Relative Mirror Symmetry and Ramifications of a Formula for Gromov-Witten Invariants Citation van Garrel, Michel (2013) Relative Mirror Symmetry and Ramifications of a Formula for Gromov-Witten Invariants. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9EQP-PD83. https://resolver.caltech.edu/CaltechTHESIS:05312013-164406051 Abstract For a toric Del Pezzo surface S, a new instance of mirror symmetry, said relative, is introduced and developed. On the A-model, this relative mirror symmetry conjecture concerns genus 0 relative Gromov-Witten of maximal tangency of S. These correspond, on the B-model, to relative periods of the mirror to S. Furthermore, for S not necessarily toric, two conjectures for BPS state counts are related. It is proven that the integrality of BPS state counts of the total space of the canonical bundle on S implies the integrality for the relative BPS state counts of S. Finally, a prediction of homological mirror symmetry for the open complement is explored. The B-model prediction is calculated in all cases and matches the known A-model computation for the projective plane. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Relative Mirror Symmetry, BPS state counts, Gromov-Witten invariants, Open Complement, Homological Mirror Symmetry, Algebraic Geometry, Symplectic Geometry Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2011. Thesis Availability: Public (worldwide access) Research Advisor(s): Graber, Thomas B. Thesis Committee: Graber, Thomas B. (chair) Ramakrishnan, Dinakar Flach, Matthias Tian, Zhiyu Defense Date: 21 May 2013 Record Number: CaltechTHESIS:05312013-164406051 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05312013-164406051 DOI: 10.7907/9EQP-PD83 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7807 Collection: CaltechTHESIS Deposited By: Michel Francois Van Garrel Deposited On: 03 Jun 2013 22:24 Last Modified: 04 Oct 2019 00:01 Thesis Files Preview PDF - Final Version See Usage Policy. 416kB Repository Staff Only: item control page

Relative Mirror Symmetry and Ramifications of a Formula for Gromov-Witten Invariants Thesis by Michel van Garrel In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Defended May 21, 2013) ii c 2013 Michel van Garrel All Rights Reserved iii To my parents Curt and Danielle, the source of all virtues that I possess. To my brothers Clément and Philippe, my best and most fun friends. iv Acknowledgements My greatest thanks go to my advisor Professor Tom Graber, whose insights into the workings of algebraic geometry keep astonishing me and whose ideas were of enormous benefit to me during the pursuit of my Ph.D. My special thanks are given to Professor Yongbin Ruan, who initiated the project on relative mirror symmetry and to whom I am grateful many enjoyable discussions. I am grateful that I am able to count on the continuing support of Professor Eva Bayer and Professor James Lewis. Over the past few years I learned a lot from Daniel Pomerleano, who, among others, introduced me to homological mirror symmetry. Mathieu Florence has been a figure of inspiration and motivation ever since he casually introduced me to algebraic geometry. Finally, my thanks go to the innumerable enjoyable math conversations with Roland Abuaf, Dori Bejleri, Adam Ericksen, Anton Geraschenko, Daniel Halpern-Leistner, Hadi Hedayatzadeh, Clément Hongler, Brian Hwang, Khoa Nguyen, Rom Rains, Mark Shoemaker, Zhiyu Tian, Nahid Walji, Tony Wong and Gjergji Zaimi. v Abstract For a toric Del Pezzo surface S, a new instance of mirror symmetry, said relative, is introduced and developed. On the A-model, this relative mirror symmetry conjecture concerns genus 0 relative Gromov-Witten of maximal tangency of S. These correspond, on the B-model, to relative periods of the mirror to S. Furthermore, two conjectures for BPS state counts are related. It is proven that the integrality of BPS state counts of KS , the total space of the canonical bundle on S, implies the integrality for the relative BPS state counts of S. Finally, a prediction of homological mirror symmetry for the open complement is explored. The B-model prediction is calculated in all cases and matches the known A-model computation when S = P2 . vi Contents Acknowledgements iv Abstract v 1 Introduction 1 2 Local and relative BPS state counts 3 2.1 A formula relating the invariants . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Integrality of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Mirror geometries to P2 13 3.1 Mirror symmetry statements for P2 and KP2 . . . . . . . . . . . . . . . . . . 14 3.2 The SYZ conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Batyrev’s construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Local mirror symmetry for P2 . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Relating two families by coordinate change . . . . . . . . . . . . . . . . . . . 22 4 Relative mirror symmetry 27 4.1 Reflexive polytopes and local mirror symmetry . . . . . . . . . . . . . . . . . 28 4.2 Proof of relative mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . 33 5 A prediction of homological mirror symmetry 39 A Local mirror symmetry for P2 45 vii B Local mirror symmetry for P1 × P1 and S1 49 C Some Gromov-Witten invariants 52 Bibliography 62 1 Chapter 1 Introduction The study of mirror symmetry in mathematics originated in theoretical physics and has, since the 1980s, led to an intensive interaction between the two fields. An essential ingredient of mirror symmetry is the curve counting theory of Gromov-Witten invariants. Denote by S a Del Pezzo surface and by D a smooth effective anti-canonical divisor on it. Furthermore, let β ∈ H2 (S, Z) be an effective curve class. This data is associated with the genus 0 GromovWitten invariants Iβ (KS ) of KS , the total space of the canonical bundle on S. Local mirror symmetry for KS asserts that these Gromov-Witten invariants are computed via periods on its mirror variety. Alternatively, one considers genus 0 relative Gromov-Witten invariants of S relative to D. Such invariants are virtual counts of genus 0 curves in S with a variety of tangency conditions along D. These are indexed by the weight partition of the cohomology of D. One such choice governs the genus 0 relative Gromov-Witten invariants of maximal tangency, denoted by Nβ (S, D). This requires that the curves meet D in exactly one point, thus assures that the relevant moduli space is zero-dimensional and hence removes the need for insertions. In chapter 4, we introduce a conjecture relating these relative invariants of maximal tangency to relative periods on the mirror to S. We call this new instance relative mirror symmetry and prove it when S is toric. A formulation of relative mirror symmetry for general genus 0 relative Gromov-Witten invariants is under development by the author. The proof we present for the invariants of maximal tangency relies on the following theorem, which was proven for P2 by Gathmann in [1], and then extended by Graber-Hassett to all Del Pezzo surfaces (unpublished): 2 Theorem 1. (Gathmann, Graber-Hassett) With the above notation, Nβ (S, D) = (−1)β·D (β · D) Iβ (KS ). We explore further ramifications of this formula in chapters 2 and 5. In chapter 2 we consider BPS state counts, which are refinements of Gromov-Witten invariants. Whereas Gromov-Witten invariants are rational numbers in general, BPS state counts are expected to be integers. For Calabi-Yau three-folds, of which KS are examples, this was conjectured by Gopakumar-Vafa in [2] and [3]. Relative BPS state counts for log Calabi-Yau surface pairs, of which (S, D) are examples, were introduced by Gross-Pandharipande-Siebert in [4]. The authors conjecture that these invariants are integers as well. We prove that the conjecture for KS implies the conjecture for (S, D). In chapter 5, we are interested in the homological mirror symmetry conjecture for the open complement S − D. The conjecture states that the derived Fukaya category of S − D ought to be equivalent to the bounded derived category of coherent sheaves of its mirror MS . Taking Hochschild cohomology on both sides yields the expectation that the Hochschild cohomology of MS is isomorphic to the symplectic cohomology of S − D. As a step towards verifying this prediction of homological mirror symmetry, we calculate the Hochschild cohomology of MS as a module over the polynomial ring. For P2 − D, this matches up with the calculation of its symplectic cohomology by Nguyen-Pomerleano. 