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©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Two-Point Gromov-Witten Formulas for Symplectic Toric Manifolds

A Dissertation Presented

Alexandra Mihaela Popa

The Graduate School

in Partial Fulﬁllment of the

Requirements

for the Degree of

Doctor of Philosophy

Mathematics

Stony Brook University

August 2012 Copyright by Alexandra Mihaela Popa 2012 Stony Brook University The Graduate School

Alexandra Mihaela Popa

We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation.

Aleksey Zinger- Dissertation Advisor Associate Professor, Department of Mathematics

Jason Starr- Chairperson of Defense Associate Professor, Department of Mathematics

Radu Laza Assistant Professor, Department of Mathematics

Martin Rocek Professor, C.N. Yang Institute for Theoretical Physics

This dissertation is accepted by the Graduate School.

Charles Taber Interim Dean of the Graduate School

ii Abstract of the Dissertation

Two-Point Gromov-Witten Formulas for Symplectic Toric Manifolds

Alexandra Mihaela Popa

Doctor of Philosophy

Mathematics

Stony Brook University

2012

We show that the standard generating functions for genus 0 two-point twisted Gromov- Witten invariants arising from concavex vector bundles over symplectic toric manifolds are explicit transforms of the corresponding one-point generating functions. The latter are, in turn, transforms of Givental’s J-function. We obtain closed formulas for them and, in particular, for two-point Gromov-Witten invariants of non-negative toric complete intersections. Such two-point formulas should play a key role in the computation of genus 1 Gromov-Witten invariants (closed, open, and unoriented) of toric complete intersections as they indeed do in the case of the projective complete intersections.

iii Contents

Index of notation v

Acknowledgements vii

1 Introduction 1 1.1 Someresults...... 2 1.2 Outlineofthedissertation ...... 5

2 Overview of symplectic toric manifolds 7 2.1 Definition,charts,andKählerclasses ...... 7 2.2 Cohomology, Kähler cone, and Picard group ...... 18 2.3 Torusactionandequivariantnotation...... 21 2.4 Examples ...... 24

3 Explicit Gromov-Witten formulas 27 3.1 Notation and construction of explicit power series ...... 27 3.2 Statements...... 34

4 Equivariant theorems 38 4.1 Constructionofequivariantpowerseries ...... 39 4.2 Equivariantstatements ...... 43

5 Proofs 48 5.1 Outline...... 48 5.2 Notationforﬁxedpointsandcurves...... 51 5.3 Recursivity, polynomiality, and admissible transforms...... 52 5.4 Torus action on the moduli space of stable maps ...... 56 5.5 RecursivityfortheGWpowerseries ...... 59 5.6 MPCfortheGWpowerseries...... 61 5.7 Recursivity and MPC for the explicit power series ...... 66

Bibliography 72

A Derivation of (5.2) from [LLY3] 75

iv Index of notation

Symbol Description Tk, TN C˚ k, C˚ N e TN -equivariantp q p Eulerq class N 1, 2,...,N r s t u M,MJ k N integer matrix, k J submatrix of M with columns ˆ indexed byˆ|J | N ; Section 2.1 τ Ďr s XM symplectic toric manifold given by a minimal toric pair M,τ ; (2.2), Definition 2.1, Proposition 2.2 p p1 pk p q k k ˚ τ t t1 ...tk if p p1,...,pk Z , t t1,...,tk T / t HTN XM M1 “p MN qP k “p qP NP p q t z t z1,...,t zN if t T , z z1,...,zN C ; (2.1) ¨ p q P “p N NqP α α1,...,αN weights of the standard action of T on C ; Section 2.3 “ p q M0,m X,A the moduli space of stable maps from genus 0 curves p q with m marked points into X respresenting A; [MirSym, Chapter 24] evi :M0,m X,A X the evaluation map at the i-th marked point; p qÝÑ [MirSym, Chapter 24] E E` E´ X concavex vector bundle; Chapter 1 “ ‘ ÝÑ ˚ ` VE` M0,m X,A VE` π˚ev E ; (1.1) ÝÑ p q ” 1 ˚ ´ VE´ M0,m X,A VE´ R π ev E ; (1.1) ÝÑ p q ” ˚ VE M0,m X,A VE VE´ VE` ; (1.1) İ İİ İ ÝÑ p q ” ‘ ˚ ` ˚ ´ VE, VE M0,m X,A see (1.7), Chapter 4; e VE ev e E e VE ev e E , İİ 1 1 ÝÑ p q ˚ p ` q p q“˚ p ´ q p q e VE ev2 e E e VE ev2 e E 2 p q p q“ p q p q ψi H M0,m X,A the first Chern class of the universal cotangent line bundle P p p qq for the i-th marked point 2 ψi HTN M0,m X,A and its equivariant version; Chapters 1, 4 P p p Y qq p1 p2 ψ η1,ψ η2 A two-point GW invariant of Y ; (1.3) τ τ KM the closed Kähler cone of XM ; Proposition 2.16 @ D τ τ Λ d H2 X ; Z : ω, d 0 ω K ; (1.6), Footnote 2 P p M q ě @ P M d R Λ R ring, R Λ@ D adQ : ad (R d Λ , rr ss rr ss” dPΛ P @ P 1 " 1 * Qd Qd Qřd`d ; Chapter 1 R ~ R ring, R ~¨ ”R ~´1 R ~ ; Section 3.1 ” rr ss ` r s A A1,..., Ak formal variables; Section 3.1 ”p V W q V W R A p the space of homogeneous polynomials of degree p in A r s

v with coefficients in the ring R; Section 3.1 k N supp v j : vj 0 if v R (or v C ) τ p q t τ ‰ u P P XM J z XM : supp z J ; Remark 2.10 V pτ q tr sP p qĎ u N τ M set of k-tuples in N indexing the T -fixed points of XM ; (2.4),r (2.13),s Corollary 2.20(a) 1, if j J, τ V τ J z1,...,zN XM , zj P with J M ;(2.13) r s r sP ” #0, otherwise, P ˚ τ ˚ J , :H N X H N restriction induced by the point J ; (2.28) ¨p q ¨ J T p M qÝÑ T r s φJ the equivariant Poincarédual of the point J ; (2.35) ˇ τ τ ´1 p r s k ˇ Lp X Lp X C , where z,c t z,t c , t T ; (2.14) ÝÑ M ” M ˆ {„ p q „ p ¨ q @ P γi Lei ; (2.16) ` ´ τ Li ,Li XM non-negativer and non-trivial line bundle, negative line bundle ÝÑ a b ` ` ´ ´ such that E Li , E Li ; (1.5) ” i“1 ” i“1 ℓ˘ ˚ℓ˘ ˘ ˘ ˚ 1i Àki À ℓri Z Li γ1 ... γk ; (3.5), Proposition 2.17 P “ b b k ˘ 2 τ ˘ ˘ λi HTN XM e Li ℓrixr; (4.4) P p q p q“ r“1 2 τ Uj H XM ; Z c1 L´řMj ; (2.16) P 2p τ q p ˚ q Hi H XM ; Z c1 γi ; (2.16) P 2p τ q p q N uj HTN XM c1 Dj where Dj ET j L´Mj ; (2.27), (2.24) P 2 p τ q pr sq˚ r s” N ˆ xi H N X c1 γ where γ ET triv γi; (2.27), (2.23) P T p M q p i q i ” ˆ τ τ EM J N : Dj , Dj z XM : zj 0 ; (2.18) # Ďr s jPJ “H+ ”tr sP “ u Dj d Ş Uj, d ; (3.3) ˘p q ˘ Li d c1 Li , d ; 3.3 p q N @a p qD pb q ` ´ νE d Dj d @ Li d D Li d ; (3.3) p q j“1 p qí“1 p qì“1 p q τ 1 IJ XM I řJ , the P řpassing throughř I and J ; (5.8) pτ Y q V τ r s r s Ij XM I j if I M and j N I; Section 5.2 p Yt uqV τ P Pr s´ v I, j the element of M so that v I, j I and v I, j I j p q V τ p q‰ p qĂ Yt u whenever I M and j N j ; Section 5.2 f the coefficient of qPd in f wheneverPr s´tf uR Λ ; Section 3.1 q;d P rr ss f A;p the degree p homogeneous part of f R A ; Section 3.1 p p1 pk P rr ss J K d d d ě0 k A ~ q A1 ~ q1 ... Ak ~ qk p p1,...,pk Z ; `J K dq ` dq1 ` dqk @ “p qPp q ! ) ! ) ! Remarks) 3.2, 4.2

vi Ackowledgements

I am indebted to Aleksey Zinger for teaching me localization techniques and for suggesting all the questions that I have worked on so far. I thank him for his thorough comments on and corrections of countless drafts of my papers, in particular this dissertation. He spent a very generous amount of time on my mathematical guidance and shared his Gromov-Witten expertise and talent in many discussions with me; I am very grateful. I thank Melissa Liu for explaining parts of Givental’s work needed for this dissertation, for her answers to my questions on toric manifolds, and for an inspiring discussion about a preliminary stage of the results in this dissertation. I am indebted to Iulic˘aPopa, my teacher, for over twenty years of wonderful lessons of mathematics and guidance. In particular, I thank him for the computations without paper and pen and the discovery-type problems in my childhood, and for his teaching me in high school the proofs of the fundamentals ahead of the time they were taught in class. I thank Professor Starr for his advice, support, and help over the past four years at Stony Brook. I thank both of my parents, my grannie, and my sister for supporting my study of mathematics in so many ways.

vii Chapter 1

Introduction

Torus actions on moduli spaces of stable maps into a smooth projective variety facilitate the computation of equivariant Gromov-Witten invariants [Gi1] via the Localization The- orem [ABo], [GraPa]. Equivariant formulas lead to other interesting consequences beyond the computation of non-equivariant Gromov-Witten invariants. In the case of the projective spaces, two-point equivariant Gromov-Witten formulas in [PoZ] lead to the confirmation of mirror symmetry predictions concerning open and unoriented genus 1 Gromov-Witten invariants in the same paper and to the computation of closed genus 1 Gromov-Witten invariants in [Po]. In this dissertation we obtain equivariant formulas expressing the standard two-point closed genus 0 generating function for certain twisted Gromov-Witten invariants of symplectic toric manifolds in terms of the corresponding one-point generating functions. We also obtain explicit formulas for the latter. In particular, we show that the standard generating function for these two-point invariants is a fairly simple transform of the well-known Givental’s J-function. The formulas obtained in this dissertation compute, in particular, the twisted/un-twisted Gromov-Witten numbers (1.2)/(1.3) below. For a smooth projective variety X and a class A H2 X; Z , M0,m X,A denotes the moduli space of stable maps from genus 0 curves with mP markedp q points intop Xq representing A. Let evi : M0,m X,A X p q ÝÑ be the evaluation map at the i-th marked point; see [MirSym, Chapter 24]. All cohomology groups in this dissertation will be with rational coefficients unless otherwise specified. For 2 each i 1, 2,...,m, let ψi H M0,m X,A be the first Chern class of the universal cotangent“ line bundle for thePi-th markedp p point.qq Let

π : U M0,m X,A ÝÑ p q be the universal curve and ev : U X the natural evaluation map; see [MirSym, Sec- tion 24.3]. ÝÑ A holomorphic vector bundle E X is called concavex if ÝÑ E E` E´, with H1 P1,f ˚E` 0, H0 P1,f ˚E´ 0 f : P1 X. “ ‘ “ “ @ ÝÑ

Such a vector bundle induces a vector` orbi-bundle˘ VE over` M0,m ˘X,A : p q ˚ ` 1 ˚ ´ VE V ` V ´ , where V ` π ev E , V ´ R π ev E . (1.1) ” E ‘ E E ” ˚ E ” ˚ 1 ˚ Given a class A H2 X; Z and classes η1,η2 H X , the corresponding genus 0 twisted two-point Gromov-WittenP p q (GW) invariants of PX are:p q

X p1 p2 p1 ˚ p2 ˚ V Q ψ η1,ψ η2 A,E ψ1 ev1 η1 ψ2 ev2 η2 e E . (1.2) ” vir p q P żrM0,2pX,Aqs In particular,@ if E E`, theD twisted Gromov-Witten` invariants˘` (1.2)˘ are the genus 0 two- point Gromov-Witten“ invariants of a complete intersection Y s´1 0 ã X deﬁned by a generic holomorphic section s:X E`: ” p q Ñ ÝÑ p1 p2 X p1 p2 Y p1 p2 Y ˚ ψ η1,ψ η2 ψ η1,ψ η2 ψ η1,ψ η2 η1,η2 H Y ; (1.3) A,E` “ A ” A,0 @ P p q the ﬁrst@ equality followsD from@ [El, TheoremD 0.1.1,@ RemarkD 0.1.1]. The numbers (1.2) have been computed in the X Pn´1 case under various assumptions on E through various approaches. The case when“E is a positive line bundle is solved in [BK] and [Z1] and extended to the case when E is a sum of positive line bundles in [PoZ]. The former led to the computation of the genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces in [Z2], while the latter to the computation of the genus 1 Gromov-Witten invariants of Calabi-Yau complete intersections in [Po]. The case when E is a concavex vector bundle has been solved in [Ch] in the setting of [LLY1]. More recently, genus 0 formulas with any number of ψ classes have been obtained in [Z3]. In this dissertation we extend the approaches of [Z1] and [PoZ] to the case when X is an arbitrary compact symplectic toric manifold and E is a sum of non-negative and negative line bundles.

1.1 Some results

If n is a non-negative integer, we write n 1, 2,...,n . r s”t u Let s 1, N1,...,Ns 2 and for each i s let ě ě Pr s s ˚ 2 Nj ´1 Hi pri H H P , ” P ˜j“1 ¸ ź s Nj ´1 Ni´1 2 Ni´1 where pri : P P is the projection onto the i-th component and H H P j“1 ÝÑ P p q is the hyperplaneś class on PNi´1. ą0 s Theorem 1.1. Let d d1,...,ds Z . The degree d genus 0 two-point GW invariants s “p qPp q Ni´1 QrA1,...,As,B1,...,Bss ´1 ´1 (1.3) of P are given by the following identity in N N ~ , ~ : A i B i 1 2 i“1 i , i @ iPrss rr ss ś ´ ¯ s Ni´1 N1´1´a1 Ns´1´as N1´1´b1 Ns´1´bs P 1 a1 as b1 bs H1 ... Hs H1 ... Hs i“ A1 ... As B1 ... Bs , ś ~1 ψ ~2 ψ a1,...,asě0 B ´ ´ Fd b1,...,bÿsě0 a1 as b1 bs (1.4) 1 A1 e1~1 ... As es~1 B1 f1~2 ... Bs fs~2 p ` q p ` q p ` q p ` q . s e f “ ~1 ~2 i i a ,b ,e ,f ě0 Ni Ni ` i i i i Ai r~1 Bi r~2 ai`bÿi“Ni´1 i“1 r“1 p ` q r“1 p ` q ei`fi“di ˆ ˙ ś ś ś 2 This follows from Corollary 3.8 in Section 3.2.

