~η1 ) = θ(ξ~1 )ν1 (~γ1 ) and µ2 (~η2 ) = θ(ξ~2 )ν2 (~γ2 ). (In particular, ν1 , ν2 are subpartitions of µ1 , µ2 respectively and µ1 − ν1 = θ = µ2 − ν2 ). Proof of Theorem 4.8.1 The statement is clear if β is not a multiple of Eij for each i, j because both (4.8.1) and (4.8.2) vanish. Now fix i, j, d > 0 and let β = dEij . We learn by Proposition 4.7.2 that I(a) is the only possible contribution to (4.7.2). In other words, only Fσ0 ,b (λ0 , ρ0 ; ~σ ) := F1 ∪ F2 , ranging over all possible λ0 , ρ0 , σ0 , b, ~σ , can possibly make a contribution. Here F1 = Fλ~σ0 ,σ0 ,ρ0 ;b,0 (~σ , (0, . . . , 0) | (0, . . . , 0), ~σ )[i, j, a], F2 = Fλ0 ,ρ0 ,σ0 ;b,0 (~σ , ~σ , (0, . . . , 0) | (0, . . . , 0))[i, j, a]. (1n ) (4.8.3) (4.8.4) (With notation of Section 4.4, the admissible cover C̃ corresponding to any of these fixed loci has all those simple ramification points that are branched over simple markings in the connected component C̃0 , and each C̃k (k 6= 0) is either empty or a chain of rational curves.) 60 As mentioned earlier, in order to evaluate the invariant (4.8.1), it is enough to perform localization calculations over Fσ0 ,b (λ0 , ρ0 ; ~σ )’s because (4.8.1) is a linear combination of invariants of the form (4.7.2). We have a lemma on the inverse Euler classes of virtual normal bundles. Lemma 4.8.2. Given F ∈ M T with M ∈ F1 ∪ F2 , we have 1 o(σ̂) εk (F ) 1 , =( ) vir ) o(σ0 ) eT (NFvir ) t(e σ ) eT (NM 0 for M ∈ Fk , k = 1, 2. Here ε1 (F ) = 1 (b) + 1 (a − b) + j − i and ε2 (F ) = 1 + 2 (a − b) + j − i. Proof. All contracted connected components of the associated cover are necessarily of genus 0. The proof of Lemma 4.7.1 can be carried through. We let I(ν1 , ν2 ) and I(ν1 , ν2 ; ~σ ) be the contributions to (4.8.1) of ` σ ) and σ0 ,b,~ σ Fσ0 ,b (ν1 , ν2 ; ~ ` σ ) respectively. σ0 ,b Fσ0 ,b (ν1 , ν2 ; ~ Now we compute I(ν1 , ν2 ; ~σ ). In order for the contribution not to vanish, the partitions ν1 and ν2 must be subpartitions of µ1 and µ2 respectively. Let us assume ν1 ⊂ µ1 , ν2 ⊂ µ2 . The configurations (4.8.3) and (4.8.4) force µ1 − ν1 = µ2 − µ2 . We set θ = µ1 − ν1 . Lemma 4.8.3. Take a fixed locus M ∈ ~ σ σ ). For k = 1, 2 and each F ∈ M T , b,σ0 Fσ0 (ν1 , ν2 , ~ ` ι∗F ev∗k (µk (~ηk )) = t(e σ) X αθ(ξ~k ) (e σ ) ι∗M 0 ev∗k (νk (~γk )). (4.8.5) Pk Here Pk means that we take the sum over all possible θ(ξ~k ), νk (~γk ) satisfying the equality µk (~ηk ) = θ(ξ~k )νk (~γk ). Proof. The left side of (4.8.5) is XX P e σ αµk (~ ηk ) (δ) t(δ). By Proposition 3.2.1, it equals δ⊃e e e e = t(e αθ(ξ~k ) (e σ )αν1 (~γk ) (δe − σ e) t(δ) σ) e σ Pk δ⊃e X Pk αθ(ξ~k ) (e σ) X αν1 (~γk ) (e ) t(e ), e  which gives the right side of (4.8.5). It follows from Lemma 4.8.3 that for each F ∈ M T , ι∗F (ev∗1 (µ1 (~η1 )) · ev∗2 (µ2 (~η2 ))) coincides 61 with t(e σ )2 X ∗ (ev∗1 (ν1 (~γ1 )) · ev∗2 (ν2 (~γ2 ))). αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ )ιM 0 (4.8.6) Q In the formula, the index Q means that the sum is over all possible θ(ξ~1 ), θ(ξ~2 ), ν1 (~γ1 )) and ν2 (~γ2 )) satisfying µ1 (~η1 ) = θ(ξ~1 )ν1 (~γ1 ) and µ2 (~η2 ) = θ(ξ~2 )ν2 (~γ2 ). Applying (4.8.6) and Lemma 4.8.2, the contribution I(ν1 , ν2 ; ~σ ) is Z ∗ X ιM (ev∗1 (ν1 (~γ1 )) · ev∗2 (ν2 (~γ2 )) t(e σ) X 0 , αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ) H(e σ) vir ) a! eT (NM M0 Q where H(e σ ) := σ0 ,b,M 0 0 Qr+1 σ ) is simplified k=1 H(σk , σk ) is a product of Hurwitz numbers. Thus, I(ν1 , ν2 ; ~ to H(e σ )t(e σ) X conn αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ ) hν1 (~γ1 ), ν2 (~γ2 )i(a,dEij ) . Q Adding up all possible I(ν1 , ν2 ; ~σ )’s, we obtain I(ν1 , ν2 ) = XX Q σ e hθ(ξ~1 )|θ(ξ~2 )i = X conn H(e σ )t(e σ )αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ ) hν1 (~γ1 ), ν2 (~γ2 )i(a,dEij ) . Moreover, αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ )he σ |e σi = σ e X αθ(ξ~1 ) (e σ )αθ(ξ~2 ) (e σ )H(e σ )t(e σ ). σ e This implies that I(ν1 , ν2 ) = X conn hθ(ξ~1 )|θ(ξ~2 )i hν1 (~γ1 ), ν2 (~γ2 )i(a,dEij ) . Q Consequently, by taking into account of all I(ν1 , ν2 )’s, we deduce that (4.8.1) equals X conn hθ(ξ~1 )|θ(ξ~2 )i hν1 (~γ1 ), ν2 (~γ2 )i(a,dEij ) , where the sum is taken over all possible choices stated in the theorem. This finishes the proof. g For partitions µ, ν of n, we denote the Hurwitz number H(µ, ν, (2)b ) by Hµ,ν , where g = 1 2 (b + 2 − `(µ) − `(ν)) is determined by the Riemann-Hurwitz formula, and we refer to it as the 62 g double Hurwitz number. In general, it is not easy to obtain a closed formula for Hµ,ν . However, when ν = (n), we have the following fact due to Goulden, Jackson, and Vakil. Proposition 4.8.4 ([GJV]). Given any partition µ = (µ1 , . . . , µ`(µ) ) of n. The so-called oneg is the coefficient of t2g in the power series expansion of part double Hurwitz number Hµ,(n) `(µ) Y sinh(µi t/2) (2g + `(µ) − 1)! n2g+`(µ)−2 t/2 . |Aut(µ)| sinh(t/2) i=1 µi t/2 Given nonnegative integers a1 , a2 , let ga1 , ga2 , g(a) be integers satisfying ak = 2gak − 1 + `(µk ), k = 1, 2 g(a) = 1 (a − `(µ1 ) − `(µ2 ) + 2). 