3 Chapter 2 Local and relative BPS state counts This chapter are the results of joint work with Tony W. H. Wong and Gjergji Zaimi. Denote by S a Del Pezzo surface, by D a smooth effective anti-canonical divisor on it and by KS the total space of the canonical bundle on S. Furthermore, let β ∈ H2 (S, Z) be an effective non-zero curve class. On one side are the genus 0 degree β relative Gromov-Witten invariants of maximal tangency Nβ (S, D) of (S, D). On the other side are the genus 0 degree β local Gromov-Witten invariants Iβ (KS ) of KS . Theorem 1 relates these two sets of invariants via Nβ (S, D) = (−1)β·D (β · D) Iβ (KS ). (2.1) In general, Gromov-Witten invariants are rational numbers, since the relevant moduli spaces are Deligne-Mumford stacks. Since KS is Calabi-Yau, generically genus 0 curves are embedded with normal bundle isomorphic to O(−1) ⊕ O(−1). For d ≥ 1, a degree d cover of such a curve will contribute to the degree dβ invariant Idβ (KS ). This contribution is quantified by the Aspinwall-Morrison formula to be 1/d3 , proven by Manin in [5]. The BPS state counts n(β) are the rational numbers defined via Iβ (KS ) = X n(β/k) k|β k3 . (2.2) If all embedded genus 0 curves were of the above form, the n(β) would count actual genus 0 degree β curves in KS . This is false in general. It is nevertheless conjectured by Gopakumar- 4 Vafa that n(β) ∈ Z for all Calabi-Yau 3-folds. This was proven in the case where KS is toric by Peng in [6]. On the other side, in [4] Gross-Pandharipande-Siebert introduce relative BPS state counts for log Calabi-Yau surface pairs, of which (S, D) are examples. Assume that β is primitive and set w = D · β. For d ≥ 1, consider the relative GW invariant Ndβ (S, D). Adopting the same notation as in [4], we write NS [dw] = Ndβ (S, D). (2.3) The authors consider the generating series NS = ∞ X NS [dw] q d . (2.4) d=1 Computing multiple cover contributions leads the authors to define the relative BPS numbers nS [dw] ∈ Q via NS = ∞ X nS [dw] d=1   ∞ X 1 k(dw − 1) − 1 k2 k=1 k−1 q dk . (2.5) Analogously to the local case, Gross-Pandharipande-Siebert conjecture that the nS [dw] are integers for all d ≥ 1. In this chapter, we prove the following theorem. Theorem 2. (Garrel-Wong-Zaimi) Let β ∈ H2 (S, Z) be an effective non-zero primitive curve class. For d ≥ 1, consider the two sequences of rational numbers Ndβ (S, D) and Idβ (KS ) and assume that they satisfy equation (2.1) for all dβ. Define two sequences of rational numbers nS [dw] and n(d) by means of the equations (2.2), (2.3), (2.4) and (2.5). Then: nS [dw] ∈ Z, ∀d ≥ 1 ⇐⇒ dw · n(d) ∈ Z, ∀d ≥ 1. An immediate consequence then is: Corollary 3. The conjecture on the integrality of the local BPS invariants of KS implies the conjecture on the integrality of the relative BPS invariants of (S, D). Moreover, the result in [6] on the integrality for the toric local case implies: 5 Corollary 4. If S is toric, then its relative BPS numbers are integers. We prove theorem 2 in the next two sections. 2.1 A formula relating the invariants Assuming that β is primitive, formula (2.2) applied to dβ gives ∞ X d Idβ (KS ) q = d=1 ∞ X X 1 d=1 k|d k3 d n( β) q d . k Combining this with formula (2.1) and noting that dβ · D = dw yields NS = ∞ X NS [dw] q d d=1 = ∞ X (−1)dw dw Idβ (KS ) q d d=1 = ∞ X q d (−1)dw dw d=1 X 1 k|d k3 d nS ( β). k Lemma 5. The nS [dw] are related to the n(dβ) by the formula ∞ X X 1 k( d w − 1) − 1 d k nS [ w] NS = q 2 k k−1 k d=1 d k|d = ∞ X q d (−1)dw dw d=1 X 1 k|d k3 d nS ( β). k Proof. This follows from the change of variable d˜ = dk: ∞ X   ∞ X 1 k(dw − 1) − 1 q dk 2 k k − 1 d=1 k=1   ˜ ∞ X X 1 k( kd w − 1) − 1 d˜ d˜ = q n [ w]. S k−1 k2 k ˜ ˜ NS = d=1 nS [dw] k|d 6 Fix a positive integer m. We write the formula of lemma (5), capped in degree m + 1 and larger, in matrix form: Let row d encode the terms in q d , and let kd parametrize the columns. Then, the above m equations turn into:   R [nS [dw]]d = A · L · A−1 (−1)dw dw n(d) d , (2.6) where Rij :=    i/j (jw−1)−1 1 (i/j)2 i/j−1  0
if j|i, else; Aij := (−1)iw iw · δij ;    1 3 if j|i, (i/j) Lij :=  0 else. Note that multiplying with A−1 yields a matrix of determinant ±1. Notation. For a square-free integer n, let #p (n) denote the number of primes in the prime factorization of n. Moreover, for integers k and m, write k ∈ I(m) to mean that k divides m and that m/k is square-free. Lemma 6. Define the m × m matrix C as follows. If t|s, let   (−1)sw X #p (s/kt) ktw k(tw − 1) − 1 Cst := (−1) (−1) , (s/t)2 k−1 (2.7) k∈I(s/t)  If t does not divide s, set Cst = 0. Then the invariants {nS [dw]} and (−1)dw dw n(d) , for 1 ≤ d ≤ m, are related via   C · [nS [dw]]d = (−1)dw dw n(d) d . 7 Moreover, C has determinant ±1 and is lower triangular. It follows by Cramer’s rule that C integral ⇐⇒ C −1 integral. Proof. We start by writing L = B · L̃ · B −1 , where L̃ij =   1 if j|i,  0 else; Bij = 1 · δij . i3 By Möbius inversion, the inverse of L̃ is given by L̃−1 ij =   (−1)#p (i/j) if j|i and i/j is square-free,  0 else. Moreover, w (AB)ij = (−1)iw 2 · δij , i and
i2 (AB)−1 ij = (−1)iw · δij . w It now follows from formula (2.6) that the matrix C is given by C = AB · L̃−1 · (AB)−1 · R. We calculate that  AB · L̃ −1  = sr   (−1)sw w2 (−1)#p (s/r) s if r|s and s/r is square-free,  0 else; 8 and that (AB) −1
· R rt =   (−1)rw r2 r/t (tw−1)−1 1 w (r/t)2 r/t−1
 0 if t|r, else. If t does not divide s, then there is no integer r such that t|r|s, so that Cst = 0. If, however, t|s, then   r/t (tw − 1) − 1 Cst = (−1) (−1) (−1) r/t − 1   X 1 sw #p (s/r) rw r/t (tw − 1) − 1 = (−1) (−1) (−1) , (s/t)2 r/t − 1 X sw #p (s/r) rw 1 (s/t)2 where the sum runs over all r such that t|r|s and such that s/r is square-free. Set k = r/t, so that, for t dividing s, Cst = (−1) sw 1 (s/t)2 X #p (s/kt) (−1) k∈I(s/t) ktw (−1)   k (tw − 1) − 1 , k−1 finishing the proof. Lemma 6 reduces theorem 2 to proving that the coefficients of the matrix C are integers. We show this in lemmas 9 and 10. 2.2 Integrality of C The following lemma follows directly form the proof of lemma A.1 of [6].1 Lemma 7. (Peng) Let a, b and α be positive integers and denote by p a prime number. If p = 2, assume furthermore that α ≥ 2. Then  α   α−1 
p a−1 p a−1 2α ≡ mod p . pα b − 1 pα−1 b − 1 1 In the combinatorics literature, there are a significant number of results on the divisibility of binomial coefficients by prime powers, see e.g., [7]. 9 In the case that p = 2 and α = 1, we have the following lemma: Lemma 8. Let k ≥ 1 be odd and let a be a positive integer. Then     2ka − 1 a ka − 1 ≡0 + (−1) k−1 2k − 1 mod (4) . Proof. Note that   2ka − 1 2ka − 1 2ka − 2 2ka − 2k + 2 2ka − 2k + 1 = · ··· · 2k − 1 2k − 1 2k − 2 2 1 2ka − 1 ka − 1 ka − k + 1 2ka − 2k + 1 = · ··· · 2k − 1 k − 1 1 1 (ka − 1)(ka − 2) · · · (ka − k + 1) (2ka − 1)(2ka − 3) · · · (2ka − 2k + 1) = · (k − 1)(k − 2) · · · 1 (2k − 1)(2k − 3) · · · 1   ka − 1 (2ka − 1)(2ka − 3) · · · (2ka − 2k + 1) = · , k−1 (2k − 1)(2k − 3) · · · 1 and hence     2ka − 1 a ka − 1 + (−1) 2k − 1 k−1    ka − 1 (2ka − 1)(2ka − 3) · · · (2ka − 2k + 1) a = (−1) + . k−1 (2k − 1)(2k − 3) · · · 1 It thus suffices to show that (2ka − 1)(2ka − 3) · · · (2ka − 2k + 1) ≡ (−1)a+1 (2k − 1)(2k − 3) · · · 1 mod (4) . (2.8) Suppose first that a is even, so that the left-hand-side of (2.8) is congruent to (−1)(−3) · · · (−(2k − 3))(−(2k − 1)) 1 · 3 · · · (2k − 3)(2k − 1) ≡(−1)k ≡ (−1)a+1 mod (4), where the last congruence follows form the fact that k is odd. Suppose now that a is odd. 10 Then the left-hand-side of the expression (2.8) is congruent to 2k(a − 1) + 1 2k(a − 1) + (2k − 1) 2k(a − 1) + (2k − 3)) · ··· 2k − 1 2k − 3 1 ≡1 ≡ (−1)a+1 mod (4) . We return to the proof of theorem 2. If s = t, then Cst = ±1 and is therefore an integer. We assume henceforth that t|s, but t 6= s. Let p be a prime number and α a positive integer. For an integer n, we use the notation pα ||n, to mean that pα |n, but pα+1 - n. In order to show that Cst ∈ Z, we show, for every prime number p, that if s pα || , t then 2α p | X #p (s/kt) (−1) (−1) ktw k∈I(s/t)   k (tw − 1) − 1 . k−1 Fix a prime number p and a positive integer α such that s pα || . t For k ∈ I(s/t) to be square-free, it is necessary that pα−1 |k. This splits into the two cases k = p · l, or k = l, where l ∈ I(s/pt). 11 Regrouping the terms of the expression of (2.7) accordingly yields X X #p (s/kt) (−1) ktw (−1) l∈I(s/pt) k∈{l,pl}   k (tw − 1) − 1 . k−1 Thus, it suffices to prove that for all l ∈ I(s/pt), f (l) := X #p (s/kt) (−1) ktw (−1) k∈{l,pl}   k (tw − 1) − 1 ≡0 k−1
mod p2α , which we proceed in showing. There are two cases: either the sign (−1)ktw in the above sum changes or not. The only case where the sign does not change is when p = 2, α = 1, and both t and w are odd. Lemma 9. Assume that either p 6= 2 or, if p = 2, that α > 1. Then f (l) ≡ 0