´1 ´1 Remark 1.2. The sums on both sides of (1.4) are power series in ~1 and ~2 . In order to see that this is the case for the right-hand side of (1.4), divide both the numerator s s Niei Nifi i“1 i“1 and denominator of each ai,bi,ei,fi-summand by ~1ř ~2ř . Part of the statement of Theorem 1.1 is that the right-hand side sum in (1.4) is divisible by ~1 ~2 (i.e. it vanishes ` when evaluated at ~1, ~2 ~, ~ ). Identity (1.4) should be interpreted by first dividing p q “ p ´ q this sum by ~1 ~2 and then setting it equal to the left-hand side. ` τ The results below concern the GW invariants of a compact symplectic toric manifold XM defined by (2.2) from a minimal toric pair M,τ as in Definition 2.1. We assume that the vector bundle E splits p q

a b ` ´ τ ` ` ´ ´ E E E XM , where E Li , E Li , (1.5) ” ‘ ÝÑ ” i“1 ” i“1 à à ` ´ 1 Li are non-trivial, non-negative line bundles and Li are negative line bundles. Theorem 1.3 and Remark 1.4 below describe two-point twisted GW invariants in terms of one-point ones. As is usually done, the twisted GW invariants will be assembled into a generating function in the formal variables Q Q1,...,Qk “ p q with powers indexed by

τ τ Λ d H2 X ; Z : ω, d 0 ω K , (1.6) ” P p M q ě @ P M τ τ 2 @ D ( where KM is the closed K¨ahler cone of XM . A ring R and the monoid Λ induce an R-algebra denoted R Λ : to each d we associate a basis element denoted Qd and set rr ss

d R Λ adQ : ad R d Λ . rr ss ” #dPΛ P @ P + ÿ Addition in R Λ is defined naturally; multiplication is defined by rr ss 1 1 Qd Qd Qd`d d, d1 Λ ¨ ” @ P 1 τ 2 τ Recall that a line bundle L ÝÑ XM is called positive (respectively negative) if c1pLq P H pXM ; Rq τ τ (respectively ć1pLq) can be represented by a Kähler form on XM . A line bundle L ÝÑ XM is called 2 τ non-negative if c1pLq P H pXM ; Rq can be represented by a 2-form ω satisfying ωpv,Jvq ě 0 for all v. The ` ´ ´ assumptions that the line bundles Li are non-trivial and that Li are negative (that is, c1pLi qă0 as opposed ´ to just c1pLi qď0) are only used in the theorems that rely on the one-point mirror theorem (5.2) of [LLY3], that is Theorem 3.5, Corollary 3.7, Corollary 3.8, Theorem 4.7, Corollary 4.8, and Corollary 4.9. 2By [Br, Theorem 4.5], a non-empty closed convex subset of Rd is the intersection of its supporting half- spaces. The supporting half-spaces of a closed convex cone C in Rd are all sets of the form tv PRd : v,w ě0u for some w PRd such that v,w ě0 for all v PC. This implies that @ D @ D τ d d ω PKM ðñ ω, ě0 @ PΛ. @ D

3 and extended by R-linearity. τ For each m 1 and each d Λ 0 , let σi : M0,m XM , d U be the section of the universal curveě given by the i-thP marked´t u point, p q ÝÑ

İ 0 ˚ ` 1 ˚ ´ τ VE R π˚ ev E σ1 R π˚ ev E σ1 M0,m XM , d , and İİ ” p´ q ‘ p´ q ÝÑ p q (1.7) 0 ˚ ` 1 ˚ ´ τ VE R π˚ èv E σ2 ˘ R π˚ èv E σ2 ˘ M0,m XM , d whenever m 2. ” p´ q ‘ p´ q ÝÑ p q ě ` İ ˘ İİ ` ˘ If m 3 and d 0, VE and VE are well-defined as well and they are 0. We next define the ě “ İ genus 0 two-point generating function Z:

İ İ ~1~2 d e VE Z ~1, ~2,Q Q ev1 ev2 p q , (1.8) ~ ~ ˚ ~ ψ ~ ψ p q” 1 2 dPΛ p ˆ q « 1 1 2 2 ff ` ÿ p ´ q p ´ q τ τ where ev1, ev2 :M0,3 XM , d XM are the evaluation maps at the first two marked points. This is used - in thep case ofqÝÑ the projective spaces - for the computation of the genus 1 GW invariants of Calabi-Yau complete intersections. τ τ With ev1, ev2 : M0,2 XM , d XM denoting the evaluation maps at the two marked points and for all η H2pXτ ,q let ÝÑ P p M q İ İ ˚ d e VE ev2 η ˚ τ ´1 Zη ~,Q η Q ev1˚ p q H XM ~ Λ , p q” ` ~ ψ1 P p qr srr ss dPΛ´0 « ´ ff ÿ İİ (1.9) İİ ˚ d e VE ev2 η ˚ τ ´1 Zη ~,Q η Q ev1 p q H X ~ Λ . ˚ ~ ψ M p q” `dPΛ´0 « 1 ff P p qr srr ss ÿ ´ τ τ τ Theorem 1.3. Let pri : XM XM XM denote the projection onto the i-th component ˚ τ ˆ ÝÑ and let ηj, ηj H X be such that P p M q s ˚ ˚ 2pN´kq τ τ q pr1 ηjpr2 ηj H XM XM j“1 P p ˆ q ÿ τ is the Poincarédual to the diagonal class,q where N k is the complex dimension of XM . Then, ´ s İ İ İİ 1 ˚ ˚ Z ~1, ~2,Q pr1 Zηj ~1,Q pr2 Zηj ~2,Q . p q“ ~1 ~2 p q p q ` j“1 ÿ q This follows from Theorem 4.5 below, which is an equivariant version of Theorem 1.3. Remark 1.4. The genus 0 two-point twisted GW invariants (1.2) are assembled into

˚ d e VE Z ~1, ~2,Q Q ev1 ev2 p q , (1.10) ˚ ~ ψ ~ ψ p q”dPΛ´0 p ˆ q 1 1 2 2 ÿ „p ´ q p ´ q τ τ where ev1, ev2 :M0,2 X , d X . By the string relation [MirSym, Section 26.3], p M qÝÑ M

˚ ~1~2 e VE ˚ τ τ ´1 ´1 Z ~1, ~2,Q ev1 ev2 p q H X X ~ , ~ Λ , ~ ~ ˚ ~ ψ ~ ψ M M 1 2 p q“ 1 2 dPΛ´0p ˆ q 1 1 2 2 P p ˆ qr srr ss ` ÿ „p ´ qp ´ q 4 τ τ where ev1, ev2 :M0,3 X , d X . By (1.7) and (1.1), p M qÝÑ M İ ˚ ` ˚ ´ e VE ev e E e VE ev e E . p q 1 p q“ p q 1 p q The last two equations imply that

İ ˚ ˚ ` ˚ ˚ ´ Z ~1, ~2,Q pr e E Z ~1, ~2,Q pr e E , p q 1 p q“ p q 1 p q İ İ ˚ 0 τ τ τ where Z is obtained from Z by disregarding the Q term and pr1 : XM XM XM is the projection onto the ﬁrst component. This together with Theorem 1.3ˆ expressesÝÑ Z˚ in İ İİ ` ˚ terms of Zη, Zη in the E E case. In all other cases, Z can be expressed in terms of one- point GW generating functions“ which can be computed under one additional assumption; see Remark 3.9.

2 2 Remark 1.5. If E OP2 1 OP2 2 and H H P is the hyperplane class, then “ p´ q‘ p´ q P p q 2d ! V ˚ 2 ˚ V ˚ ˚ 2 d e E ev1 H ev2 H e E ev1 Hev2 H 1 p q 2 d 1. 2 p q “ 2 p q “ p´ q 2d d! @ ě żM0,2pP ,dq żM0,2pP ,dq p q 2 2 If E OP2 1 OP2 1 OP2 1 and H H P is the hyperplane class, then “ p´ q‘ p´ q‘ p´ q P p q d`1 ˚ 2 ˚ 2 1 e VE ev1 H ev2 H p´ q d 1. 2 p q “ d @ ě żM0,2pP ,dq 2 1 If E OP1 1 OP1 1 and H H P is the hyperplane class, then “ p´ q‘ p´ q P p q

˚ ˚ 1 e VE ev1 Hev2 H d 1. 1 p q “ d @ ě żM0,2pP ,dq These follow from (3.36) in Section 3.2 which relies on Theorem 4.5, the equivariant version of Theorem 1.3 above. The first of these equations implies the first statement in [KlPa, Proposition 2] by the divisor relation of [MirSym, Section 26.3], the second recovers the first statement in [PaZ, Lemma 3.1], and the third implies the Aspinwall-Morrison formula.

1.2 Outline of the dissertation

Chapter 2 presents the facts about symplectic toric manifolds needed for the Gromov- Witten theory parts of the dissertation. These are Proposition 2.2 and Remark 2.10 in Section 2.1, Propositions 2.14, 2.16, and 2.17 in Section 2.2, and Corollary 2.20 and Propo- sition 2.21 in Section 2.3. This chapter is inspired by the view in [Gi2] of a symplectic toric manifold as given by a matrix and the choice of a certain regular value together with the holomorphic charts of [Ba]. It contains proofs of all statements or references to the ones that are omitted. The reader interested only in the Gromov-Witten theory part may want to skip all proofs in Chapter 2. İ İİ Section 3.2 gives formulas for the one-point GW generating functions Zη, Zη of (1.9) under an additional assumption in terms of explicit formal power series constructed in Section 3.1. It begins with a short setup.

5 The explicit GW formulas of Section 3.2 and Theorem 1.3 above follow from the equivari- İ İİ ant statements of Section 4.2. In particular, equivariant versions of Zη and Zη are expressed in terms of explicit power series constructed in Section 4.1. Chapter 4 also begins with a short setup. An outline of the proofs of the equivariant theorems of Section 4.2 is given in Section 5.1. The remaining sections of Chapter 5 provide the details.

6 Chapter 2

Overview of symplectic toric manifolds

This chapter reviews the basics of symplectic toric manifolds and sets up notation that will be used throughout the rest of the dissertation. It combines the perspectives of [Au, Chapter VII], [McDSa, Section 11.3], [Ba, Section 2], [CK, Section 3.3.4], [Gi2], [Gi3], and [Sp, Sections 5,6]. Sections 2.1-2.2 give the deﬁnition and describe the basic properties of a compact symplectic toric manifold. Section 2.3 is a preparation for localization computations in a toric setting; it describes the ﬁxed points and curves and the equivariant cohomology.

2.1 Deﬁnition, charts, and K¨ahler classes

Throughout this dissertation, k and N denote ﬁxed positive integers such that k N and ď N 1, 2,...,N . r s” k N If v R (or v C ) and j k (or j N ), let vj R (or( vj C) denote the j-th component of vPand deﬁneP Pr s Pr s P P supp v j : vj 0 . p q” ‰ If J N , let ( Ďr s RJ v RN: supp v J R|J|, CJ z CN: supp z J C|J|. ” P p qĎ – ” P p qĎ –

If A aij iPrks,j PrNs is a k N matrix( and J N , denote by AJ the k J( submatrix of A consisting“p q of the columnsˆ indexed by the elementsĎr s of J. Let ˆ| |

i N ω dz dz std 2 j j ” j“1 ^ ÿ be the standard symplectic form on CN . Let

N N 2 2 µstd : C R , µstd z1,...,zN z1 ,..., zN ÝÑ p q” | | | | ` ˘ 7 N ˚ N N be the moment map for the restriction of the standard action of T C on C , 2 ωstd , ”p q p ´ q

t1,...,tN z1,...,zN t1z1,...,tN zN , p q¨p q “ p q to S1 N TN . p q Ă k ˚ k N An integer k N matrix M mij i k ,j N induces an action of T C on C , 2ωstd , ˆ “p q Pr s Pr s ”p q p ´ q m11 m21 mk1 m1N m2N mkN t1,...,tk z1,...,zN t t ...t z1,...,t t ...t zN ; (2.1) p q¨p q “ p 1 2 k 1 2 k q the moment map of its restriction to S1 k Tk is p q Ă N k µM M µstd : C R . ” ˝ ÝÑ If in addition τ Rk, let P P τ M ´1 τ Rě0 N , M ” p q X p q τ N J N supppzq ´1 τ τ k XM C C z C : C µ τ , XM XM T ; (2.2) ” ´ “ P X M p q‰H ” JĎrNs CJ Xµď´1pτq“H ( L r M r τ see diagram (2.3). By Proposition 2.2 below, XM is a compact projective manifold if the ´1 τ 1 k pair M,τ is toric in the sense of Deﬁnition 2.1. In this case, µstd PM S has a unique smoothp structureq making the projection p q{p q

µ´1 P τ µ´1 P τ S1 k stdp M q ÝÑ stdp M q{p q ´1 τ 1 k τ a submersion. With this smooth structure, µstd PM S is diﬀeomorphic to XM via a ´1 τp ã q{p τ q diﬀeomorphism induced by the inclusion µstd PM XM . We summarize this setup in a diagram: p q Ñ r P τ M ´1 τ Rě0 N M ” p _qXp q

´1 ´1 / / µstd / N / µ τ µ P τ / Xτ / CN ❲❲ / Rě0 / RN M std M M ❲❲❲❲❲ (2.3) p q” p q ❲❲❲❲❲ p q ❲❲❲❲❲ projection projection ❲❲❲❲ µ ❲❲❲❲ M r M ❲❲❲❲❲ 1 ❲❲❲ µ´ pP τ q diffeo ❲❲❲+ std M / Xτ Rk τ pS1qk M Q Given a pair M,τ consisting of an integer k N matrix M and a vector τ Rk, we define p q ˆ P V τ τ J M J N : J k, PM R ” Ďr s | |“ X ‰H (2.4) ! ) J N : J k, v M ´1 τ Rě0 N s.t. supp v J . ” Ďr s | |“ D P p qXp q p qĎ ! ) Definition 2.1. A pair M,τ consisting of an integer k N matrix M and a vector τ Rk is toric if p q ˆ P

8 τ (i) τ is a regular value of µM and P ; M ‰H τ (ii) det MJ 1 for all J V ; Pt˘ u P M (iii) P 0 0 ( P τ is bounded). M “t u ðñ M A toric pair M,τ is minimal if p q (iv) P τ RrNs´tju for all j N . M X ‰H Pr s If a pair M,τ satisfies (ii) in Definition 2.1 above, then p q N τ k z C , supp z J for some J V t T such that t z j 1 j J. P p qĚ P M ùñ D P p ¨ q “ @ P N τ If M,τ is a toric pair, then a point z C lies in XM if and only if supp z J for somep J q V τ and the TN -fixed points ofPXτ are indexed by V τ ; see Lemmap 2.4(q Ěi) and P M M M Corollary 2.20(a). r

τ Proposition 2.2. If M,τ is a toric pair, then XM is a connected compact projective manifold of complex dimensionp q N k endowed with a TN-action induced from the standard action of TN on CN . ´

τ Proof of Proposition 2.2. By Lemmas 2.5(a), (g), and (h) below, XM is a connected, compact complex manifold. It admits a positive line bundle by Lemmas 2.13, 2.7(b), and 2.9 τ below. By the Kodaira Embedding Theorem [GriH, p181], XM is then projective. Remark 2.3. If X is a compact symplectic toric manifold in the sense of [Ca, Deﬁni- tion I.1.15], then the image of its moment map is a Delzant polytope P (a polytope with certain properties [Ca, Deﬁnition I.2.1]); see [At, Theorem 1] or [GuS, Theorem 5.2]. This

polytope P determines a fan ΣP , which in turn determines a compact complex manifold XΣP ;

see [Au, Section VII.1.ac]. This complex manifold XΣP is endowed with a symplectic form, a torus action, and a moment map with image P making it into a symplectic toric manifold; see [Au, Theorem VII.2.1]. Moreover, this symplectic form is K¨ahler with respect to the complex structure, as stated in [Gi2, Section 3] and can be deduced from [Au, Chapter VII].