2 For k = 1, 2, we put [µk (~γk )] = |Aut(µk )| . |Aut(µk (~γk ))| Our connected invariants can be expressed in terms of one-part double Hurwitz numbers. Theorem 4.8.5. Assume that γ1k , γ2` ’s are E1 , . . . , Er or 1 and β is a nonzero effective curve class. If β = dEij for some d, i, j and all γ1k , γ2` ’s are either Ei or Ej , the invariant conn hµ1 (~γ1 ), µ2 (~γ2 )i(a,β) (4.8.7) is given by g g a1 a2 (t1 + t2 )(−1)g(a) (−1 − δ1,r )`(µ1 )+`(µ2 ) da−1 [µ1 (~γ )][µ2 (~γ2 )] X Hµ1 ,(n) Hµ2 ,(n) , na−2 a1 !a2 ! a +a =a 1 (4.8.8) 2 where δ1,r is the Kronecker delta (which is 1 if r = 1 and 0 otherwise). Otherwise, (4.8.7) vanishes. Thus, by Proposition 4.8.4, there is an explicit closed formula for the invariant (4.8.7). Proof. Let r > 1. Each insertion of (4.8.7) is a linear combination of fixed-point classes with coefficients zero or having nonnegative (t1 + t2 )-valuation. According to Lemma 4.5.1, (4.8.7) is divisible by t1 + t2 . As mentioned earlier, (4.8.7) is a polynomial in t1 , t2 . So if at least one of γ1k , γ2` ’s is 1, the invariant must be zero because the sum of the degrees of the insertions is at most `(µ1 )+`(µ2 )−1, which is the virtual dimension. 63 Assume that all γ1k , γ2` ’s are E1 , . . . , Er , in which case (4.8.7) is proportional to (t1 + t2 ). By Lemma 4.5.1 again, the invariant is zero if β is not a multiple of Eij for all i, j. Now we assume further that β = dEij for some d, i, j. We may evaluate (4.8.7) modulo (t1 + t2 )2 , so any T-fixed locus that contributes a factor (t1 + t2 )k for some k ≥ 2 may be ruled out. That is, it is enough to investigate those T-fixed loci defined in (4.4.1) (s = a), which we denote by F ’s. However, in order for the contributions of these loci to (4.8.7) not to vanish, the ramification points lying above the distinguished markings must map to xi or xj+1 . As a result, (4.8.7) vanishes if one of γ1k , γ2` ’s is Ek for some k 6= i, j. This completes the proof of the second assertion. Now we show the first assertion. Let P =− 1 1 [xi ], Q = − [xj+1 ]. Li Rj+1 It remains to evaluate (4.8.7) with γ1k , γ2` ’s from Ei or Ej . We check that Ei ∼ P, Ei ∼ Q (“∼” means that the difference between the left side and the right side can be written in terms of classes [xi+1 ], . . . , [xj ] and 1 as long as we are working modulo (t1 + t2 )); similarly, Ej ∼ P, Ej ∼ Q. By the vanishing claims just verified, we can replace all γ1k ’s with P and all γ2` ’s with Q. The invariant (4.8.7) is not exactly the resulting invariant hµ11 (P ) · · · µ1`(µ1 ) (P ), µ21 (Q) · · · µ2`(µ2 ) (Q)iconn (a,dEij ) . Instead, it is congruent modulo (t1 + t2 )2 to J := [µ1 (~γ1 )][µ2 (~γ2 )]hµ11 (P ) · · · µ1`(µ1 ) (P ), µ21 (Q) · · · µ2`(µ2 ) (Q)iconn (a,dEij ) . We can thus execute localization calculations over those F ’s with one more constraint on the source curve C0 : CL0 carries the marking corresponding to µ1 , and CR0 carries the marking corresponding to µ2 because the ramification points associated to µ1 (resp. µ2 ) are mapped to 64 xi (resp. xj+1 ). This means that in (4.4.1), λ0 = µ1 , ρ0 = µ2 , and σ0 = (k). To summarize, we only have to consider such T-fixed loci, denoted by Fa1 ,a2 , where the source curve C decomposes into three pieces Ca1 ∪ Σ ∪ Ca2 : Cak is a contracted component carrying ak simple markings, and its unique distinguished marking corresponds to µk ; the intersection Ca1 ∩ Ca2 is empty; the cover C̃ak associated to Cak is of genus gak ; and Σ is a chain of noncontracted components, which connects Ca1 and Ca2 , and the two twisted points of intersection have stack structures given by the monodromy (n). Note that Cak ’s are twisted points whenever they contain less than three special points and are otherwise twisted curves. In this way, we reduce our calculation to the integral over M (BSn , µ1 , (n); a1 ) × M (BSn , µ2 , (n); a2 ), followed by division by the product of the automorphism factor dj−i+1 and the distribution factor a1 !a2 ! of simple marked points. Let 1 : M (BSn , µ1 , (n); a1 ) → M 0,a1 +2 be the natural morphism mapping Ca1 to its coarse moduli space Ca1 (the node Ca1 ∩Σ is mapped to the marking Q1 ) and L1 the tautological line bundle formed by the cotangent space TQ∗ 1 Ca1 . Let ψ1 = c1 (L1 ). We define 2 : M (BSn , µ2 , (n); a2 ) → M 0,a1 +2 and ψ2 in a similar way. To proceed, let us summarize the contributions of virtual normal bundles. Set θ = (r + 1)t1 . • Contracted components: For k = 1, 2, Cak contributes (−1)gak −1 θ2gak −2 mod (t1 + t2 ). • A chain of noncontracted components: The contribution of each node smoothing is just ( t1 + t2 −1 ) . d All other node contributions are Lk Rk , k = i, . . . , j + 1, 65 each of which equals −θ2 mod t1 + t2 . Furthermore, all noncontracted curves contribute ( t1 + t2 j−i+1 ) −θ2 mod (t1 + t2 )2 . Hence the total contribution equals −θ2 dj−i (t1 + t2 ) mod (t1 + t2 )2 . • Smoothing nodes joining a contracted curve to a noncontracted curve: The contributions are given by 1 1 nRi ∗ n ( d − 1 ψ 1 ) , 1 1 nLj+1 − ∗2 ψ2 ) n( d . The contribution Ia1 ,a2 of the fixed locus Fa1 ,a2 to J is congruent modulo (t1 + t2 )2 to a1 [µ1 (~γ1 )][µ2 (~γ2 )] `(µ1 ) `(µ2 ) (−1) (−θ) · θ dj−i+1 a1 !a2 ! θ4 Z 2 Y d 1 (−1)gak θ2gak a −1 ( )ak ∗k ψkak −1 . n k θ M (BSn ,µk ,(n); ak ) −θ2 dj−i (t1 + t2 ) × k=1 Note that each factor in the second line is replaced with 1 in case ak = 0. Simplifying the expression yields 2 (t1 + t2 )(−1)g(a)+`(µ1 )+`(µ2 ) da−1 [µ1 (~γ1 )][µ2 (~γ2 )] Y na−2 a1 !a2 ! Z k=1 M (BSn ,µk ,(n);ak ) ∗k ψkak −1 . For ak > 0, Z M (BSn ,µk ,(n); ak ) ∗k ψkak −1 = deg(k ) Z M 0,ak +2 ga ψkak −1 = Hµkk,(n) . We conclude that g g a1 a2 (t1 + t2 )(−1)g(a)+`(µ1 )+`(µ2 ) da−1 [µ1 (~γ1 )][µ2 (~γ2 )] Hµ1 ,(n) Hµ2 ,(n) · Ia1 ,a2 ≡ na−2 a1 !a2 ! mod (t1 + t2 )2 . Keeping in mind that (4.8.7) is a multiple of t1 + t2 , so we obtain (4.8.8). The case r = 1 is similar, and so we omit the proof. By applying the intersection matrix with respect to the curve classes E1 , . . . , Er , we arrive at the following statement which is a little shorter than Theorem 4.8.5. Corollary 4.8.6. Let γ1k , γ2` ’s be 1 or divisors on Ar and β a nonzero effective curve class. 