mod p2α . Proof. In this situation,     pl(tw − 1) − 1 l(tw − 1) − 1 f (l) = ± − k−1 k−1
≡ 0 mod p2α , by lemma 7. Lemma 10. Assume that p = 2 and that α = 1. Then f (l) ≡ 0 mod (4) . 12 Proof. In this case,     2l(tw − 1) − 1 tw−1 l(tw − 1) − 1 f (l) = ± + (−1) l−1 2l − 1 ≡0 follows from lemma 8. mod (4) , 13 Chapter 3 Mirror geometries to P2 Broadly speaking, mirror symmetry for a Calabi-Yau mirror pair (X, X̌) states that the association X̌ o mirror symmetry /X , exchanges the complex and symplectic geometries of X and X̌. The complex side is called the B-model, while the symplectic side is referred to as the A-model. The classical formulation of mirror symmetry in addition produces a recipe for computing Gromov-Witten (GW) invariants of X in terms of the periods of X̌, schematically: Periods of X̌ o / GW invariants of X. The first part of this chapter consists in reviewing some of the mirror constructions and mirror symmetry statements relating to P2 and KP2 . This will serve to motivate the result on relative mirror symmetry of the next chapter, and the result on the consequence of homological mirror symmetry in the chapter after that. Additionally, in section 3.5 we explore how two families of affine elliptic curves, serving as mirrors for KP2 , are related. Throughout this chapter we emphasize an enumerative perspective. From this viewpoint, local mirror symmetry calculates the genus 0 local Gromov-Witten invariants of P2 , which are the genus 0 Gromov-Witten invariants of KP2 . This terminology is justified by the fact that genus 0 stable maps into KP2 factor through P2 . 14 3.1 Mirror symmetry statements for P2 and KP2 ∂ Denote by θz the logarithmic differential z ∂z . For holomorphic functions f : ∆× → C, consider the differential equation L f = 0, (3.1) where L = θz3 + 3zθz (3θz + 1)(3θz + 2). This is called the A-hypergeometric differential equation associated to P2 . Chiang- Klemm-Yau-Zaslow in [8] show that (the Taylor coefficients of) solutions of this equation, via change of variable and analytic continuation, calculate the local GW invariants of P2 . The solutions to the above equation are expressed as periods of various mirror geometries, which in turn are based on the mirror constructions developed by Batyrev in [9] and [10]. In [8], the authors consider the family of affine elliptic curves  Bφ := Pφ (x, y) := xy − φ(x3 + y 3 + 1) = 0 | x, y ∈ C× , (3.2) for φ ∈ C× and periods of the 1-form ResPφ =0 (log Pφ ) dxdy . xy Using the same family, Takahashi in [11] expresses mirror symmetry with relative periods. These are integrals of the relative cohomology class d xxyd y ∈ H2 (T2 , Bφ ; Z) over relative homology classes in H2 (T2 , Bφ ; Z). Based on [9], Stienstra in [12] proves that periods of the family Mq̌0 :=   q̌ 2 0 = 1 + t1 + t2 + | (t1 , t2 ) ∈ T , t1 t2 (3.3) for q̌ ∈ C× , correspond to solutions of (A.1). Here, the author considers integrals over 15 the cohomology class d xxyd y ∈ H2 (T2 , Mq̌0 ; Z). We show how the two families are related in lemma 11 and proposition 13 below. Hori-Iqbal-Vafa in [13] consider the related family of open 3-folds  Zq̌ :=  q̌ 2 2 xy = Fq̌ (t1 , t2 ) := 1 + t1 + t2 + | (x, y) ∈ C , (t1 , t2 ) ∈ T , t1 t2 (3.4) and integrals of the holomorphic 3-form Res 1 d t1 d t2 d x d y ∈ H3 (Zq̌ , Z). xy − Fq̌ (t1 , t2 ) t1 t2 Hosono in [14] proves that these periods yield solutions to the differential equation (A.1). The mirror to KP2 of this form was constructed by Gross-Siebert in [15], see also [16], by passing through tropical geometry. A construction avoiding tropical geometry and emphasizing semiflat coordinates was elaborated by Chan-Lau-Leung in [17]. Geometrically, Mq̌0 is the fixed locus of the C× -action on Zq̌ given by λ · (x, y, t1 , t2 ) = (λx, λ−1 y, t1 , t2 ). By instead taking the GIT-quotient of the same action, one gets a correspondence of the periods of each family. This was described by Gross in [18] and thoroughly developed by Konishi-Minabe in [19]. The latter authors describe the variation of mixed Hodge structures on Mq̌0 and Zq̌ , yielding a natural language for the local B -model. Denote by φ a third root of q̌. After the coordinate change ti 7→ φ ti , the family Mq̌0 is described as     1 × 0 = 1 + φ t1 + t2 + | t1 , t2 ∈ C . t1 t2 16 Setting W0 (t1 , t2 ) := t1 + t2 + 1 , t1 t2 we can rewrite Mq̌0 as the family parametrized by the fibers W0 (t1 , t2 ) = −1/φ, of the superpotential W0 : T2 −→ C. (3.5) A Landau-Ginzburg model consists of a complex manifold and a holomorphic function on it, called the superpotential. The present Landau-Ginzburg model, namely (T2 , W0 ), is the mirror to P2 constructed by Givental in [20]. In terms of periods, mirror symmetry in this setting sets up a correspondence between oscillatory integrals on (T2 , W0 ) and GromovWitten invariants of P2 . Implementing the SYZ conjecture, the Gross-Siebert program relates both sides of this correspondence to tropical disk counts of tropical P2 , see, e.g., [21]. Mirror symmetry for P2 also states that the quantum cohomology ring Q H∗ (P2 ) of P2 is isomorphic to the Jacobian ring Jac(W0 ) of W0 . In terms of the homological mirror symmetry (HMS) conjecture introduced by Kontsevich at the ICM in Zürich (cf. [22]), this is categorified as follows. Define M by compactifying the fibers of W0 , i.e.,  M = t21 t2 + t1 t22 + t33 − st1 t2 t3 ⊂ P2C[s] , and denote by W the extension of W0 to M . The generic fiber of W is an elliptic curve (mirror to an elliptic curve in P2 ) and the three singular fibers are simple nodal curves of arithmetic genus 1. Homological mirror symmetry then predicts a set of correspondences (M, W ) ↔ P2 , Complex geometry of Crit(W ) ↔ Symplectic geometry of P2 , Symplectic geometry of Crit(W ) ↔ Complex geometry of P2 , 17 where Crit(W ) denotes the critical locus of W .1 On one side, the symplectic geometry of Crit(W ) is described by the derived category of Lagrangian vanishing-cycles D Lagvc (W ), while its complex geometry is encoded in its category of matrix factorizations MF(W ). On the other side, the symplectic geometry of P2 is described by its Fukaya category F(P2 ), whereas its derived category of coherent sheaves D Coh(P2 ) encodes its complex geometry. Kontsevich’s HMS conjecture states that these categories ought to be equivalent:2 MF(W ) ≃ F(P2 ), D Lagvc (W ) ≃ D Coh(P2 ). The second correspondence was proven by Auroux-Katzarkov-Orlov in [23], building on work by Seidel of [24].3 A proof of the first correspondence was announced by Abouzaid-FukayaOh-Ohta-Ono. The data of Gromov-Witten invariants is encoded by these categories as well, albeit too abstractly to allow for calculations: Q H∗ (P2 ) is the Hochschild cohomology of F(P2 ); and Jac(W ) = Jac(W0 ) is the Hochschild cohomology of MF(W ). 3.2 The SYZ conjecture The discussion in the previous section elaborates on an algebraic viewpoint of mirror symmetry. That is, we introduced different varieties that function as mirrors, i.e., their periods calculate Gromov-Witten invariants of the A-model. These statements do not explain why mirror symmetry holds. Enters the Strominger-Yau-Zaslow (SYZ) conjecture, see [26], which provides a geometric explanation of mirror symmetry in terms of dualizing Lagrangian torus fibrations. This conjecture is not expected to hold in full generality, but adaptations of it have led to a geometric understanding of mirror symmetry. The mirror constructions for P2 and KP2 are schematically summarized as in the diagram below. As usual, denote by D 1 Concerning the second equivalence, if Crit(W ) is not symplectic, the statement is adapted. Kontsevich originally formulated the conjecture for pairs of mirror Calabi-Yau manifolds. The statements here are an adaptation to Fano varieties. 3 See also the paper [25] by Auroux-Katzarkov-Orlov. 2 18 a smooth effective anti-canonical divisor on P2 , i.e., an elliptic curve, and denote by T the toric divisor on P2 , namely the union of the coordinate axes. (M, OW ) o Homological Compactification of fibers (T2 , W ) o O Smoothing of divisor / (P2 , T ) O Gross-Siebert program C× -fixed locus Zq̌ o / (P2 , D) O Mirror Symmetry 0-section and C× -fixed locus Local Mirror Symmetry / K 2. P The geometric construction of the family Zq̌ was carried out by Gross-Siebert using tropical geometry in [16]. The Gross-Siebert program aims at explaining mirror symmetry in terms of the SYZ-conjecture by passing through tropical geometry, cf. [15], [21] for P2 and the book [27]. Chan-Lau-Leung in [17] give an alternative construction of the family Zq̌ without passing through tropical geometry: The SYZ conjecture describes how to construct a mirror for an open subvariety V of KP2 . In order to get a mirror to the entire space, that mirror needs to be deformed by quantum/instanton corrections. These are encoded by open GromovWitten invariants that arise in the Lagrangian torus fibration constructed by Gross for V , see [18]. 