Since X and XΣP have the same moment polytope (i.e. image of the moment map), they are isomorphic as symplectic toric manifolds by Delzant’s uniqueness theorem [De, Theo- τ rem 2.1]. On the other hand, XΣP XM for some minimal toric pair M,τ by the proof of [Au, Theorem VII.2.1]. Thus, a“ compact symplectic toric manifold pX2n,ω,q S1 n,µ in the sense of [Ca, Definition I.1.15] admits a complex structure J so thatp X,ω, Jp isq Kählerq and X, J is isomorphic to Xτ for some minimal toric pair M,τ . p q p q M p q Lemma 2.5 relies on parts (i) and (j ) of Lemma 2.4 below which in turn rely on the other parts of Lemma 2.4. Lemma 2.9 is based on Lemma 2.8 and Lemma 2.7(d). Lemma 2.7(b) follows from Lemma 2.7(a), while the proof of Lemma 2.7(d) uses Lemma 2.7(c). k k For t t1,t2,...,tk T and p p1,p2,...,pk Z , let “p qP “p qP tp tp1 tp2 ...tpk . ” 1 2 k Lemma 2.4. Let M,τ be a toric pair. p q 9 ( a) The subset P τ Rě0 N is a polytope (i.e. the convex hull of a finite set of points). M Ăp q k η ( b) Let η R be any regular value of µM . If w P , then P P M M : v RN : supp v supp w Rk t P p qĎ p qu ÝÑ is onto. In particular, if w P η , then supp w k. P M | p q|ě ( c) If J V τ , then J supp y for some y µ´1 τ . P M “ p q P M p q ( d) If J N and supp v J for some v P τ , then supp w J for some w P τ . Ďr s p qĎ P M p q“ P M ( e) The polytope P τ has dimension N k. M ´ ( f) If v is a vertex of P τ , then supp v V τ . M p qP M τ τ ( g) If VerticesM is the set of vertices of the polytope PM , the map supp : Verticesτ V τ , v supp v , M ÝÑ M ÝÑ p q is a bijection. ( h) If y µ´1 τ , then supp y J for some J V τ . P M p q p qĚ P M ( i) Let z CN . Then, z Xτ if and only if supp z J for some J V τ . P P M p qĚ P M V τ pnq k pnq pnq ( j) Let I,J M and t rT . If t and there exists δ 0 such that ti δ for P pnq MP´1M | | ÝÑ8 ą | |ě all i k , then t I j is unbounded for some j J. Pr s |p q | P Proof. (a) By [Zi, Theorem 1.1], a subset of RN is a polytope if and only if it is a bounded intersection of half-spaces. Thus, the claim follows from (iii) in Definition 2.1. (b) This is immediate from the surjectivity of dwµM . (c) This follows from the second statement in (b). 1 τ 1 (d) Assume that supp v I J and that there exists v PM with supp v I. Let I1 I p qĎ Ř P τ p q“ Ą with I1 J and I1 I 1. We show that there exists w PM with supp w I1. By the first Ď | |“| |` 1 ´1 N P 1 p q“ 1 1 statement in (b), there exists w M τ R with supp w I1. Let w 1 λ v λw with λ R satisfying P p qĂ p q“ “ p ´ q ` P 1 1 wj λw 0 if j I1 I and λ 1 1 j I. j ą P ´ ´ v1 ă @ P ˆ j ˙ (e) By (d) together with the second condition in (i) in Definition 2.1, supp w N for τ τ p q“r s some w PM and thus dim PM N k, since M has rank k by (b). (f ) By (Pe), supp v k; the opposite“ ´ inequality follows from the second statement in (b). | p q|ďV τ τ (g) By (f ), supp v M for every vertex v of PM . The map supp is injective by (ii) in Definition 2.1 andp surjectiveq P by (c) and (ii) in Definition 2.1. τ (h) By [Zi, Proposition 2.2], every polytope is the convex hull of its vertices; since µstd y PM τ p qP and PM is a polytope by (a), r

µstd y λsvs p q“ s“1 ÿ 10 τ ą0 for some vertices v1,v2,...,vr PM and λ1,λ2,...,λr R . Then, supp y supp v1 and V τ P P p qĚ p q supp v1 M by (f ). p qP τ supppzq ´1 V τ (i) If z XM , there exists y C µM τ . By(h), there exists J M with J supp y . Since suppP y supp z , it followsP thatX J p suppq z . The converse followsP from (cĎ). p q p qĎ p q ą0 k Ď p q (j ) By (cr), there exist v,w R such that MI v τ MJ w. By (ii) in Deﬁnition 2.1, it P pą0 kq ´1 “ “ ą0 k follows that there exists a Z such that MI MJ a Z . Pp q pnq M ´1M Pp q Assume by contradiction that t I j is a bounded sequence for all j J. By passing |p q pnq M|´1M P to subsequences, we may assume that t I j is convergent for all j J. It follows that |p q | P pnq M ´1M aj pnq M ´1M a t I j t I J (2.5) jPJ p q “ p q ź ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ is also convergent. On the other hand, by passing to some subsequences, we may assume pnq that for each i k , ti has a limit (possibly ). Since at least one of these limits is and none is 0, thePr right-hands | | side of (2.5) diverges8 leading to a contradiction. 8

N For z C and J j1 j2 ... jn N , let P “t ă ă ă uĎr s

zJ zj ,zj ,...,zj . ” p 1 2 n q For z Xτ , let z Xτ denote the corresponding class. P M r sP M Lemma 2.5. Let M,τ be a toric pair. r p q τ ( a) The space XM is path-connected.

k τ ( b) The torus Tr acts freely on XM . ( c) The subset Tk µ´1 τ of CN is open. ¨ M p q r ( d) The subset Tk µ´1 τ of Xτ is closed. ¨ M p q M ( e) There is a unique map r ρτ : X˜ τ Rą0 k Tk s.t. ρτ z z µ´1 τ z Xτ . M M ÝÑ p q Ă M p q¨ P M p q @ P M Furthermore, this map is smooth. r ( f) The quotient µ´1 τ S1 k is a compact and Hausdorﬀ. M p q{p q ( g) The inclusion µ´1 τ ã Xτ induces a homeomorphism M p q Ñ M µ´1 τ S1 k Xτ . (2.6) r M p q{p q ÝÑ M τ In particular, XM is compact and Hausdorﬀ. ( h) The space Xτ is a complex manifold of complex dimension N k. M ´

11 τ N Proof. (a) This holds since XM is the complement of coordinate subspaces in C . k τ V τ (b) Let t T and z XM be such that tz z. By Lemma 2.4(i), there exists J M such that P P r ¨ “ P J j1 ... jk supp z . r ”t ă ă uĎ p q By (ii) in Deﬁnition 2.1, the group homomorphism

k k M M T T , t t j1 ,...,t jk , ÝÑ ÝÑ p q is injective and so t 1, 1,..., 1 . (c) For each z CN ,“p let q P z1 0 | | . Mz M .. . ” ¨ ˛ 0 zN ˚ | | ‹ ´1 V τ˝ ‚ If z µM τ , supp z J for some J M by Lemma 2.4(h). Since MJ is invertible by (ii) in P p q p qĚ P tr Definition 2.1, so are Mz J and Mz Mz . Since the differential of the map p q p q ą0 k k R R , t µM t z , ÝÑ ÝÑ p ¨ q ą0 k k tr at t 1,..., 1 R T` is 2M˘ z Mz , the differential of the map “p qPp q Ă p q k ´1 k T µ τ R , t,z µM t z , ˆ M p q ÝÑ p q ÝÑ p ¨ q ´1 is surjective at 1,z for all z µM τ . Since the restriction of this differential to the second component vanishes,p q the differentialP p q of the map

Tk µ´1 τ CN , t,z t z, (2.7) ˆ M p q ÝÑ p q ÝÑ ¨ ´1 is surjective at 1,z for all z µM τ and so, by the Inverse Function Theorem, the image of (2.7) containsp anq open neighborhoodP p q of µ´1 τ in CN . M p q (d) Let zpnq Xτ and tpnq Tk be sequences such that P M P pnq τ pnq pnq pnq ´1 r lim z z XM and y t z µM τ . nÝÑ8 “ P ” ¨ P p q

pnq ´1 By (iii) in Deﬁnition 2.1, we canr assume that y y µM τ . By Lemma 2.4(i), there exist J y ,J z V τ such that ÝÑ P p q p q p qP M

J y j1 ... jk supp y and J z supp z ; p q”t ă ă Ď p q p qĎ p q we can assume that J y ,J z supp( ypnq supp zpnq for all n. By (ii) in Deﬁnition 2.1, p q p qĎ p q“ p q MJpyq is invertible and so

´1 pnq pnq pnq M pnq y ji t t˜ Jpyq i , t˜ i ,...,k. i p q where i p pnqq 1 “ “ z ji @ “ ` ˘ ` ˘ p q pnq pnq pnq ą0 Since y j yj 0 for all j J y and z j zj, t˜ i δ for some δ R and forp allqn ÝÑand i.‰ If t˜pnq is notP boundedp q above,p q afterÝÑ passing|p q to| ě a subsequence weP can |p q| 12 assume that t˜pnq . By Lemma 2.4(j ), there exists j J z such that, after passing to a subsequence,| | ÝÑ8 P p q ´1 n M n M Mj tp q j t˜p q Jpyq . p q “ p q ÝÑ 8 pnq pnq pnq Since t z y, it follows thatˇ z ˇ j ˇ 0 and so j ˇ supp z , contrary to the assumption. ˇ ˇ ˇ ˇ Thus, t˜p¨nq isÝÑ a compact subset ofp Tk.q AfterÝÑ passing toR a subsequence,p q we can thus assume that tptnq u t Tk. It follows that ÝÑ P t z lim tpnq lim zpnq lim tpnq zpnq lim ypnq y. ¨ “ nÝÑ8 ¨ nÝÑ8 “ nÝÑ8 ¨ “ nÝÑ8 “

k ´1 Thus, z T µM τ . (e) By theP proof¨ p ofq (c), τ is a regular value of the smooth map

ą0 k τ k Φ: R X R , t,z µM t z , ˆ M ÝÑ p q ÝÑ p ¨ q ` ´˘1 τ and the projection map π2 : Φ τ r XM is a submersion. By (a), (c), and (d), this map is surjective. We show that it isp q also ÝÑ injective; by (a), (c), and (d), this is equivalent to showing that r

k r1 rk ´1 r1,...,rk R , z, e ,..., e z µ τ ri 0 i 1,...,k, p qP p q¨ P M p q ùñ “ @ “ where the action of er1 ,..., erk Tk on z is deﬁned by (2.1) as above. We present the argument in the proofp of [Ki, 7.2q Lemma]. P Let

ur1 urk f : R R, f u µM e ,..., e z , r1,...,rk u R. ÝÑ p q” rp q¨ s p q @ P A E 1 Since f 0 f 1 , there exists u0 0, 1 such that f u0 0. Since p q“ p q Pp q p q“ N 2 1 2u0xpr1,...,rkq,Mj y 2 f u0 2 e r1,...,rk ,Mj zj , p q“ j“1 p q | | ÿ A E 1 f u0 0 implies that r1,...,rk ,Mj zj 0 for all j N . By Lemma 2.4(i), there p q “ V τ xp q y “ P r s exists J M such that J supp z and so r1,...,rk ,Mj 0 for all j J. By (ii) in P Ď p q xp q y“τ ´1 P Definition 2.1, this implies that ri 0 for all i k . The map ρM is π2 composed with the ą0 k τ ą0 k“ Pr s projection R XM R . p´1 q ˆ ÝÑp q ´1 1 k (f ) Since µM τ is compact by (iii) in Definition 2.1, so is the quotient space µM τ S . If p is the quotientp q projection map and A µ´1 τ is a closed subset, p q{p q Ă M p q p´1 p A S1 k A t z : z A, t S1 k p q “ p q ¨ ” ¨ P Pp q ` ˘ 1 k ´1 ( is the image of the compact subset S A in µM τ under the continuous multiplication map p q ˆ p q S1 k µ´1 τ µ´1 τ p q ˆ M p q ÝÑ M p q ´1 ´1 and thus compact. Since µM τ is Hausdorff, it follows that p p A is a closed subset ´1 p q p p qq ´1 of µM τ . We conclude the quotient map p is a closed map. Since µM τ is a normal topologicalp q space, by [Mu, Lemma 73.3] so is µ´1 τ S1 k. p q M p q{p q 13 ´1 ã τ (g) The map (2.6) is well-defined, since the inclusion µM τ XM is equivariant under the inclusion S1 k ã Tk, and is continuous by the defining propertyp q Ñ of the quotient topology. p q Ñ The map r Xτ µ´1 τ , z ρτ z z, M ÝÑ M p q ÝÑ M p q ¨ is equivariant with respect to the natural projection Tk S1 k by the uniqueness property r ÝÑp q τ in (e) and thus induces a continuous map in the opposite direction to (2.6). Since ρM pµ´1pτqq | M “ 1,..., 1 , the two maps are easily seen to be mutual inverses. p q τ V τ (h) We cover XM by holomorphic charts as in [Ba, Propositions 2.17, 2.18]. For each J M , let P

N k N J i1 i2 ... iN k , UJ z C : supp z J , UJ UJ T , r s´ ”t ă ă ă ´ u ” P p qĚ ” { ( N´k zi1 zi2 ziN´k hJ : UJ C , hJ z r , ,..., r . (2.8) M ´1M M ´1M M ´1M ÝÑ r s” ¨ J i1 J i2 J iN´k ˛ zJ zJ zJ ˝ ‚ V τ V τ τ τ By Lemma 2.4(i), the collections UJ : J M and UJ : J M cover XM and XM , t P u ´1 t P u respectively. The map hJ is well-defined. First, MJ exists and is an integer matrix by (ii) k in Definition 2.1. Second, if t T , z rUJ , and J j1 j2 ... jk , then r P P ”t ă ă ă u ´1 ´1 ´1 ´M Mis M ´ M Mis M ´ M Mis M J j1 J r 1 jk J k is t z J t z is t zj1 p q ... t zjk p q t zis p ¨ q p ¨ q “ ˆ ˙ ´1 ´M ´1M ´M ´1M ´`MJ M ˘Mis `Mis J ìs ˘ J is t J z zi z zi , s N k . “ p q J s “ J s @ Pr ´ s ´1 The map hJ is the composition of the continuous maps

´1 N´k hJ projection ´1 zs, if i is, C UJ UJ , hJ z “ i N . ÝÝÑĄ ÝÝÝÝÝÝÑ p q i “ #1, if i J, @ Pr s ´ ¯ P r Ą N´k hJ N´k The composition C UJ C is obviously the identity. The other relevant ÝÑ ÝÑ composition is given by UJ z y UJ , where Qr s ÝÑr sP ´1 ´MJ Mis zJ zis , if i is, yi ¨ “ j N . “ #1, if i J, @ Pr s P ´1 ´pMJ qr Let tr zJ for all r k ; it follows that t z y. ” 1 V τ Pr s ¨ “ ´1 If in addition J M , the domain and image of the overlap map hJ hJ1 are complements of some the coordinateP subspaces in CN´k, and every component of˝ this map is a ratio of monomials in the complex coordinates. In particular, this map is holomorphic.

τ τ Remark 2.6. Let M,τ be a toric pair. The projection π : XM XM is a holomorphic submersion; this canp be seenq using the charts (2.8). ÝÑ

τ r Let KM be the connected component of τ inside the regular value locus of µM . Lemma 2.7. Let M,τ be a toric pair. p q 14 k ě0 |J| ( a) Let η R . Then, η is a regular value of µM if and only if η MJ R for every J NP with J k 1. R p q Ăr s | |ď ´ τ k k τ ( b) The subset KM of R is an open cone (i.e. an open subset of R such that λη KM whenever λ 0 and η Kτ ). P ą P M ( c) For every η Kτ , V η V τ . P M M “ M ( d) For every η Kτ , M,η is a toric pair and Xη Xτ . P M p q M “ M ě0 |J| Proof. (a) If η is a regular value of µM , η MJ R for every J N with J k 1 R p q ě0 |J| Ăr s | |ď ´ by the second statement in Lemma 2.4(b). Suppose η MJ R for every J N with J k 1. We prove that for every v P η there existsR J p suppq v such that ĂrJ sk and | |ď ´ P M Ď p q | |“ det MJ 0. Suppose not, i.e. det MJ 0 for all J supp v with J k. We show that there ‰1 η 1 “ Ď p q | |“ exists v PM with supp v k; this contradicts the assumption on η. If supp v k, P ´1| p Nq| ă | p q| ě there exists w M 0 R such that supp w supp v and wj0 0 for some j0 supp v . Let P p qĂ p qĎ p q ą P p q vj λ min : j supp v such that wj 0 . ” w P p q ą " j * η It follows that v λw PM and supp v λw supp v . Continuing in this way, we obtain 1 η ´ 1 P p ´ qŘ p q v PM with supp v k. (bP) This follows| immediatelyp q|ă from (a). τ V η V τ τ (c) We show that the set η KM : M M is open and closed in KM and thus equals Kτ . It suﬃces to show thatt forP any P “J uN : J k the set M Ďt Ďr s | |“ u η Kτ : V η P η Kτ : P η RJ η Kτ : P η RJ t P M M “ u“ P M M X ‰H X P M M X “H JPP JĎrNs,|J|“k č ( JčRP ( is open. We show that the set

η Kτ : P η RJ P M M X ‰H 1 ( η1 J with J N and J k is open. Let η be any of its elements and let w PM R . By the Ďr s | |“ ąP0 k X surjectivity of dwµM , supp w J and det MJ 0; this shows that MJ R is open and p q“ ‰ p q 1 ą0 k τ τ η J η MJ R K η K : P R . P X M Ďt P M M X ‰Hu The set ` ˘ τ η J τ ě0 |J| η K : P R K MJ R P M M X “H “ M ´ p q with J N and J k is open as well. ( Ďr sτ | |“´1 V τ V τ V η (d) Since PM , µM τ and so M by Lemma 2.4(h). Since M , M ‰Hη p q‰H ‰H ‰H ‰H by (c) and so PM . Since M,τ satisﬁes (ii) in Deﬁnition 2.1, by (c) so does M,η . Thus, M,η is toric. The‰H equalitypXη q Xτ follows from (c) together with Lemmap 2.4(iq). p q M “ M Lemma 2.8. Let M,τ be a toric pair. p q

15 ( a) The quotient µ´1 τ S1 k admits a unique smooth structure such that the projection M p q{p q ´1 ´1 1 k πτ : µ τ µ τ S (2.9) M p q ÝÑ M p q{p q is a submersion.