66 conn If β = dEij for some d, i, j, the connected invariant hµ1 (~γ1 ), µ2 (~γ2 )i(a,β) is given by (t1 + t2 )(−1)g(a) da−1 [µ1 (~γ1 )][µ2 (~γ2 )] na−2 Q`(µ1 ) k=1 (Eij · γ1k ) g Q`(µ2 ) k=1 (Eij · δk ) X g Hµa1 1,(n) Hµa2 2,(n) a1 +a2 =a a1 !a2 ! . Otherwise, it is zero. Theorems 4.8.1 and 4.8.5 provide an effective method to compute two-point extended invariants of [Symn (Ar )] of nonzero degrees. With the equations in the following proposition, this also determines the divisor operators as a consequence of three-point extended invariants of zero degree being determined by the Gromov-Witten theory of [Symn (C2 )]. Proposition 4.8.7. Given any classes α1 , . . . , αk ∈ A∗T,orb [Symn (Ar )]. We have hhα1 , . . . , αk , (2)ii = d hhα1 , . . . , αk ii, du (4.8.9) and for each ` = 1, . . . , r, hhα1 , . . . , αk , D` ii = hhα1 , . . . , αk , D` ii|s1 ,...,sr =0 + s` d hhα1 , . . . , αk ii. ds` (4.8.10) Proof. By definition, hα1 , . . . , αk , (2)i(a,β) = (a + 1) hα1 , . . . , αk i(a+1,β) , and by the untwisted divisor equation (β 6= 0 or k ≥ 3), hα1 , . . . , αk , D` i(a,β) = (ω` · β) hα1 , . . . , αk i(a,β) . These relations yield (4.8.9) and (4.8.10). (Note, however, that (4.8.10) is read as hhα1 , . . . , αk , D` ii = s` d hhα1 , . . . , αk ii ds` for k ≥ 3.) The sine function sin(u) is a rational function of eiu , where i2 = −1. It is straightforward to verify that extended three-point functions involving (2) or D` are rational functions in t1 , t2 , eiu , s1 , . . . , sr by the above equations. 67 On the other hand, we treat (4.8.9) as a twisted divisor equation because it provides a means of pulling the twisted divisor (2) out. If we substitute q = −eiu , we immediately obtain a relation on differential operators: d d = iq . du dq With this, (4.8.9) seems quite close to the usual divisor equation. They are still different, though. 68 Chapter 5 Comparison to other theories 5.1 Relative Gromov-Witten theory of threefolds Fix k distinct points p1 , . . . , pk of P1 . Given a positive integer n and partitions λ1 , . . . , λk of n, let • M g (S × P1 , (β, n); λ1 , . . . , λk ) be the moduli space parametrizing genus g relative stable maps (cf. [L1, L2]) to S × P1 relative to S × p1 , . . . , S × pk with the following data: • the domains are nodal curves of genus g and are allowed to be disconnected; • the relative stable maps have degree (β, n) ∈ A1 (S × P1 ; Z) and have nonzero degree on any connected components; • the maps are ramified over the divisor S × pi with ramification type λi . The ramification points are taken to be marked and ordered. Given any cohomology weighed partition λi (~ηi ), i = 1, . . . , k, we have an evaluation map • evij : M g (S × P1 , (β, n); λ1 , . . . , λk ) → S corresponding to the ramification point of type λij over the divisor S × pi . The genus g relative invariant in the cohomology-weighed partitions λ1 (~η1 ), . . . , λk (~ηk ) is defined by 1 hλ1 (~η1 ), . . . , λk (~ηk )iS×P = Qk g,β 1 ηi ))| i=1 |Aut(λi (~ Z k l(λ Y Yi ) • [M g (S×P1 ,(β,n);λ1 ,...,λk )]vir T i=1 j=1 ev∗ij (ηij ). 69 Let B1 , . . . , B` be a basis for A1 (S; Z) and B1∗ , . . . , B`∗ its dual basis. We define the partition function by Z0 (S × P1 )λ1 (~η1 ),...,λk (~ηk ) = X 1 2g−2 β·B1 hλ1 (~η1 ), . . . , λk (~ηk )iS×P · · · s`β·B` . s1 g,β u g,β However, we are more interested in the following shifted generating function GW(S × P1 )λ1 (~η1 ),...,λk (~ηk ) = u2n− 5.1.1 Pk i=1 age(λi ) Z0 (S × P1 )λ1 (~η1 ),...,λk (~ηk ) . (5.1.1) Degree (0, n) case In this section, we study the following (truncated) shifted partition function GWu (S × P1 )λ1 (η~1 ),...,λm (η~m ) = GW(S × P1 )λ1 (~η1 ),...,λk (~ηk ) |s1 =0,...,s` =0 . Like the treatment of the fixed-point basis for A∗T,orb ([Symn (S)]), we denote the cohomologyweighted partition σ11 ([x1 ]) · · · σ1`(σ1 ) ([x1 ]) · · · σs1 ([xs ]) · · · σs`(σs ) ([xs ]) associated to fixed-point classes by σ e. Evidently, the shifted partition function GWu (S × P1 )λ1 (η~1 ),...,λm (η~m ) is determined by shifted partition functions GWu (S × P1 )σf1 ,...,σfm ’s. Let i be the square root of −1 appearing in the SYM-HILB correspondence introduced in Section 3.5. We can translate the three-point shifted partition functions to the three-point functions of [Symn (S)] and Hilbn (S) by the following equations. Proposition 5.1.1. For any cohomology-weighted partitions λ1 (η~1 ), λ2 (η~2 ), λ3 (η~3 ), GWu (S × P1 )λ1 (η~1 ),λ2 (η~2 ),λ3 (η~3 ) n = hλ1 (η~1 ), λ2 (η~2 ), λ3 (η~3 )i[Sym (S)] (u) = P3 (−i) k=1 age(λk ) n haλ1 (η~1 ), aλ2 (η~2 ), aλ3 (η~3 )iHilb (S) (−eiu ). 70 Proof. By Theorem 3.5.2, it suffices show that e e, νei[Symn (S)] (u). GWu (S × P1 )λ,e e µ,e ν = hλ, µ (5.1.2) Both sides are zero if the condition |λk | = |µk | = |νk | for each k = 1, . . . , s is violated. Now assume that the condition holds. Since the relative stable maps collapse along the S-direction to Qs 1 x1 , . . . , xs , the shifted partition function GWu (S × P1 )λ,e . e µ,e f k=1 GWu (S × P )λ ν is simply µk ,f νk k ,f Moreover, GWu (S × P1 )λfk ,f and GWu (Uk × P1 )λfk ,f obviously coincide. As the equality µk ,f νk µk ,f νk n fk , µ GWu (Uk × P1 )λfk ,f = hλ fk , νek i[Sym (Uk )] (u) µk ,f νk is guaranteed by results of [BP, BG] for the C2 -case, the identity (5.1.2) follows immediately from Corollary 3.4.2. Remark. The functions GWu (S × P1 )λ1 (η~1 ),λ2 (η~2 ),λ3 (η~3 ) are elements of Q(t1 , t2 , eiu ), and we may also recover the cup product of Hilbn (S) from the limits lim GWu (S × P1 )λ1 (η~1 ),λ2 (η~2 ),λ3 (η~3 ) . u→+i∞ 5.1.2 Ar case Assume that the partition functions GW(Ar × P1 )λ1 (~η1 ),...,λk (~ηk ) are defined by using the basis E1 , . . . , Er . Our results in the preceding chapter recover certain relative Gromov-Witten invariants by the following equalities. Theorem 5.1.2. For α = 1(1)n , (2) or Dk , k = 1, . . . , r, n hhλ1 (~η1 ), α, λ2 (~η2 )ii[Sym (Ar )] = GW(Ar × P1 )λ1 (~η1 ),α,λ2 (~η2 ) . (5.1.3) Proof. When specialized to s1 = · · · = sr = 0, the equality (5.1.3) follows from Proposition 5.1.1. In particular, (5.1.3) is valid for α = 1(1)n without the constraint. For α = (2) or Dk , the coefficients of ui sj11 . . . sjrr , where j1 + · · · + jr > 0, match up on both sides of (5.1.3) by a direct comparison of Proposition 4.4 in [M] with our results in Section 4.8. Hence, (5.1.3) follows as well in this case. 71 5.2 Quantum cohomology of Hilbert schemes of points 5.2.1 Quantum cup product Let ρ∗ : A1 (Hilbn (Ar ); Z) → A1 (Symn (Ar ); Z) be the homomorphism induced by the HilbertChow morphism ρ : Hilbn (Ar ) → Symn (Ar ). There are isomorphisms A1 (Hilbn (Ar ); Z) ∼ = Ker(ρ∗ ) ⊕ A1 (Symn (Ar ); Z) ∼ = Ker(ρ∗ ) ⊕ A1 (Ar ; Z). Let ` be the class dual to the divisor −a1 (1)n−2 a2 (1) on Hilbn (Ar ). It is an effective rational n curve class generating the kernel Ker(ρHC ∗ ). For any classes α1 , . . . , αk on Hilb (Ar ), we consider the k-point function n hα1 , . . . , αk iHilb (Ar ) = ∞ X Hilbn (Ar ) d β·ω1 q s1 · · · srβ·ωr . X hα1 , . . . , αk i(d`,β) (5.2.1) d=0 β∈A1 (Ar ;Z) Now given any basis {δ} for A∗T (Hilbn (Ar )) and {δ ∨ } its dual basis. Define the small quantum cup product ∗crep on A∗T (Hilbn (Ar )) by the three-point functions as follows: α1 ∗crep α2 = X n hα1 , α2 , δiHilb (Ar ) δ ∨ . δ Like the orbifold case, we define QA∗T (Hilbn (Ar )) as the vector space A∗T (Hilbn (Ar )) ⊗Q[t1 ,t2 ] Q(t1 , t2 )((q, s1 , . . . , sr )) with the multiplication ∗crep . 5.2.2 SYM-HILB correspondence In Section 4.8, we provide a combinatorial description of any divisor operator on the ring A∗T,orb ([Symn (Ar )]). In [MO1], on the other hand, any divisor operator on A∗T (Hilbn (Ar )) is expressed in terms of the action of affine Lie algebra ĝl(r + 1) on the basic representations. These two expressions are actually equivalent via the correspondence given in Section 3.5. We make the substitution q = −eiu as in Section 3.5 and consider the map L there. Put F = Q(i, t1 , t2 )((u, s1 , . . . , sr )) and K = Q(t1 , t2 )((u, s1 , . . . , sr )). 72 The map L extends to a F -linear isomorphism L : QA∗T,orb ([Symn (Ar )]) ⊗K F → QA∗T (Hilbn (Ar )) ⊗K F. Further, we have the following SYM-HILB correspondence. Theorem 5.2.1. The F -linear isomorphism L respects quantum multiplication by divisors: L(D ∗orb α) = L(D) ∗crep L(α) (5.2.2) for any class α and divisor D. Proof. For cohomology-weighted partitions λ1 (~η1 ), λ2 (~η2 ) and α = (2) or Dk , n hhλ1 (~η1 ), α, λ2 (~η2 )ii[Sym (Ar )] = GW(Ar × P1 )λ1 (~η1 ),α,λ2 (~η2 ) n = hL(λ1 (~η1 )), L(α), L(λ2 (~η2 ))iHilb (Ar ) . Indeed, the first equality is Theorem 5.1.2 while the second equality is Proposition 6.6 in [MO1]. As L preserves Poincaré pairings, it follows from the above equalities that hL(λ1 (~η1 ) ∗orb α) | L(λ2 (~η2 ))i = hL(λ1 (~η1 )) ∗crep L(α) | L(λ2 (~η2 ))i. This implies that L respects quantum multiplication by (2) and Dk ’s. The equality (5.2.2) now follows due to the fact that (2) and Dk ’s give a basis for divisor classes. 5.3 The Crepant Resolution Conjecture Before discussing the full version of Bryan-Graber Crepant Resolution Conjecture, let us study a simple example. 5.3.1 An example We would like to give an explicit expression for the divisor operator D1 ∗orb − on the quantum ring A∗T,orb ([Sym2 (A1 )]). Let us substitute q = −eiu so that sin(γu) = 1 1 ((−q)γ − ). 2i (−q)γ 73 Consider the following basis B := {1(E1 )1(E1 ), 2(E1 ), 1(1)1(E1 ), 2(1), 1(1)1(1)}, whose elements are ordered according to their orbifold degrees. The matrix representation of the operator D1 ∗orb − with respect to B is given by  1 1 1 1 2θ(1 − 1+sq − 1+s/q ) iθ( 1+sq − 1+s/q )    −2iθ( 1 − 1 ) θ(2 − 1 − 1 − 2 ) 1+sq 1+s/q 1+sq 1+s/q 1−s    2t1 t2 0     0 4t1 t2  0 0 −1 0 0 −1 −θ(1+s) 1−s 0 0 0 4t1 t2 0 0    0    1 , −2   0   0 where θ := t1 + t2 and s := s1 . This is also the matrix representation of the operator L(D1 ) ∗crep − with respect to the ordered basis L(B)(cf. [MO1]). It is straightforward to check that D1 ∗orb − has distinct eigenvalues. In particular, we have a basis {v1 , . . . , v5 } of eigenvectors. By quantum multiplication by D1 and the identity 1, we find vi ∗orb vi = ai vi , for some ai 6= 0; vi ∗orb vj = 0, ∀i 6= j. So by replacing vi with vi /ai , we may assume that {v1 , . . . , v5 } is an idempotent basis; in which case, 1= 5 X vi . (5.3.1) i=1 Moreover, the Vandermonde matrix associated to the eigenvalues of D1 ∗orb − is invertible. In other words, by (5.3.1) the set {1, D1 , D12 , D13 , D14 } is a basis for the quantum cohomology QA∗T,orb ([Sym2 (A1 )]). Similarly, L(D1 ) generates the 74 quantum ring QA∗T (Hilb2 (A1 )) ⊗K F . We conclude that L : QA∗T,orb ([Sym2 (A1 )]) ⊗K F → QA∗T (Hilb2 (A1 )) ⊗K F is indeed an F -algebra isomorphism. This simple example raises the question: Do divisor classes generate the whole quantum ring? In response to this, one may wish to examine the eigenvalues of divisor operators for bigger n. This, however, seems a difficult task to perform directly. If one of the operators (2) ∗orb −, Dk ∗orb −’s turns out to have distinct eigenvalues, the ring structure will be determined, and L will be an F -algebra isomorphism. The hypothesis has yet to be entirely verified and may seem a little too good to be true. It is reasonable to expect something weaker (maybe certain combinations of these operators work). 