3.3 Batyrev’s construction In this section, we describe in more detail Batyrev’s mirror constructions relating to P2 , following the exposition of Konishi-Minabe in [19]. The families Mq̌0 of A.3 and Zq̌ of A.4 are affine open subsets of the families we introduce here. Start with  Lreg = (a0 , a1 , a2 , a3 ) ∈ C4 | a1 a2 a3 6= 0,  a30 + 27 6= 0 , a1 a2 a3 19 and consider the T3 -action (λ0 , λ1 , λ2 ) · (a0 , a1 , a2 , a3 ) = λ0 (a0 , λ1 a1 , λ2 a2 , 1 a3 ), λ1 λ2 as well as the character χ : T3 → C× , (λi ) 7→ λ30 . Denote the associated GIT-quotient by MC . All points of MC are stable and MC is identified to P1 \{−1/27} via MC → P 1 , (ai ) 7→ [a30 , a1 a2 a3 ]. For a = (ai ) ∈ MC , define the Laurent polynomial Fa (t1 , t2 ) = a0 + a1 t1 + a2 t2 + a3 . t1 t2 Denote by Ma0 → MC , the family of affine 3-folds whose fibers are given by  Fa (t1 , t2 ) + xy = 0 | (t1 , t2 ) ∈ T2 , (x, y) ∈ C2 ; and by Za → MC , 20 the family of affine curves whose fibers are given by  Fa (t1 , t2 ) = 0 | (t1 , t2 ) ∈ T2 . Consider the affine open {a0 6= 0} ⊂ MC with coordinate q̌ := a1 a2 a3 , a30 so that q̌ parametrizes C× \ {−1/27}. This will yield the families Mq̌0 and Zq̌ with the singular fiber removed. Consider the change of variable ti 7→ a0 ti , ai to get   q̌ Fa (t1 , t2 ) = a0 1 + t1 + t2 + . t1 t2 If a0 6= 0, then Mq̌0 → C× \ {−1/27} , is given by   q̌ 2 0 = 1 + t1 + t2 + | (t1 , t2 ) ∈ T , t1 t2 as in (A.3). On the other hand, by dividing, e.g., y by a0 , the family Zq̌ → C× \ {−1/27} , is described as   q̌ 2 2 | (x, y) ∈ C , (t1 , t2 ) ∈ T , xy = 1 + t1 + t2 + t1 t2 which agrees with (A.4). Remark. Two points are in order: 1. Described this way, the GIT-quotient MC may seem arbitrary. In fact, the authors 21 in [19] start by considering the respective families over L and then consider the GIT quotient induced by the action Fa (t1 , t2 ) 7→ λ0 Fa (λ1 t1 , λ2 t2 ), (λ1 , λ2 , λ3 ) ∈ T3 . The resulting families are the same as the ones described above. 2. Also, as detailed in [19], the construction readily generalizes to other toric Del Pezzo surfaces. 3.4 Local mirror symmetry for P2 We overview the mirror construction to KP2 given by Chan-Lau-Leung in [17]. The authors consider as complex moduli of Zq̌ the punctured unit open disk ∆× with complex parameter q̌. The Kähler moduli of KP2 is isomorphic to ∆× and we denote by q ∈ ∆× the Kähler parameter. Denote by c(q) a certain generating series of open Gromov-Witten invariants. The instanton-corrected4 mirror to KP2 is any one member of the family parametrized by q of non-compact Calabi-Yau varieties  xy = 1 + t1 + t2 +  q 2 2 | (x, y) ∈ C , (t1 , t2 ) ∈ T . c(q)3 t1 t2 The complex parameter q ∈ ∆× parametrizes the symplectic structure of KP2 . The authors provide conjectural evidence that the map q 7→ q̌ := 4 q , c(q)3 The SYZ conjecture describes how to construct a mirror for an open subvariety V of KP2 . In order to get a mirror to the entire space, that mirror needs to be deformed by ’quantum/instanton corrections’. These are encoded by open Gromov-Witten invariants that arise in the Lagrangian fibration predicted by SYZ for V . (More precisely, a slightly different fibration needs to be considered.) 22 provides an isomorphism between the Kähler moduli of KP2 and the complex moduli of Zq̌ , thus being inverse to the mirror map q̌ 7→ exp(−I2 (q̌)), where I2 (q̌) is defined in the next chapter to be the logarithmic solution to the differential equation (A.1). 3.5 Relating two families by coordinate change We end this chapter by describing how the families of (A.2) and (A.3) are related. Recall that, for q̌, φ ∈ C× , these two families were given by  Bφ = 0 = xy − φ(x3 + y 3 + 1) | (x, y) ∈ T2 , and Mq̌0 =   q̌ 2 | (t1 , t2 ) ∈ T . 0 = 1 + t1 + t2 + t1 t2 Consider the following embeddings into projective space: • Denote by Nφ ⊂ P2 × C× the family given by XY Z − φ(X 3 + Y 3 + Z 3 ) = 0. • Then Bφ is the affine variety Nφ − {XY Z = 0}, so that Bφ ⊂ T2 × C× , Bφ : xy − φ(x3 + y 3 + 1) = 0, , y = YZ . with affine coordinates x = X Z (3.6) 23 • On the other hand, denote by Mq̌ ⊂ P2 × C the family given by T12 T2 + T1 T22 + T33 − q̌T1 T2 T3 = 0. • Then Mq̌0 is the affine variety Mq̌ − {T1 T2 T3 = 0}. Choosing affine coordinates t1 = TT31 , t2 = TT23 , we get Mq̌0 ⊂ T2 × C× , Mt0 : t21 t2 + t1 t22 + 1 − q̌t1 t2 = 0. Comparing the Weierstrass forms of each families, we obtain the following result. Lemma 11. For φ ∈ C× , φ3 6= 1/27; and for q̌ ∈ C× , q̌ 3 6= 27, the families Nφ3 and Mq̌3 , parametrized by φ3 and q̌ 3 , have isomorphic fibers via the identification q̌ 3 = −1 φ3 − 1/27 . More precisely, the isomorphism of the fibers is given by the projective change of variable   √ √  √  X = 2 3 i T1 + 3 + 3 i T2 − 1 + 3 i q̌ T3 ,   √ √  √  Y = X̄ = −2 3 i T1 + 3 − 3 i T2 + −1 + 3 i q̌ T3 , Z = 6 φ q̌ T3 ; 24 or, alternatively, by the projective change of variable   √  √  T1 = −3 1 + 3 i φ q̌ X + 3 −1 + 3 i φ q̌ Y + 2 q̌ Z, T2 = 6 φ q̌ (X + Y ) + 2 q̌ Z, T3 = 6 Z. Proof. By direct verification. We remove the fibers above φ3 = 1/27 and q̌ 3 = 27, so that Mq̌ and Nφ have isomorphic fibers. Moreover, lemma 11 gives an explicit isomorphism between the families. Consider now the family Nφ to be embedded into P2 × C× , via the map Mq̌ → Nφ ⊂ P2 × C× . We proceed to removing the divisor {XY Z = 0} , from both families. On Nφ , we get the above family Bφ with affine coordinates X , Z Y y= . Z x= Denote by Aq̌ ⊂ Mq̌ the resulting family. Since Z is never zero on Aq̌ , T3 is never zero either and we we can use affine coordinates T1 , T3 T2 t2 = . T3 t1 = Lemma 12. For q̌ ∈ C× , q̌ 3 6= 27 and affine coordinates t1 and t2 , the fiber Aq̌ is obtained 25 by removing the union of two lines given by 3(t1 + t2 )2 − 3(t1 + q̌)(t2 + q̌) + 4q̌ 2 . This corresponds to the union of the coordinate axes xy = 0 on a fiber of Bφ such that −1 q̌ 3 = φ3 −1/27 . Denote by ζ a primitive third root of unity and consider the affine coordinates x, y on 2 ×  Bφ ⊂ T × C \  1 k ζ | k = 0, 1, 2 , 27 as well as t1 , t2 on  Mq̌0 ⊂ T2 × C× \ 27 ζ k | k = 0, 1, 2 . For Γφ a 2-cycle in T2 with its boundary supported in Bφ , Takahashi in [11] considers the relative periods dx ∧ dy , xy Γφ Z I(φ) := and shows that I(φ3 ) satisfies the A-hypergeometric differential equation (A.1) associated to P2 . It follows that I(φ3 ) calculates the local Gromov-Witten invariants of P2 . Using lemma 11 to move the above integral to the family Mq̌0 , we get the following result: Proposition 13. Denote by ∆q̌ a 2-cycle in T2 with boundary supported in Mq̌0 . Consider the relative periods √ 3 3 i d t1 d t2 Ψ(q̌) := . 2 2 ∆q̌ 3(t1 + t2 ) − 3(t1 + q̌)(t2 + q̌) + 4q̌ Z −1 Then the functions Ψ( φ3 −1/27 ), for φ ∈ C× , φ3 6= 0, and for various 2-cycles ∆q̌ , correspond to solutions of A.1. 26 Conclusion Starting from the family Mq̌ , we can either remove the divisor {T1 T2 T3 = 0} and consider relative periods of d tt11 td2t2 and get the result on relative periods from [12]. Or we can remove the divisor 
3(T1 + T2 )2 − 3(T1 + q̌)(T2 + q̌) + 4q̌ 2 T3 = 0 , and consider the above relative periods to get the same result. 27 Chapter 4 Relative mirror symmetry The goal of this chapter is to introduce relative mirror symmetry. We start by stating the theorem and then verify it separately for all toric Del Pezzo surfaces. In the appendix, we describe the Gromov-Witten invariant calculations and the solutions to the system of A-hypergeometric differential equation more explicitely in the cases of P2 , P1 × P1 and P2 blown up in one point. Recall the notation ∆× for the closed punctured unit disk. Definition. For a Landau-Ginzburg model (T2 , W ) and x ∈ C× , let Γ = Γ(x) be a coherent choice of relative 2-cycles of the fibers of W , i.e., Γ(x) ∈ H2 (T2 , W −1 (x); Z). Consider moreover the integrals fΓW (x) := Z ω0 , Γ(x) where ω0 = h d t1 d t2 t1 t2 i ∈ H2 (T2 , W −1 (x); Z). Then the relative periods associated to W and Γ are defined, for x ∈ ∆× , by IΓW (x) := fΓW (1/x). Theorem 14. Let S be a toric Del Pezzo surface and let D be a smooth effective anticanonical divisor on it. Denote by (T2 , W S ) its mirror Landau-Ginzburg model and by r the dimension of the Kähler moduli of S. Then there is an (r − 1)-dimensional family WφS r−1 of deformations of the superpotential W S , parametrized by φ ∈ (∆× ) , with the following 28 property. For coherent choices of relative 2-cycles Γφ = Γφ (x) ∈ H2 (T2 , (WφS )−1 (x); Z), consider the relative periods WS IΓφφ (x), where (x, φ) ∈ ∆× × (∆× )r−1 . Then these periods, via change of variable and analytic continuation, calculate the genus 0 relative Gromov-Witten invariants of maximal tangency of (S, D). Remark. Denote by KS the total space of the canonical bundle on S. Theorem 1 yields a correspondence between the genus 0 Gromov-Witten invariants of KS and the relative Gromov-Witten invariants of the conjecture. The solutions to the A-hypergeometric system of differential equations associated to S calculate the genus 0 Gromov-Witten invariants of KS . Therefore, it is enough to show that, via change of variable and analytic continuation, the above relative periods yield the solutions to the A-hypergeometric system of differential equations. In the next section, we review some aspects of local mirror symmetry for toric Del Pezzo surfaces. In the sections thereafter, we prove theorem 14 separately for each toric Del Pezzo surface. Denote by Si , for i = 1, 2 and 3 the toric Del Pezzo surface obtained by blowing up P2 in i general points. 