´1 1 k ( b) There exists a unique symplectic form ωτ on µ τ S such that M p q{p q π˚ω ω , τ τ std ´1 “ µM pτq ˇ ˇ where πτ is the projection (2.9). ˇ ( c) The map (2.6) is a diffeomorphism. Proof. (a) By [tD, Proposition 5.2], if G is a compact Lie group acting freely and smoothly on a manifold M, then the quotient M G carries a unique differentiable structure such that the projection { M M G ÝÑ { is a submersion. Thus, the claim follows from (i) in Definition 2.1 and Lemma 2.5(b). (b) This follows from the Marsden-Weinstein symplectic reduction theorem [MW, Theo- rem 1]. (c) By (a) and Lemma 2.5(g), it is enough to show that the restriction

π : µ´1 τ Xτ ´1 M M µM pτq p q ÝÑ ˇ ˇ of the projection π :Xτ Xτ is aˇ submersion. This follows from the fact that the map M ÝÑ M Tk µ´1 τ Xτ , t,z z , r ˆ M p q ÝÑ M p q ÝÑr s k is a submersion whose diﬀerential at t,z vanishes on TtT 0. This map is a submersion because it is the composition of two submersions,p q ˆ

Tk µ´1 τ Xτ , t,z t z and π : Xτ Xτ . ˆ M p q ÝÑ M p q ÝÑ ¨ M ÝÑ M The former map is a submersionr by the proof of Lemma 2.5(c),r while π is a submersion by Remark 2.6.

If M,τ is a toric pair, we abuse notation and denote by ωτ not only the form on ´1 p 1 qk τ µM τ S defined by Lemma 2.8(b), but also the form it induces on XM via the diffeo- morphismp q{p (2.6)q of Lemma 2.8(c). In this case, by Lemma 2.7(d) and Lemma 2.8(b), for τ τ every η K , ωη is the unique symplectic form on X satisfying P M M ˚ τ τ π ω 1 ω 1 , π X X η µ´ pηq std µ´ pηq where : M M (2.10) M “ M ÝÑ ˇ ˇ is the projection; see alsoˇ diagram (2.3).ˇ r τ Lemma 2.9. Let M,τ be a toric pair. For every η KM , ωη is Kähler with respect to the p τq P complex structure on XM .

16 τ Proof. The form ωη is positive with respect to the complex structure on XM by (2.10) together with the equality Tk µ´1 η Xτ (justiﬁed by Lemmas 2.5(g) and 2.7(d)), Re- ¨ M p q“ M mark 2.6, and the positivity of ωstd. r Remark 2.10. If M,τ is a toric pair and J N , the pair MJ ,τ is toric if and only P τ p q Xτ Ďr s p q if MJ . In this case, MJ is a connected compact projective manifold of complex dimension‰ HJ k by Proposition 2.2. It is biholomorphic to | |´ Xτ J z Xτ : supp z J M p q”tr sP M p qĎ u via the map

z , if j j , Xτ z ι z Xτ J , ι z r r MJ J M where J j “ (2.11) Qr s ÝÑr p qsP p q p q ” #0, if j J, ` ˘ R if J j1 j2 ... jr . In particular, if M,τ is a minimal toric pair and Mj is the matrix obtained“t ă fromăM byă deletingu the j-th column,p q then Xτ is a connected compact projective Mj τ p manifold of complex dimension N 1. The map (2.11) identifies XM with the hypersurface ´ p j τ τ p X N j Dj z X : zj 0 . (2.12) M pr s´t uq ” ” tr s P M “ u If J V τ with V τ defined by (2.4), then Xτ J is the point P M M M p q 1, if j J; J z1,...,zN , where zj P (2.13) r s”r s ” #0, otherwise. This follows from Lemma 2.4(i) and (ii) in Definition 2.1. τ J τ If J N is such that PM R and J k 1, then XM J is a one-dimensional Ďr s X ‰H | |“ ` V τ p q complex manifold and there exist exactly 2 multi-indices I M with I J. The latter I V τ I J P ι Ă V τ follows since multi-indices M with correspond bijectively via J to elements of MJ , P Ă P τ P τ which in turn correspond to the vertices of MJ by Lemma 2.4(g); MJ has dimension 1 by Lemma 2.4(e). Remark 2.11. If M,τ is a toric pair with M a k N matrix, then VM,Vτ is a toric p q V τ V Vτ ˆ τ p q Vτ pair whenever V GLk Z . In this case, M VM and XM is biholomorphic to XVM . The pair VM,Vτ satisfiesP p theq first condition of“ (i) in Definition 2.1, since V is an isomorphism. p τ Vτq V τ V Vτ Since PM PVM , M VM and so VM,Vτ satisfies the second condition of (i), (ii), and (iii) in Definition“ 2.1“ as well. p q

τ Remark 2.12. If M,τ is a toric pair with M a k k 1 matrix, then XM is biholomorphic 1 p q V τ ˆp ` q to P . In order to see this, note ﬁrst that M 2 by Lemma 2.4(g) and Lemma 2.4(e). By | |“ V τ Remark 2.11, we can assume that MJ Idk for some J M . The claim now follows from τ “ P (2.8): XM is a compact manifold covered by two charts „ „ hJ : UJ C, hI : UI C Ñ Ñ ˚ ´1 ˘1 satisfying hJ UJ UI hI UJ UI C since I J k 1 and hI hJ z z by (ii) in Deﬁnition 2.1.p X q“ p X q“ Y “r ` s ˝ p q“

17 2.2 Cohomology, K¨ahler cone, and Picard group

Throughout the remaining part of this dissertation, M,τ is a toric pair. In order to τ p q complete the proof in Section 2.1 that XM is projective, we describe some holomorphic line bundles over it. For each p Zk, let P τ τ ´1 p k Lp X p C X C , where z,c t z,t c , t T . (2.14) ” M ˆ ” M ˆ „ p q „ p ¨ q @ P τ τ L k Since π : XM r XM withr π z z is a T -principal bundle by Lemma 2.5(b) and Re- mark 2.6, ÝÑ p q”r s τ r Lp X , z,c z , ÝÑ M r s ÝÑr s is a holomorphic line bundle. Furthermore,

˚ L0 OXτ , L L´p, Lp Lr Lp`r. “ M p “ b “

The line bundle L´Mj admits a holomorphic section

τ sj : X L M , z z,zj . (2.15) M ÝÑ ´ j r s ÝÑr s ´1 Since sj is transverse to the zero set by (2.8) and sj 0 Dj by (2.12), c1 L´Mj PD Dj . For all j N and i k , let p q“ p q“ p q Pr s Pr s ˚ Uj c1 L M , γi Le , Hi c1 γ , (2.16) ” p ´ j q ” i ” p i q k where ei : i k Z is the standard basis. Thus, t Pr suĂ k ˚bm1j ˚bm2j ˚bmkj L´Mj γ1 γ2 ... γk Uj mijHi j N . (2.17) “ b b b ùñ “ i“1 @ Pr s ÿ Lemma 2.13 below is used in the proof of Proposition 2.2 in Section 2.1 and to describe the τ Kähler cone of XM in Proposition 2.16 below. Lemma 2.13. For every η Zk Kτ , P X M 1 c1 L η ωη , p ´ q“ π r s where ωη is the Kähler form defined by (2.10). Proof. We follow closely the proof of [Au, Proposition VII.3.1]. Let

LR µ´1 η S1 k ´η ÝÑ M p q p q L be the pull-back of L´η via the diﬀeomorphism (2.6) of Lemma 2.8(c) and

´1 S1 ´1 1 p µM η L µ η η S p q ´η ” M p qˆ´ ÝÑ S1 k p q

18 be its sphere bundle. Let p µ´1 η S1 / µ´1 η M p qˆ M p q r

q πη 1 p ´1 S / µM pηq L´η pS1qk be the natural projections. # ´1 1 k Let ei be the fundamental vector ﬁeld on µM η S corresponding to ei R for the Tk-action given by (2.14) with p η. Thus, p qˆ P “´ # d e exp itei x1 iy1,...,xN iyN ,x iy i ” dt p p q ¨ p ` ` ` qq ˇt“0 N ˇ ˇ ˇmij yj B xj B ηi y B x B , x y x y “ j“1 j ´ j ` ´ ÿ ˆ B B ˙ ˆ B B ˙ N 2 N 2 where xj,yj,x,y are the standard coordinates on C R and C R , respectively. Let ”p q ” N 1 ´1 1 1 α xjdyj yjdxj Ω µM η and σ xdy ydx Ω S . ” j“1p´ ` q P p q ” ´ P p q ÿ ` ˘ ´1 1 Since ιe# α σ 0 on µM η S for all i k , α σ descends to a 1-form α σ S1 on i p ‘ q“ p qˆ Pr s ‘ p ‘ q S1 1 S1 L´η. This form is a connection 1-form for the principal S -bundle L´η because it satisﬁes

L # α σ 1 0, ι # α σ 1 1, X p ‘ qS “ X p ‘ qS “ where X# y B x B “´ x ` y B B 1 S1 1 is the fundamental vector ﬁeld for the S -action on L´η as a principal S -bundle; see [Au, Exercises V.4,V.5]. Let β denote the curvature form associated to α σ S1 . By [Au, Section V.4.c], it is uniquely determined by p ‘ q

˚ p β d α σ 1 . “ p ‘ qS ˚ ˚ Since q d α σ 1 2p ωstd, β 2ωη by the uniqueness of reduced symplectic form pp ‘ qS q“´ “ ´ ωη of Lemma 2.8(b). Thus, by [Au, Proposition VI.1.18] and [Au, Section VI.5.b], r R 1 1 2 ´1 1 k c1 L ´ β ωη H µ η S , p ´ηq“ 2π r s“ π r sP deRp M p q{p q q as claimed. We deﬁne

τ ´1 ą0 |Jc| EM J N : Dj J N : MJc τ R ; (2.18) ” # Ďr s jPJ “H+“ Ďr s p qXp q “H č ( the second equality follows from Lemma 2.4(i)(c)(d) and (2.12).

19 Proposition 2.14. If M,τ is a toric pair, p q

˚ τ Q H1, H2,..., Hk, U1, U2,..., UN H XM r s . p q– k τ Uj mijHi, 1 j N Uj : J E ´ ď ďr s ` P M ˆ i“1 ˙ ˜jPJ ¸ ř ś 2 τ If, in addition M,τ is minimal, H X ; Z is free with basis H1, H2,..., Hk . p q p M q t u Proof. This follows from [McDSa, Section 11.3] together with Lemma 2.8(c).

˚ τ Remark 2.15. By Proposition 2.14, H XM is generated as a Q-algebra by H1,..., Hk . Along with (ii) in Deﬁnition 2.1, this impliesp q that H˚ Xτ is generated as a tQ-algebra byu p M q U1,..., UN . t u

Proposition 2.16. If M,τ is a minimal toric pair, there is a basis c1 L´ηi : i k for 2 τ p q t p q Pr su H XM formed by the ﬁrst Chern classes of ample line bundles, with L´ηi as in (2.14). In p q τ τ particular, the K¨ahler cone KM of XM has dimension k.

k Proof. By Lemmas 2.13, 2.9, and 2.7(b)(d), there exists a subset η1,...,ηk Z , linearly t uĎ independent over Q, such that the line bundles L´ηj are positive. The ﬁrst Chern classes of these line bundles form a Q-basis of H2 Xτ by the last statement in Proposition 2.14. p M q Proposition 2.17. If M,τ is a minimal toric pair, the Picard group of Xτ is free of rank p q M k and has a Z-basis given by γ1,...,γk deﬁned by (2.16).

0,1 τ 0,2 τ Proof. The ﬁrst Chern class homomorphism is an isomorphism because h XM h XM 0 which in turn follows from Proposition 2.14. p q“ p q“

Remark 2.18. If M,τ is a toric pair, there is a short exact sequence p q N ‘k F G τ 0 O τ L´M TX 0. (2.19) XM j M ÝÑ ÝÑ j“1 ÝÑ ÝÑ à Speciﬁcally, we can take

τ F z ,ei z,mi1z1,mi2z2,...,miN zN i k , z X , pr s q” @ Pr s r sP M N “ ‰ τ G z,y1,...,yN yjdzπ B , z XM , y1,...,yN C, z z p q” j“1 j @ P P ÿ ˆB ˇ ˙ ˇ r ˇ k τ τ where ei : i k is the standard basis for C and π :X X is the projection. Thus, t Pr su M ÝÑ M N τ r c1 TXM Uj. (2.20) p q“ j“1 ÿ

20 2.3 Torus action and equivariant notation

The equivariant cohomology of a topological space X endowed with a continuous TN-action is ˚ ˚ N H N X H ET TN X , T p q” ˆ where ETN C8 0 N is the classifying space` for TN˘. In particular, the equivariant cohomology of” a p point´ isq

˚ ˚ ˚ 8 N H N point H N H P Q α1,...,αN Q α , T p q– T ” p q “ r s” r s ˚ 8 N `8 ˘ where αj c1 πj OP8 1 , πj : P P is the projection onto the j-th component and ” p p qq p q ÝÑ 8 OP8 1 is dual to the tautological line bundle over P . The equivariant Euler class of an orientedp q vector bundle V X endowed with a lift of the TN -action on X is ÝÑ N ˚ e V e ET TN V H N X . p q” p ˆ q P T p q A TN -equivariant map f : X Y between compact oriented manifolds induces a push- forward map ÝÑ s s`dim Y ´dim X f :H N X H Y ˚ T p q ÝÑ TN p q characterized by

˚ 1 1 ˚ 1 ˚ f η η η f η η H N Y ,η H N X . (2.21) p q “ p ˚ q @ P T p q P T p q żX żY s s´dim X If Y is a point, f˚ is the integration along the fiber homomorphism X :HTN X HTN . The push-forward map f extends to a homomorphism between the modulesp ofqÝÑ fractions with ˚ ş denominators in Q α ; in particular, the integration along the fiber homomorphism extends to r s ˚ : H N X Q α Q α Q α , where Q α Q α1,...,αN T p qb r s p q ÝÑ p q p q” p q żX is the field of fractions of Q α . If X is a compact oriented manifold on which TN acts smoothly, then, by the classicalr s Localization Theorem [ABo]

η ˚ Q α η Q α , η H N X , (2.22) N T r s Q X “ TN F e F {X P p q @ P p q ż F ĂÿX ż p q where the sum runs over the components of the TN pointwise ﬁxed locus XTN of X.