5.3.2 Nonderogatory Conjecture We name the following nonderogatory conjecture, but we claim no originality for the statement. The reader is urged to consult [MO1] for a partial evidence of the conjecture. Conjecture 5.3.1 ([MO1]). Let L be as in Section 5.2.2. The commuting family of the operators L((2)) ∗crep −, L(D1 ) ∗crep −, . . . , L(Dr ) ∗crep − on the quantum cohomology of Hilbn (Ar ) is nonderogatory. That is, its joint eigenspaces are one-dimensional. Let us briefly explain some consequences of the nonderogatory conjecture on our quantum cohomology rings. We set R = Q(i, t1 , t2 , q, s1 , . . . , sr ) and q = −eiu . Since the quantum ring A∗T (Hilbn (Ar )) ⊗Q[t1 ,t2 ] R is semisimple, it admits a basis, say {v1 , . . . , vm }, of idempotent eigenvectors summing to the identity 1. Note that the basis elements are also the simultaneous eigenvectors for L((2)) ∗crep −, L(D1 ) ∗crep −, . . . , L(Dr ) ∗crep −. Suppose that e0k , e1k , . . . , erk are respectively the eigenvalues of the operators L((2)) ∗crep −, L(D1 ) ∗crep −, . . . , L(Dr ) ∗crep − 75 corresponding to the eigenvector vk . The nonderogatory property ensures that we can find numbers a0 , a1 , . . . , ar such that r X aj ej1 , . . . , j=0 r X aj ejm j=0 is a sequence of distinct elements. Therefore, the Vandermonde argument given earlier shows Pr that the element a0 · L((2)) + j=1 aj · L(Dj ) generates A∗T (Hilbn (Ar )) ⊗Q[t1 ,t2 ] R. This implies Pr that a0 · (2) + j=1 aj · Dj generates the quantum cohomology of [Symn (Ar )] over R as well. We thus obtain the following “corollary”.1 “Corollary” 5.3.2. The divisor classes (2) and D1 , . . . , Dr generate the quantum cohomology ring QA∗T,orb ([Symn (Ar )]), and any extended three-point function is a rational function in t1 , t2 , eiu , s1 , . . . , sr . Under the substitution q = −eiu , the map L : QA∗T,orb ([Symn (Ar )]) ⊗K F → QA∗T (Hilbn (Ar )) ⊗K F gives an isomorphism of F -algebras. On the other hand, we can match the orbifold Gromov-Witten theory with the relative Gromov-Witten theory. “Corollary” 5.3.3. The equality hhλ1 (~η1 ), λ2 (~η2 ), λ3 (~η3 )ii = GW(Ar × P1 )λ1 (~η1 ),λ2 (~η2 ),λ3 (~η3 ) holds for any cohomology-weighted partitions λ1 (~η1 ), λ2 (~η2 ), λ3 (~η3 ) of n. 5.3.3 Multipoint functions Once the nonderogatory conjecture holds, all extended three-point functions are known by “Corollary” 5.3.2. In this situation, we are actually able to generalize “Corollary” 5.3.2 to cover multipoint invariants. This can be done by proceeding in an analogous manner to Okounkov and Pandharipande’s determination of multipoint invariants of Hilbn (C2 ) (cf. [OP1]). 1 Whenever we put a double quotation mark “ ”, we emphasize that the statement or word inside comes with the hypothesis of the nonderogatory conjecture. 76 Let B be a basis for the Chen-Ruan cohomology A∗T,orb ([Symn (Ar )]). We recall the WDVV equation from [AGV2], but we write it in terms of extended functions to better suit our needs. For the time being, we drop the superscript [Symn (Ar )]. Proposition 5.3.4 ([AGV2]). Given Chen-Ruan classes α1 , α2 , α3 , α4 , β1 , . . . , βk . Let S be the set {1, . . . , k}, we have X S1 ` S1 ` hhα1 , α2 , βS1 , γiihhγ ∨ , βS2 , α3 , α4 ii S2 =S γ∈B X = X X hhα1 , α3 , βS1 , γiihhγ ∨ , βS2 , α2 , α4 ii. S2 =S γ∈B Here, for instance, hhα1 , α2 , βS1 , γii := hhα1 , α2 , βi1 , . . . , βi` , γii if S1 = {i1 , . . . , i` }. “Proposition” 5.3.5. All extended multipoint functions of [Symn (Ar )] can be determined from extended three-point functions and are rational functions in t1 , t2 , eiu , s1 , . . . , sr . Proof. We may see this by induction. Suppose that any extended m-point function with m ≤ k is known and is a rational function in t1 , t2 , eiu , s1 , . . . , sr . To determine extended (k + 1)-point function, it suffices to study N := hhα0 , α1 , . . . , αk ii for α0 = (2)` ∗orb D1m1 ∗orb · · · ∗orb Drmr , where `, m1 , . . . , mr are nonnegative integers. We may assume that ` + m1 + · · · + mr ≥ 2 in light of Proposition 4.8.7 and the fundamental class axiom. Let us write α0 = D ∗orb δ for some D = (2) or Dj . Clearly, N= X hhD, δ, γiihhγ ∨ , α1 , . . . , αk ii. γ∈B Let S = {1, . . . , k − 2}. By the WDVV equation, X hhD, δ, γiihhγ ∨ , αS , αk−1 , αk ii + γ∈B = X γ∈B X hhD, δ, αS , γiihhγ ∨ , αk−1 , αk ii γ∈B ∨ hhD, αk−1 , γiihhγ , αS , δ, αk ii + X hhD, αk−1 , αS , γiihhγ ∨ , δ, αk ii γ∈B + (terms with extended m-point functions, 3 ≤ m ≤ k). This says that N is determined by lower-point functions and extended (k + 1)-point functions with a δ-insertion. By replacing D ∗orb δ with δ if necessary and continuing the above procedure, we conclude that N can be calculated from lower-point functions and is a rational function in 77 t1 , t2 , eiu , s1 , . . . , sr . By induction, our claim is thus justified. “Corollary” 5.3.6 (The Crepant Resolution Conjecture). Let q = −eiu and k ≥ 3. For any Chen-Ruan classes α1 , . . . , αk on [Symn (Ar )], we have n n hhα1 , . . . , αk ii[Sym (Ar )] = hL(α1 ), . . . , L(αk )iHilb (Ar ) . n n In particular, hα1 , . . . , αk i[Sym (Ar )] = hL(α1 ), . . . , L(αk )iHilb (Ar ) |q=−1 . Proof. We suppress the indices [Symn (Ar )] and Hilbn (Ar ). The proof of “Proposition” 5.3.5 works as well for multipoint functions on Hilbn (Ar ). What makes things nice is that we get exactly the same set of WDVV equations on both [Symn (Ar )] and Hilbn (Ar ) sides via L provided that we have the equalities: hhα1 , α2 , α3 , Dii = hL(α1 ), L(α2 ), L(α3 ), L(D)i for D = (2) and Dj (j = 1, . . . , r). But these are clear by divisor equations and “Corollary” 5.3.2. Thus by a recursive argument, we conclude that L preserves (extended) multipoint functions, and the first claim follows. The second claim is now clear. 