4.1 Reflexive polytopes and local mirror symmetry We start by reviewing the mirror constructions by Batyrev of [10], as well as parts of the local mirror symmetry calculations of Chiang-Klemm-Yau-Zaslow in [8]. For a toric Del Pezzo surface S, denote by ∆S the 2-dimensional integral reflexive polytope such that S = P∆ S . 29 Then these polytopes are as follows. Note that the number of 2-simplices equals the number of independent solutions to the associated system of A-hypergeometric differential equations. P2 P1 × P1 S2 S3 S1 Starting from ∆S , Batyrev in [10] describes how to construct a family of affine elliptic curves whose mirror family is induced by the dual ∆∗S . For the periods of the family induced by ∆S , Chiang-Klemm-Yau-Zaslow in [8] derive associated systems of Picard-Fuchs equations. The authors moreover show how the solutions to these equations yield, via change of variable and analytic continuation, the genus 0 Gromov-Witten invariants of KS . We proceed to describing how these families of affine elliptic curves are constructed. Denote by m the number of vertices of ∆S . We label the vertices counterclockwise v1 , . . . , vr , starting with the vertex at (1, 0). Let moreover v0 = (0, 0) and set v i = (1, vi ), i = 0, . . . , m. Denote by r the dimension of the Kähler moduli of S. For j = 1, . . . , r, consider an integral  j basis of linear relations lj = (l0j , . . . , lm ) among the v i . That is, such that r X lij v i = 0. i=0 30 In addition, the lj are required to span the Mori cone of P∆S . This condition uniquely determines the lj for all S except for S3 . The authors obtain the following relations: 1. For P2 : l1 = (−3, 1, 1, 1). 2. For P1 × P1 : l1 = (−2, 1, 0, 1, 0), l2 = (−2, 0, 1, 0, 1). 3. For S1 : l1 = (−2, 1, 0, 1, 0), l2 = (−1, 0, 1, −1, 1). 4. For S2 : l1 = (−1, 1, −1, 1, 0, 0), l2 = (−1, 0, 1, −1, 1, 0), l3 = (−1, −1, 1, 0, 0, 1). 5. For S3 we use the following choice: l1 = (−1, 1, −1, 1, 0, 0, 0), l2 = (−1, 0, 1, −1, 1, 0, 0), l3 = (−1, −1, 2, −1, 0, 1, 0), l4 = (−1, −1, 1, 0, 0, 0, 1). The authors furthermore describe how to obtain from this data the system of A-hypergeometric differential equations associated to S. We do not recall it, as it is not relevant here. Nevertheless: Theorem 15. (Chiang-Klemm-Yau-Zaslow in [8]) The solutions to the A-hypergeometric system of differential equations associated to S compute, via change of variable and analytic continuation, the genus 0 Gromov-Witten invariants of KS . Following [19] and [10], let L(∆S ) be the space of Laurent polynomials with Newton polytope ∆S . Write vi = (vi1 , vi2 ). Then Fa ∈ L(∆S ) if Fa (t1 , t2 ) = m X i=0 v1 v2 ai t1i t2i , 31 where (t1 , t2 ) ∈ T2 and a = (a0 , . . . , ar ). In the case of P2 for instance, this yields Fa (t1 , t2 ) = a0 + a1 t1 + a2 t2 + a3 . t1 t2 Moreover, Fa is defined to be ∆S -regular, written Fa ∈ Lreg (∆S ), if in addition the following condition is satisfied. For any 0 < m ≤ 2 and any m-dimensional face ∆0 ⊂ ∆S , the equations 0 Fa∆ = 0, 0 ∂Fa∆ = 0, ∂t1 0 ∂Fa∆ = 0, ∂t2 have no common solutions. Here, 0 Fa∆ := X v1 v2 ai t1i t2i . vi ∈∆0 Denote by ZS → Lreg (∆S ) the family of affine elliptic curves with fibers given by {Fa (t1 , t2 ) = 0} ⊂ T2 . Consider moreover the T3 -action, Fa (t1 , t2 ) 7→ λ0 Fa (λ1 t1 , λ2 t2 ), where (λ0 , λ1 , λ2 ) ∈ T3 , as well as the GIT-quotient ZS /T3 → M(∆S ) := Lreg (∆S )/T3 . 32 Then each of lj determines a T3 -invariant complex structure coordinate zj := r Y lj aii , i=0 for S. This yields: 1. For P2 : z= a1 a2 a3 . a30 2. For P1 × P1 : z1 = a1 a3 a2 a4 , z2 = 2 . 2 a0 a0 z1 = a1 a3 a2 a4 , z2 = . 2 a0 a0 a3 3. For S1 : 4. For S2 : z1 = a1 a3 a2 a4 a2 a5 , z2 = , z3 = . a0 a2 a0 a3 a0 a1 5. For S3 : z1 = a1 a3 a2 a4 a2 a5 a2 a6 , z2 = , z3 = 2 , z4 = . a0 a2 a0 a3 a0 a1 a3 a0 a1 Requiring that z ∈ (∆× )r , for z = (z1 , . . . , zr ), determines a subset of M(∆S ). Denote by MzS → (∆× )r the restriction to this subset. This yields the following list.   z 0 = 1 + t1 + t2 + , t1 t2   z1 z2 P1 ×P1 Mz = 0 = 1 + t1 + t2 + + , t1 t2     1 z2 S1 Mz = 0 = 1 + t1 + z1 t2 + + , t1 t1 t2   z2 z3 S2 Mz = 0 = 1 + z1 (t1 + t1 t2 + t2 ) + + , t1 t2   z2 z3 z4 S3 Mz = 0 = 1 + z1 (t1 + t1 t2 + t2 ) + + + . t1 t1 t2 t2 2 MzP = 33 We will need the following result by Stienstra. The theorem is initially stated for the family Z, but it translates readily to the family MzS . Note that the result applies more generally. Theorem 16. (Stienstra, in [12], see also [19]) Let S be a toric Del Pezzo surface S and i h consider the relative cohomology class ω0 = d tt11 td2t2 ∈ H2 (T2 , MzS ; Z). For coherent choices of relative 2-cycles Γz ∈ H2 (T2 , MzS ; Z), the period integrals PΓS (z) := Z ω0 , Γz are in bijection with the solutions to the A-hypergeometric system of differential equations associated to S. Combining this with theorem 1 readily yields: Corollary 17. The period integrals PΓS , via change of variable and analytic continuation, compute the genus 0 relative Gromov-Witten invariants of (S, D), where D is a smooth effective anti-canonical divisor on S. 4.2 Proof of relative mirror symmetry Denote by (T2 , W S ) the Landau-Ginzburg model mirror to the toric Del Pezzo surface S. Various non-zero values can be taken for the coefficients, so we set them all to 1. Then: 1 , t1 t2 1 1 1 1 W P ×P (t1 , t2 ) = 1 + t1 + t2 + + , t1 t2 1 1 W S1 (t1 , t2 ) = 1 + t1 + t2 + + , t1 t1 t2 1 1 W S2 (t1 , t2 ) = 1 + t1 + t1 t2 + t2 + + , t1 t2 1 1 1 W S3 (t1 , t2 ) = 1 + t1 + t1 t2 + t2 + + + . t1 t1 t2 t2 2 W P (t1 , t2 ) = 1 + t1 + t2 + 34 In light of corollary 17, it suffices to show that the relative periods of (T2 , W S ), via change of variable and analytic continuation, are in bijection with the periods PΓS . We prove theorem 14 for each surface separately. Proof for P2 Denote by x a third root of the complex parameter z. The family 2 MzP =   z , 0 = 1 + t1 + t2 + t1 t2 after the coordinate change ti 7→ x ti , is described as the fibers 2 W P (t1 , t2 ) = −1/x. Note that this coordinate change does not change ω0 . Then P2 2 PΓP (z) = IΓW (−x3 ), P2 that is, the periods IΓW (x), after the change of variables z = −x3 , are in bijection with the solutions to the A-hypergeometric differential equation associated to P2 . This proves theorem 14 for P2 . Proof for P1 × P1 Denote by z = (z1 , z2 ) ∈ (∆× )2 the complex parameter for P1 × P1 . For i = 1, 2 and for a choice of square-roots φi of zi , consider the coordinate change ti φi ti . 35 1 1 Then the family MzP ×P is described as     φ2 1 1 t2 + . 0 = 1 + φ1 t1 + + t1 φ1 t2 Define, for ψ ∈ C× , 1 1 1 WψP ×P (t1 , t2 ) := t1 + + ψ t1 1   1 t2 + , t2 1 so that MzP ×P is given by n o 1 ×P1 0 = 1 + φ1 WφP2 /φ (t , t ) . 1 2 1 For a coherent choice of relative 2-cycles Γψ (z) ∈ H2 (T2 , Wψ−1 (z); Z), relative periods are 1 W P ×P IΓ ψ 1 (−, ψ) : ∆× → C, Z z 7→ ωo . Γψ (1/z) Then 1 1 W P ×P 1 PΓP ×P (z) = IΓ ψ Therefore the relative periods IΓW 1 P1 ×P1 1
−φ21 , (φ2 /φ1 )2 . , via change of variable and analytic continuation, 1 correspond to the periods PΓP ×P , proving theorem 14 for P1 × P1 . Proof for S1 Denote by z = (z1 , z2 ) ∈ ∆× × ∆× the complex parameter for S1 . By choosing a square root φ1 of z1 and by considering the change of coordinates t1 φ1 t1 , t2 1 t2 , φ1 36 the family MzS1 is described by    z2 1 1 . 0 = 1 + φ1 t1 + t2 + + t1 φ1 t1 t2 Equivalently, it is given by the fibers of WzS21/φ1 (t1 , t2 ) = − 1 , φ1 where WψS1 (t1 , t2 ) := t1 + t2 + 1 ψ + . t1 t1 t2 Let Γψ (z) ∈ H2 (T2 , Wψ−1 (z); Z) be a coherent choice of relative 2-cycles. Relative periods for S1 are W S1 IΓ ψ (−, ψ) : ∆× → C, Z z 7→ ωo . Γψ (1/z) Note that W S1 PΓS1 (z) = IΓ ψ