Lemma 2.19. If M,τ is a toric pair, TN z Tk is diffeomorphic to T|supppzq|´k for every p q p ¨ { q z Xτ . P M Proof. By Lemma 2.4(i) and (ii) in Definition 2.1, there exists J supp z with J k and r Ď p q | |“ det MJ 1 . The map Pt˘ u TN ´1 z ´MJ Ms |supppzq|´k ¨ y1,...,yN yJ ys T Tk Qr s ÝÑ s P supppzq´J P ´ ¯ 21 is a diffeomorphism with inverse

1, if j supp z , N R p q |supppzq|´k T z λs T λ t1z1,...,tN zN ¨ , where tj , if j js, Q ÝÑ r s P Tk ” $ zjs “ &’ 1 , if j J, zj P ’ and supp z J j1 ... j supp z k ; see the proof of Lemma 2.5(%h) in Section 2.1. p q´ ”t ă ă | p q|´ u Corollary 2.20. ( a) The TN-fixed points in Xτ are the points J of (2.13). M r s N τ τ ( b) The closed T -fixed curves in XM are the submanifolds XM J of Remark 2.10 with p q V τ J k 1; all such tuples J are of the form J I1 I2 with with I1,I2 M and | | “ ` “ 1 Y P I1 I2 k 1. These curves are biholomorphic to P . | X |“ ´ Proof. The first two statements follow from Lemma 2.19. The third follows from the last part of Remark 2.10. The last follows from Remarks 2.10 and 2.12.

N τ We next consider lifts of the standard action of T on XM to the line bundles Lp of τ (2.14) which will be used in describing the equivariant cohomology of XM . One such lift is the canonical one t1,...,tN z1,...,zN ,c t1z1,...,tN zN ,c (2.23) p q¨r s”r s N τ for all t1,...,tN T , z1,...,zN X , and c C. We denote by p qP p qP M P N N τ ET triv Lp ET TN X ˆr ÝÑ ˆ M the induced line bundle. Another lift is given by

t1,...,tN z1,...,zN ,c t1z1,...,tN zN ,tjc (2.24) p q¨r s”r s N τ for all t1,...,tN T , z1,...,zN X , and c C. We denote by p qP p qP M P N N τ ET j Lp ET TN X ˆr ÝÑ ˆ M the induced line bundle. These line bundles are related by isomorphisms

N N N ET triv Lp ET j L0 ET j Lp, (2.25) p ˆ qbp ˆ q– ˆ N ˚ ˚ ET j L0 pr π OP8 1 , (2.26) ˆ – 1 j p´ q N τ 8 N where pr1 :ET TN XM P denotes the natural projection. The ﬁrst of these follows by considering theˆ isomorphismÝÑ p q

τ Lp L0 Lp, z,c1 z,c2 z,c1c2 z X ,c1,c2 C b ÝÑ r sbr s ÝÑr s @ P M P N N which is T -equivariant with respect to the T action on Lp L0 obtained by tensoring b r (2.23) with (2.24) and the action (2.24) on Lp. The second is given by

N ˚ ˚ N τ ET j L0 e,z,c e,z,cej pr π OP8 1 e e1,...,eN ET ,z X ,c C. ˆ Q p q ÝÑ p q P 1 j p´ q @ “p qP P M P 22 r For all j N ,i k and with γi deﬁned by (2.16), let Pr s Pr s N N ˚ ˚ τ Dj ET j L M , γ ET triv γi; uj c1 Dj , xi c1 γ H N X . (2.27) r s” ˆ ´ j i ” ˆ ” pr sq ” p i qP T p M q For each J V τ , the inclusion J ã Xτ induces a restriction map P M r s Ñ M ˚ τ ˚ ˚ J , : H N X H N J H N . (2.28) ¨ p q ¨ J T p M q ÝÑ T pr sq – T ˇ By (2.17), (2.25) and (2.26), ˇ

uj mijxi αj j N . (2.29) “ i“1 ´ @ Pr s ÿ τ N For each j N , the section (2.15) of L´Mj XM is T -equivariant with respect to the Pr s ÝÑ N τ V τ action (2.24) and thus induces a section sj of Dj over ET TN XM . If J M and j J, N r s ˆ P P sj does not vanish on ET TN J and thus ˆ r s τ J V uj J 0 j J. (2.30) P M ùñ p q“ @ P τ τ On the other hand, if J E , with E deﬁned by (2.18), then sj is a nowhere zero P M M jPJ section of Dj and thus jPJ r s À À τ ˚ τ J EM uj 0 HTN XM . (2.31) P ùñ jPJ “ P p q ź Proposition 2.21. Let M,τ be a toric pair. p q τ ( a) If J j1 ... jk V , “p ă ă qP M ´1 x1 J x2 J ... xk J αj αj ... αj M . p q p q p q “ 1 2 k J ´ ¯ ´ ¯ ( b) With xi and uj deﬁned by (2.27),

˚ τ Q α x1,x2,...,xk,u1,u2,...,uN HTN XM r sr s . (2.32) p q“ k τ uj mijxi αj, 1 j N uj : J E ´ ` ď ď ` P M ˆ i“1 ˙ ˜jPJ ¸ ř ś ˚ τ τ If in addition P H N X , then P 0 if and only if P J 0 for all J V . P T p M q “ p q“ P M Proof. (a) This follows from (2.29) and (2.30). ě0 k p ˚ τ (b) By Remark 2.15, there exists B Z such that H : p B is a Q-basis for H XM . The map Ăp q t P u p q ˚ τ p p ˚ τ H X H x H N X p B p M q Q ÝÑ P T p M q @ P N τ 8 N defines a cohomology extension of the fiber for the fiber bundle ET TN XM P . Thus, by the Leray-Hirsch Theorem [Spa, Chapter 5], the map ˆ ÝÑ p q

˚ ˚ τ p p ˚ τ H N H X P H Px H N X p B T b p M q Q b ÝÑ P T p M q @ P 23 is an isomorphism of vector spaces. The relations in (2.32) hold by (2.29) and (2.31). We show below that there are no other relations and simultaneously verify the last claim. ˚ τ V τ Suppose P HTN XM , P J 0 for all J M . By (ii) in Deﬁnition 2.1, any element P ˚ τ P p q p q“ P V τ of HTN XM is a polynomial in u1,...,uN with coeﬃcients in Q α . If J M and j N J, then p q r s P Pr s´ uj J αj (2.33) p q αi“0 @ iPJ “´ ˇ a1 as by (2.29) and (a). By (2.30) and (2.33),ˇ whenever ui1 ...uis is a monomial appearing in P τ ˇ and J V , i1,...,is J . This shows that P M t uX ‰H ˚ τ τ 1 P H N X , P J 0 J V P H , P T p M q p q“ @ P M ùñ P where H1 is the ideal 1 τ H ui ...ui : i1,...,is J J V Q α u1,...,uN . ” 1 s t uX ‰H @ P M Ă r sr s 1 ´ τ ¯ Since H uj : J E by Lemma 2.4(i), Ďp jPJ P M q ś ˚ τ V τ τ P HTN XM , P J 0 J M P uj : J EM . P p q p q“ @ P ùñ P˜jPJ P ¸ ź ˚ τ τ By (2.31), this implies that P 0 H N X if P J 0 for all J V . “ P T p M q p q“ P M For every J V τ , let P M φJ uj. ” jPrNs´J ź By (2.19) and (2.30), τ τ φJ J e T J X , φJ I 0 I V J . (2.34) p q“ r s M p q“ @ P M ´t u Thus, by the Localization Theorem` (2.22),˘

˚ τ V τ PφJ P J P HTN XM ,J M , (2.35) Xτ “ p q @ P p q P ż M τ i.e. φJ is the equivariant Poincar´edual of the point J X . r sP M 2.4 Examples

Example 2.22 (the complex projective space PN´1 with the standard action of TN ). If M 1,..., 1 RN and τ Rą0, ” p q P P then N 2 2 τ ě0 N µM : C R, µM z z1 ... zN , P v R : v1 ... vN τ , ÝÑ p q“| | ` `| | M “ P ` ` “ ! ) M,τ is a minimal toric pair, Xτ CN 0 , ` ˘ p q M “ ´t u N τ N´1 2n´1 r 1 ˚ N´1 XM P S ?τ S , HTN P Q α1,...,αN x x αk . “ – p p qq{ p q– r sr s k“1p ´ q L ź 24 Example 2.23 (the Hirzebruch surfaces Fk P OP1 OP1 k ). If k 0, ” p ‘ p qq ě 001 1 1 M , τ , ” 1 1 0 k ” k 1 ˆ ˙ ˆ ` ˙ then k t1,t2 z1,z2,z3,z4 t2z1,t2z2,t1z3,t1t z4 , p q ¨ p q” 2 2 2 4 2 ` z3 ˘z4 µM : C R , µM z 2| | `2 | | 2 , ÝÑ p q” z1 z2 k z4 ˆ| | ` | | ` | | ˙ τ ě0 4 P v R : v3 v4 1, v1 v2 kv4 k 1 , M “ P ` “ ` ` “ ` M,τ is a minimal toric pair,! ` ˘ ) p q Xτ C4 C2 0 0 C2 , Xτ Xτ T2. M “ ´ ˆ Y ˆ M “ M ´ ¯ L The map r r τ „ bk X Fk, z1,z2,z3,z4 z1,z2 ,z3, z1,z2 z4 , (2.36) M Ñ r s ÝÑ r s p q ÝÑ 4 4 is a T -equivariant biholomorphism with respect“ to the action` of T on Fk given˘‰ by

bk bk t1,t2,t3,t4 z1,z2 ,z3,ϕ t1z1,t2z2 ,t3z3, t1y1,t2y2 t4ϕ y1,y2 , p q¨rr s s” r s p q ÝÑ p q 1 ” z1,z2 P ,z3 C,ϕ ´OP1 k ¯ı . @r s P P P p q rz1,z2s By Proposition 2.14, ˇ ˇ ˚ Q H1, H2, U1, U2, U3, U4 H Fk r s p q“ U1 H2, U2 H2, U3 H1, U4 H1 kH2 U1U2, U3U4 p ´ ´ ´ ´ ´ q ` p q Q H1, H2 2 r s . – H , H1 H1 kH2 2 p ` q Since the toric hypersurfaces` D2 and D3˘deﬁned by (2.12) intersect at one point, 2 H1H2 U2U3 1, H kH1H2 k. “ “ 1 “´ “´ The isomorphism (2.36) maps D4 onto E0 and D3 onto E8, where

E0 image of the section 1, 0 in Fk, ” p q E closure of the image of 0,σ in Fk, 8 ” p q where σ is any non-zero holomorphic section of OP1 k . Since V τ 1, 3 , 1, 4 , 2, 3 , 2, 4 , by Corollaryp q 2.20, the T4-ﬁxed points in Xτ are M “ tp q p q p q p qu M 1, 0, 1, 0 , 1, 0, 0, 1 , 0, 1, 1, 0 , and 0, 1, 0, 1 , r s r s r s r s 4 while the closed T -ﬁxed curves are all 4 toric hypersurfaces D1, D2, D3, and D4. By Proposition 2.21(b),

˚ Q α1,α2,α3,α4 x1,x2 H 4 Fk r sr s . T p q” x2 α1 x2 α2 , x1 α3 x1 kx2 α4 p ´ qp ´ q p ´ qp ` ´ q ´ ¯ 25 Example 2.24 (products). Let M1,τ1 and M2,τ2 be (minimal) toric pairs, where Mj is p q p q a kj Nj matrix. Define ˆ M1 0 M1 M2 . ‘ ” 0 M2 ˆ ˙ Then, M1 M2, τ1,τ2 is a (minimal) toric pair, p ‘ p qq P pτ1,τ2q P τ1 P τ2 , and Xpτ1,τ2q Xτ1 Xτ2 . M1‘M2 “ M1 ˆ M2 M1‘M2 “ M1 ˆ M2 N1`N2 Nj N1`N2 The projections πj :C C induce a T -equivariant biholomorphism ÝÑ r r r pτ1,τ2q „ τ1 τ2 X z π1 z , π2 z X X , M1‘M2 Qr s ÝÑ r p qs r p qs P M1 ˆ M2 TN1`N2 τ1 ´τ2 ¯ TN1 where the action of on XM1 XM2 is the product of the standard actions of on τ1 TN2 τ2 ˆ XM1 and of on XM2 . By (2.4) and Lemma 2.4(a)(g), V pτ1,τ2q V τ1 V τ2 . M1‘M2 “ M1 ˆ M2 pτ ,τ q TN1`N2 1 2 Thus, by Corollary 2.20(a), the -fixed points of XM1‘M2 are the points I1 , I2 for I V τj I pr s r sq all j Mj , with j defined by (2.13). By Corollary 2.20(b) and the second statement in P r s pτ ,τ q TN1`N2 1 2 Lemma 2.4(b), the closed -fixed curves in XM1‘M2 are all curves of the form C1 I2 Nj τj τj ˆr s and I1 C2, where Cj is any closed T -fixed curve in X and Ij V is arbitrary. r sˆ Mj P Mj In particular, PN1´1 ... PNs´1 is given by the minimal toric pair ˆ ˆ N1 columns Ns columns

1 1 ... 1 1 ... 0 0 ... 0 0 τ1 hkkikkj ... hkkikkj τ2 0 0 ... 0 0 0 0 ... 0 0 , τ ¨ ˛ Rą0 s. ¨ ˛ . (2.37) $ . .. . “ . P p q s rows’ . . . ˚ ‹ ’˚ ‹ ˚τs‹ &’˚ 0 0 ... 00 00 ... 0 0 ‹ ˚ ‹ ˚ ‹ ˝ ‚ ˚ 0 0 ... 0 0 ... 1 1 ... 1 1 ‹ ’˚ ‹ ’˝ ‚ By Proposition%’ 2.21(b), piq Q α , 1 i s, 1 j Ni x1,...,xs ˚ N1´1 Ns´1 j HTN P ... P r ď ď ď ď sr s. (2.38) N ˆ ˆ “ i piq ` ˘ xi αj , 1 i s ˜j“1 ´ ď ď ¸ ś ´ ¯ Remark 2.25. Let σ : N N be a permutation and M,τ be a (minimal) toric pair. Let r s ÝÑr s p q σ σ M miσpjq 1ďiďk M Id ” 1ďjďN ” ˝ be the matrix obtained from M by permuting` ˘ its columns as dictated by σ. Then M σ,τ σ´1 τ p τ q is a (minimal) toric pair as well and Id induces a biholomorphism between XM and XM σ σ (since µM σ µM Id ) equivariant with respect to “ ˝ N N T T , t1,t2,...,tN tσ 1 ,tσ 2 ,...,tσ N . ÝÑ p q ÝÑ p q p q p q 1 1 In particular, taking k 0 in Example 2.23 gives` - via (2.37) - P P˘ as expected, since the corresponding matrices“ diﬀer by a permutation of columns. ˆ

26 Chapter 3

Explicit Gromov-Witten formulas

τ For the remaining part of this dissertation, XM is the compact projective manifold deﬁned by (2.2), where M,τ is a minimal toric pair as in Deﬁnition 2.1. Theorem 3.5 in Section 3.2 İ İİ p q ˚ τ below computes the one-point GW generating functions Zη and Zη of (1.9) if η H XM is of p 2 τ P p q the form η H , where H1,..., Hk is the basis for H XM ; Z referred to in Proposition 2.14 and “ t u p q p p1 pk ě0 k H H ... H p p1,...,pk Z . ” 1 k @ “ p qPp q

We denote İ İ İİ İİ Zp ZHp and Zp ZHp . (3.1) ” ” İ İİ Section 3.1 constructs the explicit formal power series in terms of which Zp and Zp are expressed in Theorem 3.5. Throughout this construction, which extends the constructions n´1 τ in [Z1, Section 2.3] and [PoZ, Sections 2,3.1] from P to an arbitrary toric manifold XM , we assume that νE d 0 d Λ, (3.2) p qě @ P with νE as in (3.3), and identify τ k H2 X ; Z Z p M q– k via the basis H1,..., Hk . Via this identiﬁcation Λã Z , with Λ as in (1.6). t u Ñ 3.1 Notation and construction of explicit power series