5.3.4 Closing remarks All “results” discussed above are honestly true for the case n = 2 and r = 1 since the divisor operator D1 ∗orb − has distinct eigenvalues and determines the orbifold quantum product. Also, in the definition of the map L, we may choose −i instead of i, in which setting the correct change of variables is q = −e−iu . As a matter of fact, the transformation q 7−→ n takes hhλ1 (~η1 ), . . . , λk (~ηk )ii[Sym (Ar )] to (−1) Pk 1 q j=1 age(λj ) n hhλ1 (~η1 ), . . . , λk (~ηk )ii[Sym (Ar )] . To il- lustrate this, just look at the matrix in Section 5.3.1. There we observe that terms involving q and 1q agree up to a sign. The calculation of [Symn (Ar )]-invariants in Section 5.1.1 gives an indication that these invariants might be closer, geometrically and combinatorially, to the relative invariants of Ar × P1 than the invariants of Hilbn (Ar ). In reality, it is the form the relative invariants take that mo- 78 tivates our calculation. We do know that GW(Ar × P1 )λ1 (~η1 ),...,λk (~ηk ) can be “reduced” to the three-point case by the degeneration formula (cf. [M]). It is, however, unclear if the WDVV equation “behaves” in a similar way to the degeneration formula. At the moment, we expect that the equality n hhλ1 (~η1 ), . . . , λk (~ηk )ii[Sym (Ar )] = GW(Ar × P1 )λ1 (~η1 ),...,λk (~ηk ) n should also be true. Particularly, the usual k-point function hλ1 (~η1 ), . . . , λk (~ηk )i[Sym (Ar )] should Pk be the coefficient of u i=1 age(λi )−2n in Z0 (Ar × P1 )λ1 (~η1 ),...,λk (~ηk ) . On the other hand, for n ≥ 2, the smooth schemes Hilbn (Ar ) and Hilbn ([C2 /µr+1 ]) are two different crepant resolutions of the symmetric product Symn (C2 /µr+1 ). Thus, it is quite possible that the genus zero Gromov-Witten theories of the schemes Hilbn (Ar ) and Hilbn ([C2 /µr+1 ]) are equivalent for positive integers n and r (note that the statement is obviously true for the case where n = 1 as the schemes Hilb1 (Ar ) and Hilb1 ([C2 /µr+1 ]) coincide). We will discuss a little about the Quantum Minimal Model Conjecture, which predicts that there exists an equivalence between the Gromov-Witten theories of two crepant resolutions, in the next chapter. 79 Chapter 6 Ordinary flops 6.1 Preliminaries Let X be a nonsingular projective variety and M g,n (X, β) the moduli space parametrizing genus g, n-pointed stable map f : (C, p1 , . . . , pn ) → X of degree β. We recall some basic notions before proceeding. For i = 1, . . . , n, denote the ith evaluation map by ei : M g,n (X, β) → X. There is also a map f t : M g,n+1 (X, β) → M g,n (X, β) defined by forgetting the last marked point and contracting any resulting unstable components. For i = 1, . . . , n, the map f t carries a tautological section si : M g,n (X, β) → M g,n+1 (X, β) defined by si ([f : (C, p1 , . . . , pn ) → X]) = [f 0 : (C ∪ P1 , p1 , . . . , pi−1 , [1 : 0], pi+1 , . . . , pn , [0 : 1]) → X], where [1 : 0], [0 : 1] ∈ P1 , and f 0 := f on C but f 0 (P1 ) := f (pi ). Let ωf t be the relative dualizing sheaf of f t. We have tautological line bundle Li := s∗i ωf t , 80 whose fiber over [f : (C, p1 , . . . , pn ) → X] is the cotangent line Tp∗i C at the ith marked point, and we let ψi := c1 (Li ), the first Chern class of Li . The virtual dimension of M g,n (X, β) is given by Z c1 (TX ) + (1 − g)(dim X − 3) + n. β For α1 , . . . , αn ∈ H ∗ (X), we have the n-point Gromov-Witten invariant Z hα1 , . . . , αn ig,n,β := 6.2 [M g,n (X,β)]vir e∗1 (α1 ) ∪ . . . ∪ e∗n (αn ). The main formula The main goal of this chapter is to show the following formula. Theorem 6.2.1. With notation as above, Z [M 1,0 (Pr ,d)]vir e(R1 f t∗ e∗1 (OPr (−1)⊕(r+1) )) = (−1)(r+1)d (r + 1) . 24d When we set r = 1, we obtain the following equality: Z [M 1,0 (P1 ,d)]vir e(R1 f t∗ e∗1 (OP1 (−1)) = 1 , 12d which was first computed in physics (cf. [BCOV]) and proved later in mathematics (cf. Proposition 5.2 in [GP]). Also, Theorem 6.2.1 has been independently discovered and proved by Iwao, Lee, Lin, and Wang [ILLW]. We remark that our proof, dating back to 2007, is totally different from theirs and is much simpler. Proof of Theorem 6.2.1 For each i = 0, . . . , r, let Ti = (C× )r+1 act on Cr+1 by −1 (s0 , . . . , sr ) · (x0 , . . . , xr ) = ((s−1 i s0 ) · x0 , . . . , (si sr ) · xr ). 81 Now let qi = [0, . . . , 0, 1, 0, . . . , 0] ∈ Pr (1 is the ith component). Denote by πi : (P∞ )r+1 → P∞ the projection onto the ith factor. Let H = c1 (OP(⊕ri=0 πi∗ OP∞ (1)) (1)), ti = c1 (πi∗ OP∞ (1)). The first Ti -equivariant Chern class of OPr (−1) can be determined as follows: cT1 i (OPr (−1)) = ti − H. This means particularly that Ti acts on OPr (−1)|qi trivially. We denote Ti ’s by T if there is no danger of confusion. In order to establish Theorem 6.2.1, it suffices to show that r Y Z [M 1,0 (Pr ,d)]vir T i=0 eTi (R1 f t∗ e∗1 (OPr (−1)) = (−1)(r+1)d (r + 1) . 