−φ21 , z2 /φ21 , W S1 so that, via change of variable and analytic continuation, the relative periods IΓ ψ correspond to the solutions to the system of A-hypergeometric differential equations associated to S1 , yielding theorem 14 for S1 . Proof for S2 The family MzS2 , with complex parameter z = (z1 , z2 , z3 ), is given by   z2 z3 , 0 = 1 + z1 (t1 + t1 t2 + t2 ) + + t1 t2 37 which in turn is described as    z3 1 z2 1 + . 0 = 1 + z1 (t1 + t1 t2 + t2 ) + z1 t1 z1 t2 Setting S2 (t1 , t2 ) := t1 + t1 t2 + t2 + W(ψ 1 ,ψ2 ) ψ1 ψ2 + , t1 t2 it is given by the fibers of W(zS22 /z1 ,z3 /z1 ) (t1 , t2 ) = − 1 . z1 Hence W S2 PΓS2 (z) = IΓ (ψ1 ,ψ2 ) (−z1 , z2 /z1 , z3 /z1 ) , S W(ψ2 ,ψ ) and the periods IΓ 1 2 have the desired property. Proof for S3 Finally, denote by z = (z1 , z2 , z2 , z4 ) the complex parameter for the family MzS3 given by   z2 z3 z4 0 = 1 + z1 (t1 + t1 t2 + t2 ) + + + , t1 t1 t2 t2 or, equivalently by    z2 1 z3 1 z4 1 0 = 1 + z1 (t1 + t1 t2 + t2 ) + + + . z1 t1 z1 t1 t2 z1 t2 Now, S3 W(ψ (t1 , t2 ) := t1 + t1 t2 + t2 + 1 ,ψ2 ,ψ3 ) ψ1 ψ2 ψ3 + + , t1 t1 t2 t2 yields the description of MzS3 as the fibers of W(zS32 /z1 ,z3 /z1 ,z4 /z1 ) (t1 , t2 ) = − 1 . z1 38 Therefore S W 2 PΓS2 (z) = IΓ (ψ1 ,ψ2 ,ψ3 ,ψ4 ) (−z1 , z2 /z1 , z3 /z1 , z4 /z1 ) , which yields relative mirror symmetry for S3 , and finishes the proof of theorem 14. 39 Chapter 5 A prediction of homological mirror symmetry In this chapter, we are interested in proving a prediction of the homological mirror symmetry conjecture for the open complement, yielding evidence for this conjecture in that setting. We perform the B -model side calculation for every Del Pezzo surface. The calculation of the Amodel side in the case of P2 , which matches our calculation, was done by Nguyen-Pomerleano and will be published in a forthcoming paper. Set up Denote by S a Del Pezzo surface and by D a smooth effective anti-canonical divisor on it, i.e., in the present setting, an elliptic curve. We additionally denote by Sk the Del Pezzo surface obtained by blowing up P2 in 0 ≤ k ≤ 8 generic points. Note that every Del Pezzo surface but P1 × P1 is obtained as such. If k ≤ 3, then Sk is toric. Auroux-Katzarkov-Orlov in [25] construct the Landau-Ginzburg model Wk : Mk → A1 mirror to Sk . For S toric, we considered in the preceding chapter the Landau-Ginzburg model mirror to (S, T ), where T is the toric divisor. In this chapter though, we consider the mirror Landau-Ginzburg model with respect of the smoothing of T to D. According to the 40 construction of [25], Wk : Mk → A1 is an elliptic fibration with k + 3 nodal fibers. In the case of P1 × P1 , it is an elliptic fibration with 4 nodal fibers. For toric S, these elliptic fibrations coincide with the fiber-wise compactification the superpotentials considered in the previous chapter. We will proceed with the following abbreviations. Notation. Denote by W : M → A1 the Landau-Ginzburg model mirror to the Del Pezzo surface S. Then M is an elliptic fibration with r nodal fibers, where 3 ≤ r ≤ 11. Homological mirror symmetry for S (at least one direction) states that the derived Fukaya category of S should be equivalent to the category of matrix factorizations of (M, W ): MF(M, W ) ≃ F(S). Taking Hochschild cohomology of both categories yields that the Jacobian ring of W is isomorphic to the quantum cohomology of S: Jac(W ) ∼ = Q H∗ (S). We need to understand how this statement is modified when the divisor D is removed on S. The mirror operation consists in removing the superpotential. Thus, conjecturally, M is the mirror to the open complement S − D. The homological mirror symmetry conjecture for S − D states that the wrapped Fukaya category on S − D ought to be equivalent to the bounded derived category of coherent sheaves on M : Db Coh(M ) ≃ WF(S − D). Moreover, taking Hochschild cohomology on both sides yields the prediction that the Hochschild cohomology of M is isomorphic to the symplectic cohomology of S − D: H H∗ (M ) ∼ = S H∗ (S − D). We aim in the present chapter at investigating this statement. 41 The Hochschild cohomology of M Since M is Calabi-Yau, its Hochschild cohomology takes a particularly nice form. Denote by ΩM the cotangent bundle on M . Denote by Hi (OM ), respectively by Hi (ΩM ) the sheaf cohomology groups Hi (M, OM ), respectively Hi (M, ΩM ). Denote moreover by H Hi (M ) the Hochschild cohomology groups of M . Then, H Hi (M ) = 0 when i < 0 or i > 3, and H H0 (M ) = H0 (OM ), H H1 (M ) = H1 (OM ) ⊕ H0 (ΩM ), H H2 (M ) = H0 (OM ) ⊕ H1 (ΩM ), H H3 (M ) = H1 (OM ). Indeed, as vector spaces, we have the equality H H∗ (M ) = ⊕2p=0 R Γ(M, ∧p ΩM ). This is the direct sum of the cohomology groups of the sheaves OM , ΩM and ∧2 ΩM . Moreover, M is Calabi-Yau and thus ωM = ∧2 ΩM = OM . In degree i, H Hi (M ) = ⊕q−p=i Hq (M, ∧p ΩM ). For q > 1, the groups Hq (OM ) and Hq (OM ) are zero as M is a fibration over A1 of relative dimension 1. It follows that H H−2 (M ) = H0 (OM ), H H−1 (M ) = H1 (OM ) ⊕ H0 (ΩM ), H H0 (M ) = H0 (OM ) ⊕ H1 (ΩM ), H H1 (M ) = H1 (OM ), H Hi (M ) = 0 for i > 1 or i < −2. 42 Note that the ring structure of H H∗ (M ) is not apparent in this description. Moreover, since M is Calabi-Yau and of dimension 2, it follows that, after a translation of the degree by 2, H H∗ is isomorphic to H H∗ . Hence the above description. In this chapter we compute the Hochschild cohomology groups H H∗ (M ), as modules over the polynomial ring. Two short exact sequences and identities Denote by ΩM/A1 be the sheaf of relative differentials, forming an exact sequence W ∗ ΩA1 → ΩM → ΩM/A1 → 0. Now, ΩA1 is trivial, so that W ∗ ΩA1 ∼ = OM . Moreover, since M is smooth, the above sequence is exact: 0 → OM → ΩM → ΩM/A1 → 0. (5.1) Denote by ωM/A1 the relative dualizing sheaf, yielding a short exact sequence 0 → ΩM/A1 → ωM/A1 → r Y Opi → 0, (5.2) i=1 where r is the number of nodal fibers and where the pi are the ramification points of W . As mentioned above, ωM ∼ = OM . Moreover, ωM/A1 ∼ = ωM ⊗ W ∗ (ωA∨1 ), so that ωM/A1 is trivial as well. Then, R2 π∗ (OM ) is zero. Thus, by the theorem of cohomology and base change, for all s ∈ A1 , the natural morphism R1 W∗ (OM ) ⊗C κ(s) → H1 (Ms , Os ), is an isomorphism. As all the curves in the family are of arithmetic genus 1 (the family is flat), H1 (Ms , Os ) ∼ = C and therefore R1 W∗ (OM ) ∼ = H1 (M, OM )∼ ∼ = (C[t])∼ . By the same reasoning, H0 (M, OM ) is free of rank 1 over C[t]. Consequently, the non-zero cohomology groups of ωM and ωM/A1 are free of rank 1 over C[t] as well. 43 Cohomology of ΩM/A1 The short exact sequence of (5.2) turns into a long exact sequence 0 → H0 (ΩM/A1 ) → H0 (ωM/A1 ) ∼ = C[t] → Cr → → H1 (ΩM/A1 ) → H1 (ωM/A1 ) ∼ = C[t] → 0. For principal ideal domains, submodules of free modules are free and thus H0 (ΩM/A1 ) ∼ = C[t]. We argue that the sequence 0 → H0 (ΩM/A1 ) ∼ = C[t] → H0 (ωM/A1 ) ∼ = C[t] → Cr → 0 is exact: Elements of H0 (ωM/A1 ) correspond to the global sections of R0 W∗ (ωM/A1 ). As ωM/A1 ∼ = OM , R0 W∗ (ωM/A1 ) ∼ = R1 W∗ (OM ) is locally free of rank 1 as well. It follows that its global sections correspond to functions on A1 . Moreover, the map Y H0 (ωM/A1 ) → H0 ( Opi ), corresponds to evaluating a function at the W (pi )’s. Since a polynomial can be chosen to take on any value on any number of chosen points, this map is surjective. Hence H1 (ΩM/A1 ) ∼ = C[t]. Cohomology of ΩM Recall the short exact sequence of (5.1): 0 → OM → ΩM → ΩM/A1 → 0. Since ΩM is locally free, but not ΩM/A1 , it follows that ΩM is not the trivial extension. Hence the boundary map C[t] ∼ = H0 (ΩM/A1 ) → H1 (OM ) ∼ = C[t], 44 is non-zero. Since it is a map of C[t]-modules, it is therefore injective. It follows that (5.1) induces the long exact sequence 0 0 → H0 (OM ) ∼ − H0 (ΩM/A1 ) ∼ = C[t] → H0 (ΩM ) → = C[t] → → H1 (OM ) ∼ − H1 (ΩM/A1 ) ∼ = C[t] → H1 (ΩM ) → = C[t] → 0, so that H0 (ΩM ) ∼ = C[t]. Now, by Serre duality in families, H1 (ΩM ) ∼ = H0 (Ω∨M ⊗ ωM/A1 )∨ = H0 (Ω∨M )∨ . Finally, as Ω∨M is locally free, H0 (Ω∨M ) has no torsion. Thus the map H1 (OM ) → H1 (ΩM ), is trivial, H1 (ΩM ) → H1 (ΩM/A1 ), is an isomorphism and H1 (ΩM ) ∼ = C[t]. Conclusion We end by assembling the above cohomology groups together. Theorem 18. Let S be a Del Pezzo surface and denote by M its mirror, as constructed in [25]. As C[t]-modules, the Hochschild cohomology groups H H0 (M ) and H H3 (M ) are free of rank 1, H H1 (M ) and H H2 (M ) are free of rank 2, and H Hi (M ) = 0 for i < 0 or i ≥ 4. 