If R is a ring, we denote by R ~ the ring of formal Laurent series in ~´1 with finite principal part: RV~W R ~´1 R ~ . ” rr ss ` r s Given f,g R ~ and s Zě0, we write P P V W V W s´1 ´s í f g mod ~ if f g R ~ ai~ : ai R i s 1 . – ´ P r s` #i“1 P @ Pr ´ s+ ÿ If R is a field, we view R ~ as a subring of R ~ by associating to each element of R ~ its Laurent series at ~´1 0.p q p q “ V W 27 ˘ τ With the line bundles L as in (1.5) and Uj as in (2.16) and d H2 X ; Z , we define i P p M q ˘ ˘ Dj d Uj, d , L d c1 L , d , p q” i p q” p i q N a b @ D ` @ ´ D3 (3.3) νE d Dj d Li d Li d . p q” j“1 p qí“1 p qì“1 p q ÿ ÿ ÿ If in addition Y X is a one-dimensional submanifold, let Ă ˘ ˘ Dj Y Dj Y Xτ , L Y L Y Xτ , p q” r s M i p q” i r s M τ Y τ H X Z ` ˘ ` Y ˘ where XM 2 M ; is the homology class represented by . By (2.20), our assumption (3.2), andr s FootnoteP p 2, q

a b τ ` ´ τ c1 TXM c1 Li c1 Li KM . p q´ i“1 p q` i“1 p q P ÿ ÿ ` τ Thus, if E E , then XM is Fano. In this case, the Cone Theorem [La, Theorem 1.5.33] implies that‰ the closed R-cone of curves is a polytope spanned by classes of rational curves. By [La, Proposition 1.4.28] and [La, Theorem 1.4.23(i)], this closed cone is the R-cone spanned by Λ. Thus, L´ d 0 for all d Λ 0 and all i b .4 i p qă P ´t u Pr s Let R be a ring. Similarly to Section 1.1, we denote by R Λ 0 and R Λ; νE 0 the subalgebras of R Λ given by rr ´ ss rr “ ss rr ss

d R Λ 0 adQ R Λ : a0 0 , rr ´ ss ” #dPΛ P rr ss “ + ÿ d R Λ; νE 0 adQ R Λ : ad 0 if νE d 0 . rr “ ss ” #dPΛ P rr ss “ p q‰ + ÿ In some cases, the formal variables whose powers are indexed by Λ within R Λ will be rr ss denoted by Q Q1,...,Qk as in Section 1.1, while in other cases the formal variables will ”p q d be q q1,...,qk . If f R Λ and d Λ, we write f q;d R for the coefficient of q in f. By Proposition” p 2.16,q theP setrr sss Λ: d Ps Λ is finite for everyP d Λ; thus, t P ´ P u J K P f R Λ is invertible f R is invertible. P rr ss ðñ q;0 P d If f fdq R Λ , we define J K ” dPΛ P rr ss ř f f qd R Λ; ν 0 . νE “0 d E ” dPΛ P rr “ ss J K νEÿpdq“0 3 ´ By (1.5) and (2.20), νEpdq “ c1pT pE |Y qq, d if Y is a smooth complete intersection defined by a ` ´ ´ holomorphic section of E and T pE |Y q is the tangent bundle of the total space of E |Y . 4 1 @τ 1,1 τ D 2 τ In the notation of [La], N pXM qR “ H pXM q X H pXM ; Rq as can be seen from PoincaréDuality, Lefschetz Theorem on p1, 1q-classes, and Hard Lefschetz Theorem.

28 d Let A A1,..., Ak be a tuple of formal variables. If f fd A q R A Λ and p 0, “p q ”dPΛ p q P rr ssrr ss ě we write ř d f A;p fd A A;p q R A p Λ , ” dPΛ p q P r s rr ss J K ÿ J K where fd A R A denotes the degree p homogeneous part of fd A and R A p the p q A;p P r s p q r s space of homogeneous polynomials of degree p in A1,..., Ak with coeﬃcients in R. Finally, we writeJ K ě0 k p p1 p2 ... pk p p1,p2,...,pk Z . | |” ` ` ` @ “ p qPp q For each d Λ, let P 0 k mijAi s~ jPrNs s“Dj pdq`1 i“1 ` D pdqă0 ˆ ˙ U d;A, ~ j ś ś ř Q A ~ . (3.4) p q” Dj pdq k P r s mijAi s~ ` V W jPrNs s“1 ˆi“1 ˙ Dj pśdqě0 ś ř

˚ τ By Proposition 2.17, the line bundles γi of (2.16) form a basis for the Picard group of XM . ˘ Thus, there are well-deﬁned integers ℓri such that

ℓ˘ ˚ℓ˘ L˘ γ˚ 1i ... γ ki . (3.5) i “ 1 b b k With A and d as above, let

` ´ a L pdq k b ´L pdq´1 k İ i i ` ´ E d; A, ~ ℓriAr s~ ℓriAr s~ Z A, ~ , p q” i“1 s“1 ˜r“1 ` ¸ i“1 s“0 ˜r“1 ´ ¸ P r s ź ź` ÿ ź ź´ ÿ (3.6) a L pdq´1 k b ´L pdq k İİ i i ` ´ E d; A, ~ ℓriAr s~ ℓriAr s~ Z A, ~ . p q” i“1 s“0 ˜r“1 ` ¸ i“1 s“1 ˜r“1 ´ ¸ P r s ź ź ÿ ź ź ÿ İ İİ The formal power series computing Zp and Zp in Theorem 3.5 are obtained from

İ İ Y A, ~,q qdU d;A, ~ E d;A, ~ Q A ~´1, Λ , p q” dPΛ p q p q P r s İİ ÿ İİ ““ ‰‰ (3.7) Y A, ~,q qdU d;A, ~ E d;A, ~ Q A ~´1, Λ . p q” dPΛ p q p q P r s ÿ ““ ‰‰ We deﬁne

İ İ İİ İİ ´1 ´1 I0 q Y A, ~,q mod ~ , I0 q Y A, ~,q mod ~ , (3.8) p q” p q p q” p q

29 and so a L` d ! İ i d i“1 p q I0 q 1 δb,0 q , p q” ` śN ` ˘ dPΛ´0, ν pdq“0 E Dj d ! D d ÿ0 j N j p qě @ Pr s j“1 p p q q ś b ´ b L d ! İİ L´pdq i d i i“1 ´ p q I0 q 1 q 1 i“1 . p q” ` p´ q ř ś N`` ˘ ˘ dPΛ´0, ν pdq“0 E Dj d ! D d ÿ0 j N j p qě @ Pr s j“1 p p q q L` d 0 i a i p q“ @ Pr s ś We next describe an operator Dp acting on a subset of Q A, ~ Λ and certain associated p qrrİ ss İİ “structure coefficients” in Q Λ which occur in the formulas for Zp and Zp. Fix an element Y A, ~,q of Q A, ~ Λ suchrr ss that for all d Λ p q p qrr ss P fd A, ~ Y A, ~,q q;d p q p q ” gd A, ~ p q for some homogeneous polynomialsJ fd A, ~ K,gd A, ~ Q A, ~ satisfying p q p qP r s f0 A, ~ g0 A, ~ , deg fd deg gd νE d , gd 0 d Λ. (3.9) p q“ p q ´ “´ p q A“0 ‰ @ P İ İİ This condition is satisfied by the power series Y and Y of (3.7) andˇ so the construction below İ İİ ˇ applies to Y Y and Y Y . We inductively define “ “ ě0 p ě0 k Jp Y EndQ Λ;ν 0 Q Λ; νE 0 A p p Z , D Y A, ~,q Q A, ~ Λ p Z p q P rr E “ ss rr “ ssr s @ P p q P p qrr ss @ Pp q satisfying ` ˘ ě0 k p p (P1) for every p Z with p p, Jp Y A A ; Pp q | |“ t p qup q q;0 “ (P2) there exist Cprq Cprq Y Q ΛJ with p, r KZě0 k and s Zě0 such that p,s ” p,sp q P rr ss Pp q P 8 DpY A, ~,q ~|p| Cprq q Ar~´s , (3.10) p q“ p,sp q s“0 0 k ÿ rPpÿZě q prq prq prq Cp,s 0 if s νE d r , Cp,s δp,rδ|r|,s if s p , Cp,|r| δp,r. (3.11) r zq;d“ ‰ p q`| | r zνE “0“ ď| | r zq;0“ 0 By (3.9), we can define J0 Y Q Λ; νE 0 and D Y Q A, ~ Λ by p q P rr “ ss P p qrr ss ´1 0 ´1 J0 Y 1 Y A, ~,q mod ~ , D Y A, ~,q J0 Y 1 Y A, ~,q . (3.12) t p qup q” p q p q”rt p qup qs p q Suppose next that p 0 and we have constructed an operator Jp Y and power series Dp1 Y for all p1 Zě0 k ěwith p1 p satisfying the above properties.p Forq each p Zě0 k with p p 1,P let p q | |“ P p q | |“ `

p 1 d péi D Y A, ~,q Ai ~ qi D Y A, ~,q Q A, ~ Λ , p q” supp p ` dqi p q P p qrr ss iPsuppppq | p q| ÿ " * (3.13) r p p ´1 Jp`1 Y A D Y A, ~,q mod ~ , t p qup q” r p q zA;p`1 r 30 k where e1,...,ek is the canonical basis for Z . By (P2), t u prq p 1 prq r dCpéi,p`1 r Jp`1 Y A Cpéi,pAi A qi A t p qup q“ supp p » ¨ ` dqi fi | p q| iPsuppppq |r|“p |r|“p`1 ÿ ÿ ÿ (3.14) – prq fl 1 préiq dCpéi,p`1 r Cpéi,p qi A , “ supp p » ` dqi fi |r|“p`1 iPsuppppq ˜ ¸ | p q| ÿ ÿ – fl where we set Cpréiq 0 if i supp r . By (3.14) and (3.11), péi,p ” R p q p p p Jp 1 Y A Q Λ; νE 0 A p 1 and Jp 1 Y A A ; (3.15) t ` p qup q P rr “ ssr s ` t ` p qup q q;0 “ 1 ě0 k in particular, Jp`1 Y is invertible. With cp;p1 q JQ Λ; νE 0 Kfor p, p Z with p , p1 p 1 givenp byq p q P rr “ ss P p q | | | |“ ` ´1 p p1 Jp 1 Y A cp;p1 q A , (3.16) t ` p qu p q” p q p1 Zě0 k, p1 p 1 Pp qÿ| |“ ` we define p p1 D Y A, ~,q cp;p1 q D Y A, ~,q . (3.17) p q” p q p q p1 Zě0 k, p1 p 1 Pp qÿ| |“ ` By (3.17) and the inductive assumption (3.10), r

8 DpY A, ~,q ~p`1 Cprq q Ar~´s , where p q“ p,sp q s“0 0 k ÿ rPpÿZě q prq 1 dC 1 prq cp;p préiq p éi,s C C 1 q , (3.18) p,s 1 p éi,s´1 i “ supp p ` dqi p1 Zě0 k, p1 p 1 iPsupppp1q ˜ ¸ Pp qÿ| |“ ` | p q| ÿ

préiq 1 i s where we set Cp éi,s´1 0 if supp r or 0. By the first property in (3.11) with p 1 “ 1 R p q “ 1 prq replaced by p ei with p p 1 and i supp p , Cp,s satisfies this property as well. By ´ | |“ ` P p q 1 1 1 the second property in (3.11) with p replaced by p ei with p p 1 and i supp p , prq prq ´ | |“ ` P p prqq Cp,s 0 if s p. Since Cp,p`1 δp,r whenever r p 1 by (3.18) and (3.16), Cp,s r zνE “0 “ ď “ | |“ ` also satisfies the second property in (3.11). By the second statement in (3.15) and (3.16), 1 1 cp;p1 q;0 δp,p1 . Thus, by the third property in (3.11) with p replaced by p ei with p p 1 “ 1 prq ´ | |“ ` and i supp p , Cp,s satisfies the last property in (3.11) as well. J KP pprq q prq Define C C Y Q Λ for p, s Zě0 k and r Zě0 with s p r and r p by p,s ” p,sp qP rr ss Pp q P | |ď| |´ ď| | r r r ptq prq ě0 k Cp,sCs,|r|`r´t δp,rδr,0 r Z , r p r. (3.19) t“0 ě0 k “ @ Pp q | | ď | |´ ÿ sPpÿZ q |s|ď|p|´tr

31 prq Equations (3.19) indeed uniquely determine Cp,s, since

r r´1 ptq prq ptqr prq prq prq prq Cp,sCs,|r|`r´t Cp,sCs,|r|`r´t Cp,sCs,|r| Cp,r, (3.20) t“0 ě0 k “ t“0 ě0 k ` ě0 k ` ÿ sPpÿZ q ÿ sPpÿZ q sPpÿZ q |s|ď|p|´tr |s|ď|p|´tr |s|ă|r| r r as follows from (3.11). By (3.19) together with the ﬁrst and third statements in (3.11),

prq Cp,s δp,sδr,0 . (3.21) r zq;0 “ By (3.19), (3.20), and induction on sr, | | p0q ě0 k C q δp,s p, s Z with s p . (3.22) p,sp q“ @ Pp q | |ď| | By (3.19), (3.20), the ﬁrst statement in (3.11), and induction on s and r, r | | prq Cp,s 0 if νE d r. (3.23) r zq;d “ p q‰ İ İİ İ İİ Remark 3.1. With Y , Y as inr (3.7) and I0, I0 as in (3.8),

İ İ İ İ 0 1 J0 Y 1 I0 q , D Y A, ~,q İ Y A, ~,q , p q p q“ p q p q“ I0 q p q ! İİ ) İİ İİ p qİİ 0 1 J0 Y 1 I0 q , D Y A, ~,q İİ Y A, ~,q , p q p q“ p q p q“ I0 q p q ! ) p q by (3.12). Deﬁne

p p1 pk d d d ě0 k A ~ q A1 ~ q1 ... Ak ~ qk p p1,...,pk Z . ` dq ” ` dq1 ` dq @ “ p qPp q " * " * " k * Remark 3.2. If νE d 0 for all d Λ 0 and Y A, ~,q Q A, ~ Λ satisfies (3.9), p qą ě0 P ´t u p q P p qrr ss then Jp Y Id for all p Z by (P1) above. Along with the first equation in (3.13), (3.16), (3.17), andp q“ induction onP p , this implies that | | d p DpY A, ~,q A ~ q Y A, ~,q p q“ ` dq p q " * for all p Zě0 k. Pp q Remark 3.3. Suppose p˚ Zě0, Y A, ~,q Q A, ~ Λ satisfies (3.9), and P p qP p qrr ss ˚ deg fd A, ~ deg gd A, ~ p d Λ 0 . ~ p q´ ~ p qă´ @ P ´t u By the same reasoning as in Remark 3.2, we again find that p p d Jp Y Id, D Y A, ~,q A ~ q Y A, ~,q , p q“ p q” ` dq p q " * for all p Zě0 and p Zě0 k such that p, p p˚. P Pp q | |ď 32 τ n´1 Remark 3.4. Let M,τ be the toric pair of Example 2.22 with N n so that XM P . Let p q “ “ a b ` ´ E OPn´1 ℓi OPn´1 ℓi ” i“1 p q‘ i“1 p q à à a b ` ´ ` ´ with a,b 0, ℓi 0 for all i a , ℓi 0 for all i b , and ℓi ℓi n. Thus, ě ą Pr s ă Pr s i“1 í“1 ď ř ř a b ` ´ νE d n ℓi ℓi d p q“ ˜ ´ i“1 ` i“1 ¸ ÿ ÿ for all d Zě0. By (3.7), P ` ´ a ℓi d b ´ℓi d´1 ` ´ 8 ℓ A s~ ℓ A s~ İ i ` i ´ Y A, ~,q qd i“1 s“1 i“1 s“0 , ś ś ` d ˘ ś ś ` ˘ p q“ d“0 n ÿ A s~ s“1 p ` q ` ´ a ℓi d´1 ś b ´ℓi d ` ´ 8 ℓ A s~ ℓ A s~ İİ i ` i ´ Y A, ~,q qd i“1 s“0 i“1 s“1 . ś ś ` d ˘ ś ś ` ˘ p q“ d“0 n ÿ A s~ s“1 p ` q ś By Remark 3.3,

p İ İ İ p d Jp Y Id, D Y A, ~,q A ~ q Y A, ~,q p b, p q“ p q“ ` dq p q @ ă " *p İİ İİ İİ p d Jp Y Id, D Y A, ~,q A ~ q Y A, ~,q p a. p q“ p q“ ` dq p q @ ă " * a b ` ´ If ℓi ℓi n, then i“1 ´i“1 ă