24d (6.2.1) The rest of the section is devoted to proving (6.2.1). By the virtual localization formula, the left side of (6.2.1) equals Qr ∗ 1 ∗ X 1 Z r i=0 iΓ (eTi (R f t∗ e1 (OP (−1)) , vir |AΓ | M Γ eT (NΓ ) (6.2.2) Γ where (i) Each Γ is a labeled graph, which is actually a connected tree. Its vertices v are labeled with T-fixed loci of Pr and genera of the corresponding contracted components, and its edges e are labeled with the degrees de of their images. Moreover, M Γ is the product of moduli spaces of pointed curves associated to Γ. (ii) iΓ : M Γ → M 1,0 (Pr , d) is a finite morphism with image, realized as M Γ /AΓ , a connected component of M 1,0 (Pr , d)T . Here AΓ is a finite group of automorphisms acting on M Γ and 82 fits into the exact sequence of groups 1→ Y Z/de → AΓ → Aut(Γ) → 1. e (iii) iΓ (M Γ )’s form a complete set of connected components of M 1,0 (Pr , d)T . (iv) eT (NΓvir ) is the T-equivariant Euler class of the virtual normal bundle to M Γ . The following lemma helps us get rid of superfluous fixed loci. Lemma 6.2.2. If a labeled graph has more than one edge, then it makes no contribution to (6.2.2). Proof. Suppose that such a graph Γ has more than one edge. Let [f : C → Pr ] ∈ M Γ , and consider the normalization sequence 0 → OC → v∗ OC̃ → ⊕δj=1 Onj → 0. Here v : C̃ → C is the canonical normalization map and n1 , . . . , nδ are nodes of C. Tensoring the sequence by f ∗ OPr (−1) and taking cohomology, we get 0 φf → H 0 (C, f ∗ OPr (−1)) → H 0 (C̃, v ∗ f ∗ OPr (−1)) → ⊕δj=1 OPr (−1)|f (nj ) θf → H 1 (C, f ∗ OPr (−1)) → H 1 (C̃, v ∗ f ∗ OPr (−1)) → 0. We claim that θf is not trivial. To see this, we examine the map φf . Let C1 , . . . , Cm be (f ◦ v)-contracted components of C̃ and Cm+1 , . . . , Cn the noncontracted components. As H 0 (Ci , (f ◦ v)∗ OPr (−1)) = 0 for i ≥ m + 1, we have H 0 (C̃, v ∗ f ∗ OPr (−1)) = m M H 0 (Ci , (f ◦ v)∗ OPr (−1)) ∼ = Cm . i=1 Moreover, a simple analysis shows that a connected contracted curve has nodes at least one less than the number of its irreducible components. As Γ is not a one-edge graph, we deduce that δ > m. Therefore, φf cannot be surjective, and so θf is not a zero map. 83 By the claim, there exists a node nk such that the Ti -weight of OPr (−1)|f (nk ) divides eTi (H 1 (C, f ∗ OPr (−1)) for each i. Because f (nk ) = qs for some s, and Ts acts trivially on OPr (−1)|qs , it follows that eTs (H 1 (C, f ∗ OPr (−1)) = 0. That is, Γ does not contribute to (6.2.2). For a, b = 0, . . . , r, let Γda,b be the one-edge graph with the unique edge labeled with degree d > 0 and two endpoints labeled with points qa , qb and genera ga = 1, gb = 0: • qa (ga = 1) d • qb (gb = 0) Figure 6.1. Γda,b . This is, in fact, a graph in which the source curve C consists of a noncontracted component Ce ∼ = P1 and a genus one contracted component Ca labeled with qa (the vertex labeled with qb corresponds to a single point). By the above lemma, these are all graphs that can possibly make a contribution. To proceed, it is helpful to make use of the identification H 1 (C, f ∗ OPr (−1)) ∼ = H 1 (Ca , f ∗ OPr (−1)) ⊕ H 1 (P1 , f ∗ OPr (−1)), which, together with the following lemma, enables us to calculate eTi (H 1 (C, f ∗ OPr (−1))). Lemma 6.2.3. Let λ = c1 (H 0 (Ca , ωCa )). We have the equalities eTi (H 1 (Ca , f ∗ OPr (−1))) = −λ + ti − ta , and eTi (H 1 (P1 , f ∗ OPr (−1)) = (−1)d−1 d−1 Y sta + (d − s)tb − ti ) d s=1 ( for i = 0, . . . , r. Proof. By Serre’s duality, H 1 (Ca , f ∗ OPr (−1)) = H 0 (Ca , ωCa )∨ ⊗ OPr (−1)|qa . This shows the first equality. 84 Now set p0 = [1 : 0], p1 = [0 : 1] ∈ P1 . We know that Ti acts on (f ∗ OPr (1) ⊗ ωP1 )|p0 and (f ∗ OPr (1) ⊗ ωP1 )|p1 with weights w0 : = w1 : = tb − ta d−1 1 = ta + tb − ti , d d d ta − tb 1 d−1 tb − ti + = ta + tb − ti , d d d ta − ti + respectively, and so 0 1 ∗ eTi (H (P , f O (1) ⊗ ω ) Pr P1 = = = d−2 Y sw0 + (d − 2 − s)w1 d−2 s=0 d−2 Y (s + 1)ta + [d − (1 + s)]tb − ti ) d s=0 ( d−1 Y sta + (d − s)tb − ti ). d s=1 ( Thus, eTi (H 1 (P1 , f ∗ OPr (−1)) = eTi (H 0 (P1 , f ∗ OPr (1) ⊗ ωP1 )∨ ) = (−1)d−1 d−1 Y sta + (d − s)tb − ti ), d s=1 ( which proves the second equality. Now we study the inverse equivariant Euler class 1/eT (NΓvir d ) for all a, b and d: a,b (a) Infinitesimal automorphisms of C: At the point where Ce and Ca intersect, the tangent space to Ce contributes 1, while at the point labeled with qb , the tangent space contributes tb − ta . d (b) Infinitesimal deformations of C: It comes from the node joining Ca and Ce , and so it is given by ( ta − tb − ψ)−1 , d where ψ is the first Chern class of the tautological line bundle formed by the cotangent space of Ca at the node. (c) Infinitesimal deformations of the stable map with the domain held fixed: 85 • Vertices: The vertex labeled with qa contributes eT (H 1 (Ca , f ∗ T Pr )) . eT (H 0 (Ca , f ∗ T Pr )) As studied earlier, it is expressible in terms of the Hodge class and ti ’s: Y (1 + j6=a −λ ). ta − tj On the other hand, the vertex labeled with qb contributes Y 1 1 = . r eT (Tqb P ) tb − tj j6=b • Flags: The contribution is none other than the product eT (Tqa Pr ) · eT (Tqb Pr ), which is Y (ta − tj ) · j6=a Y (tb − tj ). j6=b • The unique edge: We need to look at the moving part of eT (H 0 (Ce , f ∗ T Pr )), which can be determined by the Euler sequence on Pr . Since a detailed discussion has been provided in [GP], we just write down the expression: Y (−1)d d2d Qd 2 2d (d!) ((ta − tb ) j6=a,b 1 sta +(d−s)tb − tj s=0 d . Now we are ready to deduce (6.2.1). By Lemma 6.2.3, we have r Y i∗Γd eTi (R1 f t∗ e∗1 (OPr (−1))) = −λ Y a,b i=0 (−λ + ti − ta ) i6=a (ta − tb )2d−2 d2d−2 d−1 Y Y sta + (d − s)tb × (−1)(d−1)(r−1) ( − ti ), d s=1 × (−1)d−1 [(d − 1)!]2 i6=a,b for all a, b, and d. d By our expression of 1/eT (NΓvir d ), we find that the contribution of each Γa,b to (6.2.1) is given a,b 86 by Qr ∗ eTi (R1 f t∗ e∗1 (OPr (−1))) i=0 iΓd a,b Z eT (NΓvir d ) M Γd a,b a,b tb − ta 1 Q = (−1)(r+1)(d+1) (t − tj )(tb − tj ) d j6=a,b a Z Y Y d d X × ( ψ)k ] (ta − tj − λ) λ (−λ + ti − ta )[ ta − tb ta − tb M 1,1 k≥0 i6=a j6=a Y tb − ta d 1 Q = (−1)(r+1)(d+1) · (−1)r (ta − tj )2 (t − t )(t − t ) d t − tb j b j a j6=a,b a j6=a = Z λ M 1,1 (−1)(r+1)d Y ta − tj . 