45 Appendix A Local mirror symmetry for P2 In [1], Gathmann describes how mirror symmetry calculates the genus 0 relative GW invariants of maximal tangency of P2 . This result, under a somewhat different form, is stated in corollary 20. We start by recalling some notions of the preceding chapters. Denote by θz the ∂ . For holomorphic functions logarithmic differential z ∂z f : ∆× → C, the A-hypergeometric differential equation associated to P2 is L f = 0, (A.1) where L = θz3 + 3zθz (3θz + 1)(3θz + 2). We start by recalling the following two families of affine elliptic curves. For φ ∈ C× , in [8] and [11] is considered the family  Bφ := 0 = xy − φ(x3 + y 3 + 1) | x, y ∈ C× . (A.2) 46 For a ∈ MC = P1 \ {−1/27}, the family of [12] reads as Ma0 :=   a 2 0 = 1 + t1 + t2 + | (t1 , t2 ) ∈ T . t1 t2 (A.3) Finally, recall the family of open 3-folds   a 2 2 Za := xy = Fa (t1 , t2 ) := 1 + t1 + t2 + | (x, y) ∈ C , (t1 , t2 ) ∈ T , t1 t2 (A.4) which was introduced in [13]. Following Konishi-Minabe in [19], we proceed to describe how the periods of the families Za and Ma0 are related. The periods of Za are given by integrals of the relative cohomology class ω0 = dxdy ∈ H2 (T2 , Ma0 ; Z), xy over relative 2-cycles. The periods of Ma0 are given by integrating the 3-form ωa = Res 1 d t1 d t2 d x d y ∈ H3 (Za , Z), xy − Fa (t1 , t2 ) t1 t2 over 3-cycles. Konishi-Minabe in [19] describe an isomorphism of mixed Hodge structures H2 (T2 , Ma0 ; Z) ∼ = H3 (Za , Z), that sends ω0 to ωa . This isomorphism respects the Gauss-Manin connection, which explains why periods of ω0 over relative 2-cycles correspond to periods of ωa over 3-cycles. On the other hand, Gross in [18] describes an isomorphism H2 (T2 , Mq̌0 ; Z) ∼ = H3 (Zq̌ , Z), where q̌ ∈ ∆× . Put together, this yields a correspondence periods of Mq̌0 ←→ periods of Zq̌ . 47 We proceed by describing a basis of solutions to the equation (A.1), following Takahashi in [11]. See also [8] and [28]. The following functions are a constant solution, a logarithmic solution and a doubly logarithmic solution satisfying (A.1) as holomorphic functions (with possibly one or two branch cuts) ∆× → C: I1 (z) = 1, (0) I2 (z) = log z + I2 (z), I3 (z) = ∂ρ2 ω(z; ρ)|ρ=0 = (log z)2 + · · · , where I2 (z) := ∞ X (−1)k (3k)! k=1 ω(z; ρ) := k (k!)3 ∞ X (3ρ)3k (1 + ρ)3k k=0 zk , (−1)k z k+ρ , (α)k := α · (α + 1) · · · (α + k − 1). Local mirror symmetry asserts that the solutions spanned by these functions are given as both periods of Zq̌ and relative periods of Mq̌0 . Translating these to Gromov-Witten invariants is achieved via the change of coordinate q := − exp(I2 (z)). Indeed: Theorem 19. (Chiang-Klemm-Yau-Zaslow, in [8]) Denote by Kd the genus 0 degree d Gromov-Witten invariants of KP2 . Written in the coordinate q and via analytic continuation, ∞ (log(−q))2 X I3 (q) = − 3d Kd q d . 2 d=1 Recall the notation Nd (P2 , D) for the genus 0 degree d relative Gromov-Witten invariants 48 of maximal tangency of (P2 , D), where D is an elliptic curve. Recall also from chapter 2 the notation nP2 [3d] for the associated relative BPS numbers (here w = 3). The two following results follow from theorem 1. The first was described by Gathmann in [1], albeit from a different perspective. Corollary 20. With the same notation as above, ∞ (log(−q))2 X − I3 (q) = (−1)d Nd (P2 , D) q d 2 d=1 calculates the genus 0 relative Gromov-Witten invariants of maximal tangency of (P2 , D). In terms of BPS state counts, this yields: Corollary 21.   ∞ ∞ X 1 k(dw − 1) − 1 dk (log(−q))2 X d (−1) nS [dw] − q . I3 (q) = 2 2 k k − 1 k=1 d=1 49 Appendix B Local mirror symmetry for P1 × P1 and S1 Following the exposition of [8] and [19], we describe a basis of solutions to the A-hypergeometric differential equation associated to P1 × P1 and S1 , which is the blow up of P2 in one point. In both of these cases, the complex moduli is of dimension 2, isomorphic to ∆× × ∆× . Denote by z = (z1 , z2 ) the complex parameter. For i = 1, 2, denote by θi the partial differential operator zi ∂z∂ i . Then the A-hypergeometric differential system associated to P1 × P1 is of order 2. For functions f : ∆× × ∆× → C, it reads L1 f = 0, L2 f = 0, where L1 = θ12 − 2z1 (θ1 + θ2 )(2θ1 + 2θ2 + 1), L2 = θ22 − 2z2 (θ1 + θ2 )(2θ1 + 2θ2 + 1). 50 A basis of solutions consists of I1 (z1 , z2 ) = 1, 1 1 1 1 I2 (z1 , z2 ) = log z1 + H P ×P (z1 , z2 ), I3 (z1 , z2 ) = log z2 + H P ×P (z1 , z2 ), 1 1 I4 (z1 , z2 ) = ∂ρ1 ∂ρ2 ω P ×P (z, ρ)|ρ1 =ρ2 =0 = log z1 log z2 + · · · , where 1 1 X H P ×P (z1 , z2 ) = n1 ,n2 ≥0 (n1 ,n2 )6=(0,0) 1 1 ω P ×P (z, ρ) = (2n1 + 2n2 )! n1 n2 1 z z , n1 + n2 (n1 !)2 (n2 !)2 1 2 (2ρ1 + 2ρ2 )2n1 +2n2 n1 +ρ1 n2 +ρ2 z1 z2 . 2 (ρ + 1)2 (ρ + 1) 1 2 n n 1 2 n ,n ≥0 X 1 2 The case of S1 proceeds analogously. Denote again by z = (z1 , z2 ) ∈ ∆× × ∆× the complex parameter. The A-hypergeometric system of differential equations associated to S1 is L1 f = 0, L2 f = 0, for functions f : ∆× × ∆× → C, where L1 = θ1 (θ1 − θ2 ) − z1 (2θ1 + θ2 )(2θ1 + θ2 + 1), L2 = θ22 + z2 (2θ1 + θ2 )(θ1 − θ2 ). 51 A basis of solutions is given by I1 (z1 , z2 ) = 1, I2 (z1 , z2 ) = log z1 + 2H S1 (z1 , z2 ), I3 (z1 , z2 ) = log z2 + H S1 (z1 , z2 ), 1 I4 (z1 , z2 ) = ( ∂ρ21 + ∂ρ1 ∂ρ2 )ω S1 (z, ρ)|ρ1 =ρ2 =0 , 2 for H S1 (z1 , z2 ) = (2n1 + n2 − 1)! (−1)n2 z1n1 z2n2 , 2 n1 !(n1 − n2 )!(n2 !) n ,n ≥0 X 1 2 n1 ≥n2 ω S1 (z1 , z2 ) = Γ(1 + ρ1 − ρ2 ) (2ρ1 + ρ2 )2n1 +n2 z1n1 +ρ1 z2n2 +ρ2 . 2 (ρ1 + 1)n1 (ρ2 + 1)n2 Γ(1 + ρ1 − ρ2 + n1 − n2 ) n ,n ≥0 X 1 2 52 Appendix C Some Gromov-Witten invariants In this appendix, we provide some Gromov-Witten invariants of Del Pezzo surfaces. We consider local and relative invariants, as well as local and relative BPS state counts. We are not concerned with calculations via geometric tools or mirror symmetry. Rather, emphasis is put on how these numbers are related to each other. Denote by S1 the (degree 8) Hirzebruch surface given by blowing up P2 in one point and by S2 the (degree 7) Del Pezzo surface given by blowing up P2 in two general points. Alternatively, S2 is obtained by blowing up P1 × P1 in one point. Denote by KP2 , KP1 ×P1 , KS1 and KS2 the total spaces of the responding canonical bundles. Most of the computations below are performed with the open-source software Sage. Hu proves the following formula in [29], which we use as a means of checking some of our calculations. Let S be a Del Pezzo surface, and assume that its blowup at one point, p : S̃ → S, is a Del Pezzo surface as well. Denote by β ∈ H2 (S, Z) an effective curve class. Denote moreover by p! (β) := P D(p∗ (P D(β)), its push-forward. Here P D stands for Poincaré dual. Then Hu proves that the Gromov Witten invariant of KS of class β equals the Gromov-Witten invariant of KS̃ of class p! (β). In examples below, pulling back the class of a line in S yields the class of a line in S̃ (away from the exceptional divisor) and the class of the exceptional divisor. We proceed to introducing some notation: • H2 (P2 ) = H2 (KP2 ) ∼ = Z. For either groups, we denote by d ≥ 0 an effective curve class 53 of degree d. • H2 (S1 ) = H2 (KS1 ) ∼ = Z ⊕ Z is generated by the classes pulled back from P2 and by multiples of the exceptional divisor. We use as a basis B, the class of a line away from the exceptional divisor, and F , the fiber class, which is the class of the exceptional divisor. Moreover, we denote by (dB , dF ) ≥ (0, 0) the effective curve class dB ·B +dF ·F of degree dB + dF . In particular, the pullback of a line in P2 is (1, 1). • Denote by l2 and l3 the classes of two lines generating H2 (P1 × P1 ). In this setting, we use the notation (d2 , d3 ) to indicate the class d2 · l2 + d3 · l3 . • Consider S2 as the blow up of P1 × P1 in one point, denote by L2 and L3 the pullbacks of l2 and l3 . Denote by E the class of the exceptional divisor. In accordance with [8], we use the notation (d1 , d2 , d3 ) to mean d1 · E + d2 · L2 + d3 · L3 . In particular, the pullback of a · l1 + b · l2 is (a + b, a, b). • For whichever surface of the above surfaces, D denotes its anti-canonical divisor. Let S, respectively KS , be one of the above varieties and let β ∈ H2 (S, Z) be an effective curve class. Recall the notation Iβ (KS ) for the genus 0 degree β Gromov-Witten invariants of KS , as well as the notation Nβ (S, D) for the genus 0 degree β relative Gromov-Witten invariants of maximal tangency. These invariants are related by theorem 1 as Nβ (S, D) = (−1)β·D (β · D)Iβ (KS ). For the Calabi-Yau threefold KS , instanton numbers or BPS state counts nγ are defined via the equation Kβ = Iβ (KS ) = X nβ/a a|β a3 . 