ř ř p p İ İİ İ İ İİ İİ p d p d Jp Y ,Jp Y Id, D Y A ~ q Y, D Y A ~ q Y p q p q“ “ ` dq “ ` dq " * " * a b ` ´ for all p by Remark 3.2. If ℓi ℓi n, then we follow [PoZ, (1.1)] and set i“1 ´i“1 “ ř ř ` ´ a ℓi d b ´ℓi d ℓ`w r ℓ´w r 8 i ` i ´ F w,q qd i“1 r“1 i“1 r“1 , p q” ś ś ` d ˘ ś ś ` ˘ d“0 n (3.24) ÿ w r r“1 p ` q

q d F wś,q p MF w,q 1 p q , Ip q M F 0,q . p q” ` w dq F 0,q p q” p q " *ˆ p q ˙ 33 By (3.13) and (3.17) above,

İ İ p p 1 p´b A Jp Y Ip´b q Id, D Y A, ~,q A M F ,q p b, p q“ p q p q“ Ip b q ~ @ ě ´ p q ˆ ˙ İİ İİ p p 1 p´a A Jp Y Ip´a q Id, D Y A, ~,q A M F ,q p a. p q“ p q p q“ Ip a q ~ @ ě ´ p q ˆ ˙ 3.2 Statements

The statements and proofs of the theorems below rely on the one-point mirror formula (5.2) below, which is proved in [LLY3]. We begin by defining the mirror map occurring in this formula. For each i k , let Pr s İ U d;A, 1 E d;A, 1 1 d B p q p q fi q İ q Q Λ 0 , (3.25) ! A ) A“0 p q” I0 q dPΛ i P rr ´ ss p νq pdq“0 B ˇ Eÿ ˇ İ ˇ with I0 q defined by (3.8). The mirror map is the change of variables q Q, where p q ÝÑ f1pqq fkpqq Q1,...,Qk q1e ,...,qke . (3.26) p q“ Finally, let ` ˘ a ` Li d ! δb,0 d i“1 p q G q İ q Q Λ 0 . (3.27) p q” śN ` ˘ P rr ´ ss I0 q dPΛ, ν pdq“1 E Dj d ! p Dq d ÿ0 j N j p qě @ Pr s j“1 p p q q ś İ İİ Theorem 3.5. If νE d 0 for all d Λ, then Zp and Zp of (3.1) and (1.9) are given by p qě P k 1 İ ´ ~ Gpqq` Hifipqq İ « i“1 ff ˚ τ ´1 Zp ~,Q e Yp H, ~,q H X ~ Λ , p q“ ř p qP p M qr srr ss k 1 İİ ´ ~ Gpqq` Hifipqq İİ « i“1 ff ˚ τ ´1 Zp ~,Q e Yp H, ~,q H X ~ Λ , p q“ ř p qP p M qr srr ss where

|p| |p|´r İ İ İ İ p prq |p|´r´|s| s Yp A, ~,q D Y A, ~,q C q ~ D Y A, ~,q Q A, ~ Λ , p q” p q` p,sp q p qP p qrr ss r“1 |s|“0 ÿ ÿ (3.28) |p| |p|´r İİ İİ İİ r İİ p prq |p|´r´|s| s Yp A, ~,q D Y A, ~,q C q ~ D Y A, ~,q Q A, ~ Λ , p q” p q` p,sp q p qP p qrr ss r“1 |s|“0 ÿ ÿ İ İİ prq prq İ prq prq İİr with Cp,s Cp,s Y and Cp,s Cp,s Y deﬁned by (3.19), Q and q related by the mirror ” p q ” p q p map (3.26), G and fi given by (3.27) and (3.25), and the operator D deﬁned by (3.13) and (3.17).r r r r

34 İ İ İ r p d p p q ` If p b, D Y A ~ q dq Y and Cp,s 0 for all r p . If p a and Li d 1 for | |ă “t ` u “ Pr|İİ |s | |ă p qě İİ İİ prq all i a and d Λ 0 , then DpY A ~ q d pY and C 0 for all r p . Pr s P ´t u “t r` dq u p,s “ Pr| |s This follows from Theorem 4.7 together with (4.18) andr (EP1) below; Theorem 4.7 is an equivariant version of Theorem 3.5.

İ İİ Remark 3.6. In the inductive construction of DpY with Y Y or Y Y , the ﬁrst equation in (3.13) may be replaced by “ “

p d péi D Y A, ~,q cp;i Ai ~ qi D Y A, ~,q Q A, ~ Λ p q” ` dqi p qP p qrr ss iPsuppppq ÿ " * r for any tuple cp;i iPsuppppq of rational numbers with cp;i 1. The endomorphism p q iPsuppppq “ p Jp 1 Y and the power series D Y defined by the secondř equation in (3.13) and (3.17) in ` p q terms of the new “weighted” DpY satisfy (P1) and (P2) by the same arguments as in the 1 case when cp;i for all i supp p . Therefore, (3.19) continues to define power series “ |suppppq| P p q prq r prq Cp,s Y in terms of the “new weighted” Cp,s Y . The resulting “weighted” power series Yp p q p q ˚ τ of (3.28) do not depend on the “weights” cp;i as elements of H X ~ Λ ; this follows p M q rr ss fromr Remark 4.10. V W Corollary 3.7. If νE d 0 or νE d p for all d Λ 0 , then p q“ p qą| | P ´t u k k 1 1 İ ´ ~ Gpqq` Hifipqq İ İİ ´ ~ Gpqq` Hifipqq İİ « i“1 ff p « i“1 ff p Zp ~,Q e D Y H, ~,q , Zp ~,Q e D Y H, ~,q , p q“ ř p q p q“ ř p q with Q and q related by the mirror map (3.26) and G and fi given by (3.27) and (3.25). This follows from Theorem 3.5 together with (3.23). Let pr :Xτ Xτ Xτ denote the projection onto the i-th component. i M ˆ M ÝÑ M ˚ p ˚ s Corollary 3.8. Let gps Q be such that gpspr1 H pr2 H is the Poincarédual to P |p|`|s|“N´k τ τ the diagonal class in XM , where N k is theř complex dimension of XM . If N k and ´ İ ą νE d N k for all d Λ 0 , then the two-point function Z of (1.8) is given by p qą ´ P ´t u p s İ İ İİ 1 ˚ d ˚ d Z ~1, ~2,q gpspr1 H ~1 q Y H, ~1,q pr2 H ~2 q Y H, ~2,q . p q“ ~1 ~2 ` dq p q ` dq p q |p|`|s|“N´k ` ÿ " * " * This follows from Theorem 1.3, Corollary 3.7, and Remark 3.2.

Remark 3.9. If a k ` ℓriAr P A i“1 ˆr“1 ˙ Q A , b k p q” ś ř ´ P r s ℓriAr i“1 ˆr“1 ˙ ś ř 35 then İ ˚ ˚ ˚ Z ~1, ~2,Q Z ~1, ~2,Q pr1 P H , p q“ p q p q İ τ τ τ ˚ ˚ where pr1 :XM XM XM is the projection onto the first component, while Z and Z are as in Remark 1.4.ˆ ViaÝÑ Theorem 1.3, this expresses the two-point function Z˚ in terms of the İ İİ ˚ one-point functions Zη, Zη. In this case and if νE d 0 for all d Λ, Z can be computed İ İİ p qě P explicitly in terms of Y and Y via Theorem 3.5. We next use an idea from [CoZ] to express Z˚ in terms of one-point GW generating functions and then show how to compute the latter in the b 0 case. If pr :Xτ Xτ Xτ ą i Mˆ M ÝÑ M and gps Q are as in Corollary 3.8, then P ˚ 1 ˚ p ˚ ˚ Z ~1, ~2,Q gps pr1 H pr2 Zs ~2,Q p q“ ~1 ~2 p q ` |p|`|s|“N´k (3.29) ÿ “ İİ ˚ ˚ ˚ pr Z ~1,Q pr Zs ~2,Q , ` 1 pp q 2 p q where ‰ ˚ p ˚ d e VE ev2 H ˚ τ ´1 Z ~,Q Q ev1 p q H X ~ Λ p ˚ ~ ψ M p q” dPΛ´0 1 P p qr srr ss ÿ „ ´  τ τ and ev1 : M0,2 XM , d XM . This follows from (4.27). p qÝÑ ˚ We next assume that b 0 and νE d 0 for all d Λ and express Zp ~,Q in terms of explicit power series. Alongą with (3.29)p andqě Theorem 3.5,P this will concludep theq computation of Z˚. With U d;A, ~ given by (3.4), we define p q ` ´ a Li pdq k b ´Li pdq k d ` ´ Y A, ~,q q U d;A, ~ ℓriAr s~ ℓriAr s~ . (3.30) p q”dPΛ p q i“1 s“1 ˜r“1 ` ¸ i“1 s“1 ˜r“1 ´ ¸ ÿ ź ź ÿ ź ź ÿ r r p p p q p q As Y satisfies (3.9), we may define D Y and Cp,s Cp,s Y by (3.17) and (3.19). We define İ ” pİ q prq prq Y A, ~,q by the right-hand side of (3.28) above, with Y replaced by Y and C by C . p p p r r p p,s p,s pLet q p ` ´ p r r a Li pdq k b ´Li pdq´1 k ˚ d ` ´ Y A, ~,q q U d;A, ~ ℓriAr s~ ℓriAr s~ . (3.31) p q”dPΛ´0 p q i“1 s“1˜r“1 ` ¸ i“1 s“1 ˜r“1 ´ ¸ ÿ ź ź ÿ ź ź ÿ r We define Eprq Q Λ by p,s P rr ss |p|´b |p|´b´s d p A ~ q Y ˚ A, ~,q Eprq Ar~s mod ~´1. (3.32) ` dq p q– p,s s“0 |r|“0 " * ÿ ÿ r prq It follows that Ep,s 0 unless p b s νE d r . Whenever b 2, r zq;d “ | |“ ` ` p q`| | ě

p |p|´b |p|´b´s ˚ ` d ˚ prq s Z ~,q e E H ~ q Y H, ~,q E ~ Yr H, ~,q . (3.33) pp q“ p q » ` dq p q´ p,s p qﬁ s“0 |r|“0 " * ÿ ÿ – r p ﬂ 36 If b 1 and Q and q are related by the mirror map (3.26), “

´ p epE qf0pqq d Z˚ ~,Q e E` e´ ~ H ~ q Y ˚ H, ~,q pp q“ p q ` dq p q « " * (3.34) |p|´b |p|´b´s ` p r 8 ´ n prq s e E H f0 q 1 e E f0 q E ~ Yr H, ~,q p q p q p q p q , ´ p,s p q ´ ~ n 1 ! ´ ~ s“0 |r|“0 ﬀ n“0 ÿ ÿ ÿ p ` q „  p where a ` Li d ! d L´pdq`1 ´ i“1 p q 5 f0 q q 1 1 L d 1 ! . (3.35) p q” p´ q ´ 1 p q´ śN ` ˘ dPΛ´0,ν pdq“0 E Dj d ! D d ÿ0 j N j p qě @ Pr s ` ˘ j“1 p p qq Identities (3.33) and (3.34) follow by setting α 0 in (4.35) andś (4.36). τ n´1 “ As in [CoZ], if XM P and b 2, (3.33) can be be replaced by a simpler formula in terms of the power series“F w,q in (3.24)ě above. Assume that E Pn´1 is as in Remark 3.4 a b p q ÝÑ ` ´ and ℓi ℓi n. Similarly to [CoZ], i“1 ´i“1 “

ř ř p İ p e E` H ~ q d Y H, ~,q H , if p b, ˚ ~ ` dq p q´ ă Zp ,q p q Mp´bF H ,q p q“ e E´ ˆ $H! p p ~) q Hp, if p b, p q & Ip´bpqq ´ ě where the right-hand side should be% ﬁrst simpliﬁed in Q H, ~ q to eliminate division by H and only afterwards viewed as an element in H˚ Pn´p 1 ~qrr´1 ssq . This follows from Remarks 4.4 and 3.4 together with Theorem 4.7. By Theoremp qr 3.5 srrandss Remark 3.4,

p İİ d İİ H ~ q Y H, ~,q , if p a, Z ~,q ` dq p q ă p M p´aF H ,q p q“ $H! p p ~ )q , if p a. & Ip´apqq ě The last two displayed equations% together with (3.29) imply that

a ` 8 ℓi dqd e V ev˚Hc1 ev˚Hc2 i“1 I q 1 and I I , (3.36) E 1 2 b c1`1´b c1`1´b c2`1´b n´1 ś d“1 M0,2pP ,dqp q “ ´ p p q´ q “ ÿ ż ℓi i“1 ś whenever c1 c2 n 2 a b and with Ip q deﬁned by (3.24) if p 0 and Ip q 1 if p 0. ` “ ´ ´ ` p q ě p q” ă

5 ´ ´ In this case, fipqq “ ℓi1f0pqq with ℓi1 PZ given by (3.5).

37 Chapter 4

Equivariant theorems

İ In this chapter we introduce equivariant versions of the GW generating functions Z, İ İİ Zη, and Zη of (1.8) and (1.9). We then present theorems about them which imply the non-equivariant statements of Section 3.2. N ˚ τ With α α1,...,αN denoting the T -weights of Section 2.3, H N X is generated ” p q T p M q over Q α by x1,...,xk ; see Proposition 2.21. The classes xi of (2.27) satisfy r s t u 2 τ restriction 2 τ ˚ H N X xi Hi H X i k , e γ xi i k , T p M qQ ÝÝÝÝÝÝÑ P p M q @ Pr s p i q“ @ Pr s ˚ N τ ˚ where e γi is defined by the lift (2.23) of the action of T on XM to the line bundle γi . Let p q p p1 pk ě0 k x x1,...,xk , x x ... x p p1,...,pk Z . ” p q ” 1 ¨ ¨ k @ “ p qP N τ τ The action of T on XM induces an action on M0,m XM , d which` lifts˘ to an action on İ İİ p q the vector orbi-bundles VE, VE, and VE of (1.1) and (1.7). It also lifts to an action on the universal cotangent line bundle to the i-th marked point whose equivariant Euler class will τ τ N also be denoted by ψi. The evaluation maps evi :M0,m XM , d XM are T -equivariant. τ τ p qÝÑ With ev1, ev2 :M0,3 XM , d XM denoting the evaluation maps at the first two marked points, let p qÝÑ İ İ ~1~2 d e VE Z ~1, ~2,Q Q ev1 ev2 p q . (4.1) ~ ~ ˚ ~ ψ ~ ψ p q” 1 2 dPΛ p ˆ q « 1 1 2 2 ff ` ÿ p ´ q p ´ q τ τ With ev1, ev2 :M0,2 XM , d XM denoting the evaluation maps at the two marked points ˚ p τ qÝÑ and for all η H N X , let P T p M q İ İ ˚ d e VE ev2 η ˚ τ ´1 Zη ~,Q η Q ev1˚ p q HTN XM ~ , Λ , p q” ` ~ ψ1 P p qrr ss dPΛ´0 « ´ ff ÿ İİ (4.2) İİ ˚ d e VE ev2 η ˚ τ ´1 Zη ~,Q η Q ev1 p q H N X ~ , Λ . ˚ ~ ψ T M p q” `dPΛ´0 « 1 ffP p qrr ss ÿ ´ İ İİ p In the η x cases, these are equivariant versions of Zp and Zp in (3.1): “ İ İ İİ İİ ˚ τ ´1 Zp ~,Q Zxp ~,Q , Zp ~,Q Zxp ~,Q H N X ~ , Λ . (4.3) p q” p q p q” p qP T p M qrr ss 38 İ İ k ˚ τ In particular, Z0 Z1, with 0 Z and 1 HTN XM . ” P P p q İ İİ Section 4.1 below constructs the explicit formal power series in terms of which Zp and Zp are expressed in Theorem 4.7. Throughout this construction, which extends the constructions n´1 τ in [Z1, Section 2.3] and [PoZ, Section 3.1] from P to an arbitrary toric manifold XM , we τ k assume that νE d 0 for all d Λ and identify H2 XM ; Z Z via the basis H1,..., Hk of H2 Xτ ; Z . Viap qě this identificationP Λã Zk. p q– t u p M q Ñ 4.1 Construction of equivariant power series

İ İİ İ İİ We begin by deﬁning equivariant versions Y and Y of the power series Y and Y in (3.7) İ İİ N as these will compute Zp and Zp in Theorem 4.7. We consider the lift (2.23) of the T -action τ ˘ on XM to the line bundles Li of (1.5) and (3.5) so that

k ˘ ˘ ˘ λi e Li ℓrixr. (4.4) ” p q“ r“1 ÿ An equivariant version of the power series U d;A, ~ in (3.4) is given by p q 0 k mijAi αj s~ jPrNs s“Dj pdq`1 i“1 ´ ` D pdqă0 ˆ ˙ u d;A, ~ j ś ś ř Q α, A ~ . (4.5) p q” Dj pdq k P r s mijAi αj s~ ´ ` V W jPrNs s“1 î“1 ˙ Dj pśdqě0 ś ř By (2.29), 0 uj s~ jPrNs s“Dj pdq`1 p ` q D pdqă0 u d; x, ~ j ś ś . (4.6) p q“ Dj pdq uj s~ jPrNs s“1 p ` q Dj pśdqě0 ś İ İİ With E d;A, ~ and E d;A, ~ as in (3.6), let p q p q İ İ Y A, ~,q qdu d;A, ~ E d;A, ~ Q α, A ~´1, Λ , p q” dPΛ p q p q P r srr ss İİ ÿ İİ (4.7) Y A, ~,q qdu d;A, ~ E d;A, ~ Q α, A ~´1, Λ . p q” dPΛ p q p q P r srr ss ÿ İ İİ In the above definitions of Y and Y and throughout the construction below, the torus weights α should be thought of as formal variables, in the same way in which A of Section 3.1 İ İİ are formal variables. With A replaced by x, Y and Y become well-defined elements in ˚ τ ´1 HTN XM ~ , Λ . However, this is irrelevant for the purposes of this section and becomes p qrr ss İ İİ İ İİ relevant only when we use Y and Y in the formulas for Zp and Zp.