24 tb − tj j6=a,b As we are dealing with one-edge graph, we have AΓda,b ∼ = Z/(d) for all a, b, and d. In other words, |AΓda,b | = d. Using the well-known fact that Z λ= M 1,1 1 , 24 we obtain r Y Z [M 1,0 (Pr ,d)]vir T i=0 = r X 1 a,b=0,a6=b |AΓda,b | (r+1)d = = (−1) 24d eTi (R1 f t∗ e∗1 (OPr (−1))) Qr ∗ eTi (R1 f t∗ e∗1 (OPr (−1))) i=0 iΓd a,b Z e(NΓvir d ) M Γd r X a,b a,b Y ta − tj a,b=0,a6=b j6=a,b tb − tj (−1)(r+1)d (r + 1) , 24d where the last equality is a consequence of the following beautiful identity. Lemma 6.2.4. Given r + 1 variables x0 , . . . , xr , r X Y x0 − xj = 1. xk − xj k=1 j6=0,k Proof. Write r X Y x0 − xj P (x) R(x) := = , xk − xj q1 · · · qr−1 k=1 j6=0,k where x = (x0 , . . . , xr ), qi = Q j≥i+1 (xi − xj ) for i = 1, . . . , r − 1, and P (x) ∈ Q[x0 , . . . , xr ]. 87 First of all, (x1 − x2 )|P (x). To see this, consider P (x) as a polynomial with coefficients in Q[x0 , x2 , . . . , xr ]. Then P (x) = (x0 − x2 ) . . . (x0 − xr )(x2 − x3 ) . . . (x2 − xr )Q1 −(x0 − x1 )(x0 − x3 ) . . . (x0 − xr )(x1 − x3 ) . . . (x1 − xr )Q1 +(x1 − x2 )S, where Q1 = q3 . . . qr−1 and S ∈ Q[x0 , . . . , xr ]. Obviously, P (x0 , x2 , x2 , x3 , . . . , xr ) = 0. Thus, (x1 − x2 )|P (x). By a similar argument, all other factors of qi divides P (x) for each i, and the product q1 . . . qr−1 divides P (x) because qi ’s are relatively prime. Hence we deduce that P (x) = h · q1 . . . qr−1 for some h ∈ Q[x0 , . . . , xr ]. Note that for each i, the degree of xi in P (x) is at most r − 1 but the degree of xi in the product q1 . . . qr−1 is exactly r − 1. This means that h is free of variables x1 , . . . , xr and is thus a polynomial in at most one variable x0 . Let us write h = h(x0 ), it remains to show that h(x0 ) = 1. We need only check the cases x0 = n ∈ Z and xi = i + x0 for i ≥ 1: h(n) = r X k=1 = r X (−1)r−1 r!/k r! = (−1)k−1 (k − 1) · · · 1 · (−1) · · · (k − r) k!(r − k)! k=1 r −(1 − 1) + 1 = 1. Therefore, the lemma is proved. 6.3 Minimal Model Conjecture The motivation behind Theorem 6.2.1 is the Quantum Minimal Model Conjecture, which is stated as follows. 88 Conjecture 6.3.1 ([R]). K-equivalent projective varieties have isomorphic quantum cohomologies. In this section, we merely focus on a special kind of K-equivalences, which we now discuss. Let X be a smooth complex projective variety and X → X̄ a flopping contraction, with the exceptional locus Z∼ = Pr , and the normal bundle NZ/X ∼ = OPr (−1)⊕(r+1) . The corresponding Pr flop f : X 99K X 0 exists and is referred to as simple ordinary Pr flop. The simple flop may be constructed in this way: Blowing up X along Z to get a morphism φ : Y → X. The exceptional divisor E of φ is PZ (NZ/X ) = Z ×C Z 0 , where Z 0 = Pr . It can be shown that there is a morphism φ0 : Y → X 0 obtained by blowing down E onto Z 0 , and the normal bundle to Z 0 is given by NZ 0 /X 0 ∼ = OPr (−1)⊕(r+1) . More details can be found in [LLW]. The varieties X and X 0 are K-equivalent since φ∗ KX = ∗ φ0 KX 0 . The invariance of the genus zero Gromov-Witten theory under simple ordinary flops has been demonstrated in [LLW]. Theorem 6.2.1 suggests a similar result concerning genus one GromovWitten theory attached to the extremal ray. In the recent paper [ILLW], a more general setting is considered, namely ancestors and descendents are both included. For later convenience of explanation, we introduce the extremal Gromov-Witten potential of X, which is defined by hα1 , . . . , αn i X = ∞ X ∞ X hα1 , . . . , αn ig,d` q d , g=0 d=0 where ` is the line class in Z ∼ = Pr . We define similarly the extremal Gromov-Witten potential 0 h−, . . . , −iX of X 0 but we replace the quantum parameter q with q −1 . 89 We consider the correspondence F = φ0∗ φ∗ and set F(q) = q −1 . Corollary 6.3.2. The extremal Gromov Witten potentials are preserved by simple Pr flops. More precisely, we have the following equality: X F hα1 , . . . , αn i = hFα1 , . . . , Fαn i X0 , (6.3.1) where X and X 0 are as above. Proof. First of all, we assume that X and X 0 are not three-dimensional. By dimension reasoning, the positive degree, genus one, extremal Gromov-Witten theory is completely determined by zero-point Gromov-Witten invariants, and the genus g theory is trivial for g ≥ 2. Moreover, we have M 1,0 (W, d`) = M 1,0 (Pr , d) for W = X, X 0 and the following equality: Z Z 1= [M 1,0 (W,d`)]vir [M 1,0 (Pr ,d)]vir e(R1 f t∗ e∗1 (OPr (−1)⊕(r+1) )). On the other hand, in [LLW] and [ILLW], the degree zero theory and the genus zero theory of X and X 0 have been determined, and the relations F(`k ) = (−1)r−k `k , k ≥ 1 have also been established. Therefore, our statement follows from Theorem 6.2.1 and a direct comparison of both sides of (6.3.1). Suppose now that X and X 0 are of dimension 3. The above argument for genus ≤ 1 also works but we also need to understand higher genus invariants, in which case, the main formula for the equality (6.3.1) to hold is simply Z [M g,0 (P1 ,d)]vir e(R1 f t∗ e∗1 (OPr (−1) ⊕ OPr (−1))) = d2g−3 |χ(Mg )| , (2g − 3)! 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