54 Relative invariants of P2 In [30], p. 43, the authors compute the GW invariants for KP2 . They get: d Kd 1 3 2 − 45 8 3 244 9 4 − 12333 64 5 211878 125 6 − 102365 6 7 64639725 343 8 − 1140830253 512 9 6742982701 243 10 − 36001193817 100 Now, Nd (P2 , D) = (−1)3d · 3d · Id (KP2 ), so that: d Nd (P2 , D) 1 −9 2 − 135 4 3 −244 4 − 36999 16 5 − 635634 25 6 −307095 7 − 193919175 49 8 − 3422490759 64 9 − 6742982701 9 10 − 108003581451 10 The instanton numbers of KP2 are calculated in [8] to be: 55 d nd 1 3 2 -6 3 27 4 -192 5 1695 6 -17064 7 188454 8 -2228160 9 27748899 10 360012150 Applying the formulae discussed in chapter 2 yields the following relative BPS state counts: d n[3d] 1 -9 2 -27 3 -234 4 -2232 5 -25380 6 -305829 7 -3957219 8 -53462160 9 -749211021 10 -10800167040 Relative invariants of S1 In [8], the authors calculate the instanton numbers of KS1 to be: 56 n(dB ,dF ) dF 0 1 2 3 4 5 6 -2 0 0 0 0 0 dB 0 1 1 3 5 7 9 11 13 2 0 0 -6 -32 -110 -288 -644 3 0 0 0 27 286 1651 6885 4 0 0 0 0 -192 -3038 -25216 5 0 0 0 0 0 1695 35870 6 0 0 0 0 0 0 -17064 The result by Hu explains why the above diagonal coincides with the instanton numbers of KP2 . We proceed to calculate the Gromov-Witten invariants of KS1 . As an illustration, we perform a few computations by hand. For instance, K(0,1) = n(0,1) = −2, since no class divides (0, 1) other than itself. The same holds for the line corresponding to (1, i). But not so for K(2,2) . Indeed K(2,2) = n(2,2) n(1,1) 3 45 + 3 = −6 + = − . 1 2 8 8 For the same reason as above, K(2,3) = n(2,3) = −32. But not so for (2, 4): K(2,4) = n(2,4) n(1,2) 5 875 + 3 = −110 + = − . 1 2 8 8 Doing these calculation with the help of Sage, we get that: 57 K(dB ,dF ) dF 0 1 2 3 4 5 6 -2 0 0 0 0 0 dB 0 1 1 3 5 7 9 11 13 2 0 0 − 45 8 −32 − 875 8 −288 − 5145 8 3 0 0 0 244 9 286 1651 185900 27 4 0 0 0 0 − 12333 64 −3038 −25220 5 0 0 0 0 0 211878 125 35870 6 0 0 0 0 0 0 − 102365 6 Since (dB , dF ) · D = 3 · (dB + dF ), using theorem 1 yields: N(dB ,dF ) dF 0 1 2 3 4 5 6 6 0 0 0 0 0 dB 0 1 -3 18 -45 84 -135 198 -273 2 0 0 − 135 2 480 − 7875 4 6048 -15435 3 0 0 0 488 -6006 39624 -185900 4 0 0 0 0 − 36999 8 82026 -756600 5 0 0 0 0 0 1271268 25 -1183710 6 0 0 0 0 0 0 -614190 Relative invariants of P1 × P1 As previously mentioned, the authors of [8] calculate the relevant instanton numbers. A very similar calculation as above yields the following relative Gromov-Witten invariants for (P1 × P1 , D): 58 d3 0 1 2 3 4 5 6 6 0 0 0 0 0 d2 0 1 6 -24 54 -96 150 -216 294 2 0 54 -390 1650 − 10395 2 13524 -30744 3 0 -96 1650 − 40832 3 74676 -313728 1089132 4 0 150 − 10395 2 74676 -654435 4139748 − 41424825 2 5 0 -216 13524 -313728 4139748 − 939030024 25 259947930 6 0 294 -30744 1089132 − 41424825 2 259947930 − 7236946916 3 Relative invariants of S2 In [8], the authors compute the following instanton invariants: d1 = 0 d3 0 1 d1 = 1 d2 0 1 d1 = 2 1 1 d3 0 1 0 1 -2 0 1 -2 3 3 0 1 2 1 -4 5 2 5 -6 0 1 2 d3 d2 d2 0 d1 = 3 d3 d1 = 4 d3 0 1 2 3 4 -8 9 -32 135 -110 d2 d2 0 0 1 1 -6 7 2 2 -6 35 -32 3 7 -32 27 3 -8 135 -400 286 4 9 -110 286 -192 59 d1 = 5 d3 0 1 2 3 4 5 -10 11 -110 385 -288 -110 1100 -2592 1651 d2 0 1 2 3 4 -10 385 -2592 5187 -3038 5 11 -288 1651 -3038 1695 The symmetry is due to the fact that we pulled back our classes from P1 ×P1 . Via similar calculations as above, we get for the local GW invariants: d1 = 0 d3 0 1 d1 = 1 d2 0 1 d1 = 2 1 1 d3 0 1 0 1 -2 0 1 -2 3 3 0 1 2 1 -4 5 2 5 − 45 8 3 4 -8 9 − 65 2 135 − 875 8 0 1 2 d3 d2 d2 0 d1 = 3 d3 d1 = 4 d3 0 1 2 d2 d2 0 0 1 1 -6 7 2 2 -6 35 -32 3 7 -32 244 9 3 -8 135 -400 286 4 9 − 875 8 286 − 12333 64 60 d1 = 5 d3 0 1 2 3 4 5 -10 11 -110 385 -288 -110 1100 -2592 1651 d2 0 1 2 3 4 -10 385 -2592 5187 -3038 5 11 -288 1651 -3038 211878 125 Applying the appropriate formula, and noting that the degree of (d1 , d2 , d3 ) is d1 +d2 +d3 , we get the following relative invariants of (S2 , D): d1 = 0 d3 0 1 d1 = 1 d2 1 d3 0 d1 = 2 1 -3 0 1 0 -3 -12 0 1 -12 -27 3 0 1 2 1 -48 -75 2 -75 -108 0 -3 2 d3 d2 d2 0 d1 = 3 d3 d1 = 4 d3 0 1 2 3 4 -192 -243 -780 -3645 − 13125 4 d2 d2 0 0 1 1 -108 -147 2 2 -108 -735 -768 3 -147 -768 -732 3 -192 -3645 -12000 -9438 4 -243 − 13125 4 -9438 − 110997 16 61 d1 = 5 d3 0 1 2 3 4 5 -300 -363 -3300 -12705 -10368 -3300 -36300 -93312 -64389 -127596 d2 0 1 2 3 4 -300 -12705 -93312 -202293 5 -363 -10368 -64389 -127596 − 1906902 25 62 Bibliography [1] A. Gathmann, “Relative Gromov-Witten invariants and the mirror formula,” Math. Ann., vol. 325, pp. 393–412, 2003. [2] R. Gopakumar and C. Vafa, “M-theory and topological strings-I,” “M-theory and topological strings-II,” http://arxiv.org/list/hepth/9809187, 1998. [3] R. Gopakumar and C. Vafa, http://arxiv.org/list/hepth/9812127, 1998. [4] M. Gross, R. Pandharipande, and B. Siebert, “The tropical vertex,” Duke Math. J., vol. 153, no. 2, pp. 297–362, 2010. [5] Y. Manin, “Generating functions in algebraic geometry and sums over trees,” Progr. Math., vol. 129, pp. 401–417, 1995, Birkhäuser Boston, Boston. [6] P. Peng, “A Simple Proof of Gopakumar-Vafa Conjecture for Local Toric Calabi-Yau Manifolds,” Commun. Math. Phys., vol. 276, pp. 551–569, 2007. [7] G. Granville, “Arithmetic Properties of Binomial Coefficients I: Binomial coefficients modulo prime powers,” Canadian Mathematical Society Conference Proceedings, vol. 20, pp. 253–275, 1997. [8] T.-M. Chiang, A. Klemm, S.-T. Yau, and E. Zaslow, “Local mirror symmetry: calculations and interpretations,” Adv. Theor. Math. Phys., vol. 3, no. 3, pp. 495–565, 1999. 63 [9] V. Batyrev, “Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori,” Duke Math. J., vol. 69, no. 2, pp. 349–409, 1993. [10] V. Batyrev, “Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties,” J. Algebraic Geom., vol. 3, pp. 493–535, 1994. [11] N. Takahashi, “Log mirror symmetry and local mirror symmetry,” Comm. Math. Phys., vol. 220, no. 2, pp. 293–299, 2001. [12] J. Stienstra, “Resonant hypergeometric systems and mirror symmetry,” Integrable systems and algebraic geometry, pp. 412–452, (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1997. [13] K. Hori, A. Iqbal, and C. Vafa, “D-branes and mirror symmetry,” 2000, arXiv:hepth/0005247. [14] S. Hosono, “Central charges, symplectic forms, and hypergeometric series in local mirror symmetry,” Mirror symmetry V, pp. 405–439, AMS/IP Stud. Adv. Math., 38, Amer. Math. Soc., Providence, RI, 2006. [15] M. Gross and B. Siebert, “From real affine geometry to complex geometry,” Ann. of Math., vol. 174, no. 3, pp. 1301–1428, 2011. [16] M. Gross and B. Siebert, An invitation to toric degenerations, vol. 16 of Surveys in Differential Geometry: Geometry of special holonomy and related topics, pp. 43-78, International Press, 2011. [17] K. Chan, S.-C. Lau, and N. C. Leung, “SYZ mirror symmetry for toric Calabi-Yau manifolds,” J. Diff. Geom., vol. 90, no. 2, pp. 177–250, 2012. [18] M. Gross, “Examples of special Lagrangian fibrations,” Symplectic geometry and mirror symmetry, pp. 81–109, Seoul, 2000, World Sci. Publ., River Edge, NJ, 2001. [19] Y. Konishi and S. Minabe, “Local B-model and Mixed Hodge Structure,” Adv. Theor. Math. Phys., vol. 14, no. 4, pp. 1089–1145, 2010. 64 [20] A. Givental, “Homological geometry and mirror symmetry,” Proc. International Congress of Mathematicians, vol. 1, 2, pp. 427–480, Zürich, 1994, Birkhäuser, Basel. [21] M. Gross, “Mirror symmetry for P2 and tropical geometry,” 2009, arXiv:0903.1378v2. [22] M. Kontsevich, “Homological algebra of mirror symmetry,” Proc. International Congress of Mathematicians, vol. 1, 2, pp. 120–139, Zürich, 1994, Birkhäuser, Basel. [23] D. Auroux, L. Katzarkov, and D. Orlov, “Mirror symmetry for weighted projective planes and their noncommutative deformations,” Ann. of Math., vol. 167, no. 3, pp. 867–943, 2008. [24] P. Seidel, “More about vanishing cycles and mutation,” Symplectic Geometry and Mirror Symmetry, pp. 429–465, Proc. 4th KIAS International Conference, Seoul (2000), World Sci., Singapore, 2001. [25] D. Auroux, L. Katzarkov, and D. Orlov, “Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves,” Invent. Math., vol. 166, no. 3, pp. 537–582, 2006. [26] A. Strominger, S.T. Yau, and E. Zaslow, “Mirror Symmetry is T-duality,” Nucl. Phys., vol. B479, pp. 243–259, 1996. [27] M. Gross, Tropical Geometry and Mirror Symmetry, Number 114 in CBMS Regional Conference Series in Mathematics. The American Mathematical Society and the Conference Board of the Mathematical Sciencecs, 2011. [28] T. Graber and E. Zaslow, “Open-string Gromov-Witten Invariants: Calculations and a Mirror Theorem,” Contemp. Math., vol. 310, pp. 107–121, 2002, Orbifolds in Mathematics and Physics (Madison, WI, 2001), AMS, Providence, RI, 2002. [29] J. Hu, “Local Gromov-Witten invariants of blowups of Fano surfaces,” J. Geom. Phys., vol. 61, no. 6, pp. 1051–1060, 2011. 65 [30] B. Lian, K. Liu, and S.-T. Yau, “Mirror Principle I,” Asian J. Math., vol. 1, no. 4, pp. 729–763, 1997.