39 p As before, Qα Q α . We next describe an operator D acting on a subset of Qα A, ~ Λ and certain associated” p q “equivariant structure coeﬃcients” in Q α Λ which occurp inqrr thess İ İİ r srr ss formulas for Zp and Zp. Fix an element Y A, ~,q Qα A, ~ Λ such that for all d Λ p qP p qrr ss P

fd A, ~ Y A, ~,q q;d p q p q ” gd A, ~ p q J K for some homogeneous polynomials fd A, ~ ,gd A, ~ Q α, A, ~ , symmetric in α, and satisfying p q p q P r s

f0 A, ~ g0 A, ~ , deg fd deg gd νE d , gd A“0 0 d Λ. (4.8) p q“ p q ´ “´ p q α“0 ‰ @ P ˇ İ İİ ˇ This condition is satisfied by the power series Y and Y of (4.7) andˇ so the construction below İ İİ p applies to Y Y and Y Y. We inductively define D Y A, ~,q in Qα A, ~ Λ satisfying “ “ p q p qrr ss (EP1) with Dp defined in Section 3.1,

DpY A, ~,q Dp Y A, ~,q ; p q α“0 “ p q α“0 ˇ ´ ˇ ¯ ˇ ˇ prq prq ˇ ě0 k ˇ ě0 prq (EP2) there exist Cp,s Cp,s Y Q α Λ with p, r Z , s Z , such that Cp,s is ” p qP r srr ss Pp q P r zq;d a homogeneous symmetric polynomial in α of degree νE d r s, ´ p q´| |` 8 DpY A, ~,q ~|p| Cprq q Ar~´s, (4.9) p q“ p,sp q s“0 r Zě0 k ÿ Ppÿ q prq ě0 k ě0 C δp,rδ r ,s p, r Z , s Z . (4.10) p,s q;0“ | | @ Pp q P q y By (4.8), (3.12), and since J0 Y 1 1 by (P1), we can deﬁne t p α“0qup q q;0 “ q y 0 ˇ ´1 D Y A, ~,q Jˇ0 Y 1 Y A, ~,q Qα A, ~ Λ . (4.11) p q” t p α“0qup q p qP p qrr ss “ ˇ ‰ p1 Suppose next that p 0 and weˇ have constructed power series D Y A, ~,q for all p1 Zě0 k with p1 p satisfyingě the above properties. For each p Zě0 k withp pq p 1, letPp q | |“ Pp q | |“ `

p 1 d péi D Y A, ~,q Ai ~ qi D Y A, ~,q Qα A, ~ Λ , p q” supp p ` dqi p qP p qrr ss iPsuppppq | p q| ÿ " * (4.12) r p p1 D Y A, ~,q cp;p1 q D Y A, ~,q , p q” p q p q p1 Zě0 k, p1 p 1 Pp qÿ| |“ ` r where cp;p1 q Q Λ; νE 0 are defined by (3.16) with Y Y and where e1,...,ek is p qP rr “ ss ” α“0 t u the standard basis of Zk. Since (EP1) holds with p replaced by any p1 with p1 p, ˇ ˇ | |“ DpY A, ~,q Dp Y A, ~,q p q α“0 “ p q α“0 ˇ ` ˇ ˘ r ˇ 40r ˇ by (3.13); thus, by the second equation in (4.12) and (3.17), DpY satisfies (EP1). It is p immediate to verify that D Y A, ~,q admits an expansion as in (4.9). Since cp;p1 δp,p1 p q q;0 “ by the second statement in (3.15) and (3.16) and (4.10) holds for p ei with i supp p instead of p, (4.10) also holds for p with p p 1. ´ J PK p q By (EP1), (3.10), and (4.9), | |“ `

prq prq Cp,s Y Cp,s Y , (4.13) p q α“0 “ α“0 ˇ ´ ˇ ¯ prq ˇ ˇ with Cp,s as in (P2). ˇ prq prq ě0 k ě0 Deﬁne Cp,s Cp,s Y Q α Λ for p, s Z and r Z with s p r and r p by ” p q P r srr ss P p q P | | ď | |´ ď | | r r r Cptq Cprq ě0 k p,s s,|r|`r´t δp,rδr,0 r Z , r p r. (4.14) t“0 ě0 k “ @ Pp q | |ď| |´ ÿ sPpÿZ q |s|ď|p|´t r prq Equations (4.14) indeed uniquely determine Cp,s, since

r r´1 Cptq Cprq Cptrq Cprq CprqCprq Cprq p,s s,|r|`r´t p,s s,|r|`r´t p,s s,|r| p,r. (4.15) t“0 ě0 k “ t“0 ě0 k ` ě0 k ` ÿ sPpÿZ q ÿ sPpÿZ q sPpÿZ q |s|ď|p|´t r |s|ď|p|´t r |s|ă|r| r r This follows from prq C δp,r if r p , (4.16) p,|r| “ | |ď| | which in turn follows from (4.13), the second equation in (3.11), and the ﬁrst property in (EP2). By (4.14) and (4.10), prq Cp,s δp,sδr,0. (4.17) r zq;0 “ By (4.13), (3.19), (3.20), (4.14), (4.15),r and induction,

prq Cprq Y C Y . (4.18) p,sp q α“0 “ p,s α“0 ˇ ` ˇ ˘ By (4.17) in the d 0 case andr (4.14),ˇ (4.15),r (EP2),ˇ and induction in all other cases, prq “ Cp,s q is a degree r νE d homogeneous symmetric polynomial in α. In particular, p q q;d ´ p q rp0q z Cp,s q Q Λ . This together with (4.18) and (3.22) implies that, r p qP rr ss p0q ě0 k r Cp,s q δp,s p, s Z with s p . (4.19) p q“ @ Pp q | |ď| | r İ İ İİ İİ Remark 4.1. By (4.11), Y Y , Y Y , and Remark 3.1, α“0 “ α“0 “ İ ˇ İ ˇ İİ İİ 0 ˇ 1 ˇ 0 1 D Y A, ~,q İ Y A, ~,q , D Y A, ~,q İİ Y A, ~,q . p q“ I0 q p q p q“ I0 q p q p q p q

41 Remark 4.2. If νE d 0 for all d Λ 0 and Y A, ~,q Qα A, ~ Λ satisfies (4.8), p qą P ´t u p q P p qrr ss then p p d ě0 k D Y A, ~,q A ~ q Y A, ~,q p p1,...,pk Z . p q“ ` dq p q @ “p qPp q " * This follows by induction on p from (4.12) since cp;p1 Y δp,p1 with cp;p1 defined by | | p α“0q“ (3.16). The latter follows since J Y Id by Remark 3.2. p α“0 ˇ p q“ ˇ ˚ ě0 Remark 4.3. Suppose p Z , Y Aˇ , ~,q Qα A, ~ Λ satisfies (4.8), and P p ˇ qP p qrr ss ˚ deg fd A, ~ deg gd A, ~ p d Λ 0. ~ p q´ ~ p qă´ @ P ´ By the same reasoning as in Remark 4.2, but using Remark 3.3 instead of 3.2,

d p DpY A, ~,q A ~ q Y A, ~,q if p p˚. (4.20) p q“ ` dq p q | |ď " * By (4.14), (4.15), and (4.19),

r´1 Cprq Cptq Cprq CprqCprq Cprq p,|r|`r p,s s,|r|`r´t p,s s,|r| p,r 0 (4.21) `t“1 ě0 k ` ě0 k ` “ ÿ sPpÿZ q sPpÿZ q |s|ď|p|´t r |s|ă|r| r r if r 1 and r p r. By (4.20) and (4.10), ě | |ď| |´ DpY A, ~,q Ap mod ~´1 if p p˚. p q– | |ď This together with (4.9) implies that whenever p p˚, | |ď Cprq 0 if r 1 and r p r. (4.22) p,|r|`r “ ě | |ď| |´ Finally, using (4.21), (4.22), and induction, we ﬁnd that

Cprq 0 if r 1, p p˚, s p r. p,s “ ě | |ď | |ď| |´ τ n´1 Remark 4.4. Let M,τr be the toric pair of Example 2.22 with N n so that XM P and E Pn´1 be asp in Remarkq 3.4. By (4.7), “ “ ÝÑ ` ´ a ℓi d b ´ℓi d´1 ` ´ 8 ℓ A s~ ℓ A s~ İ i ` i ´ Y A, ~,q qd i“1 s“1 i“1 s“0 , ś ś ` n d ˘ ś ś ` ˘ p q“ d“0 ÿ A αj s~ j“1 s“1 p ´ ` q ` ´ a ℓi d´1 ś ś b ´ℓi d ` ´ 8 ℓ A s~ ℓ A s~ İİ i ` i ´ Y A, ~,q qd i“1 s“0 i“1 s“1 . ś ś ` n d ˘ ś ś ` ˘ p q“ d“0 ÿ A αj s~ j“1 s“1 p ´ ` q ś ś 42 By Remark 4.3, p İ d İ İ DpY A, ~,q A ~ q Y A, ~,q and Cprq Y 0 p b, 1 r p, p q“ ` dq p q p,s p q“ @ ă ď ď " *p İİ d İİ İİ DpY A, ~,q A ~ q Y A, ~,q and Crprq Y 0 p a, 1 r p. p q“ ` dq p q p,s p q“ @ ă ď ď " * a b ` ´ r If ℓi ℓi n, then i“1 ´i“1 ă ř ř p p İ d İ İİ d İİ DpY A ~ q Y, DpY A ~ q Y, “ ` dq “ ` dq " * " * a b ` ´ for all p by Remark 4.2. If ℓi ℓi n, then i“1 ´i“1 “

ř b ř İ 1 d İ İ 1 d İ DbY A ~ q Y, DpY A ~ q Dp´1Y p b, “ I0 q ` dq “ Ip´b q ` dq @ ą p q " * a p q " * İİ 1 d İİ İİ 1 d İİ DaY A hq Y, DpY A ~ q Dp´1Y p a, “ I0 q ` dq “ Ip a q ` dq @ ą p q " * ´ p q " * by (4.12) and Remark 3.4.

4.2 Equivariant statements

τ τ τ Theorem 4.5. Suppose M,τ is a minimal toric pair and pri : XM XM XM is the p q ˚ τ ˆ ÝÑ projection onto the i-th component. If ηj, ηj H N X are such that P T p M q s ˚ ˚ 2pN´kq τ τ pr1 ηj pr2 ηj q HTN XM XM j“1 P p ˆ q ÿ is the equivariant Poincar´edual of the diagonal,q then s İ İ İİ 1 ˚ ˚ Z ~1, ~2,Q pr1 Zηj ~1,Q pr2 Zηj ~2,Q . (4.23) p q“ ~1 ~2 p q p q ` j“1 ÿ q s s τ Ni´1 Corollary 4.6. Let M,τ be the minimal toric pair (2.37) so that XM P ,N Ni, p q “i“1 “i“1 s ˚ Ni´1 τ τ τ ś ř and HTN P is given by (2.38). Let prj :XM XM XM denote the projection onto i“1 ˆ ÝÑ ˆ ˙ ě0 piq the j-th component.ś For all i s and r Z , denote by σr the r-th elementary symmetric Pr s P αpiq,...,αpiq polynomial in 1 Ni . Then,

s İ r İ İİ 1 i p1q psq ˚ ˚ Z ~ ~ i“1 Z ~ Z ~ 1, 2,Q 1 σr1 ...σrs pr1 pa1,...,asq 1,Q pr2 pb1,...,bsq 2,Q . p q“ ~1 ~2 p´ q ř p q p q ` ri`ai`bi“Ni´1 @ iPrss ri,ai,biÿě0 @ iPrss

43 This follows from Theorem 4.5 as the equivariant Poincar´edual to the diagonal in s PNi´1 is i“1

ś s ri p1q psq ˚ a1 a ˚ b1 b 1 i“1 σ ...σ pr x ...x s pr x ...x s . p´ q ř r1 rs 1 p 1 s q 2 p 1 s q ri`ai`bi“Ni´1 @ iPrss ri,ai,biÿě0 @ iPrss

İ Theorem 4.7. Let M,τ be a minimal toric pair. If νE d 0 for all d Λ, then Zp and İİ p q p qě P Zp of (4.3) and (4.2) are given by

k N 1 İ ´ ~ Gpqq` xifipqq` αj gj pqq İ « i“1 j“1 ﬀ ˚ τ Zp ~,Q e Yp x, ~,q H N X ~ Λ , p q“ ř ř p qP T p M q rr ss (4.24) k N 1 İİ ´ ~ Gpqq` xifipqq` αj gj pqq İİ V W « i“1 j“1 ﬀ ˚ τ Zp ~,Q e Yp x, ~,q H N X ~ Λ , p q“ ř ř p qP T p M q rr ss where V W

|p| |p|´r İ İ İ İ p prq |p|´r´|s| s Yp x, ~,q D Y x, ~,q C q ~ D Y x, ~,q , p q” p q` p,sp q p q r“1 |s|“0 ÿ ÿ (4.25) |p| |p|´r İİ İİ İİ r İİ p prq |p|´r´|s| s Yp x, ~,q D Y x, ~,q C q ~ D Y x, ~,q , p q” p q` p,sp q p q r“1 |s|“0 ÿ ÿ İ İİ r İ İİ prq prq prq prq with Cp,s Cp,s Y and Cp,s Cp,s Y defined by (4.14), Q and q related by the mirror map ” p q ” p q 6 (3.26), G, fi and gj Q Λ 0; νE 0 given by (3.27), (3.25), and (5.1), and the operator P rr ´ “ ss İ İİ p r r r r d prq prq D defined by (4.12). The coefficient of q within each of Cp,s and Cp,s is a degree r νE d homogeneous symmetric polynomial in α ,...,α . ´ p q 1İ N İ İ pY d pY Cprq r r ` If p b, D A ~ q dq and p,s 0 for all r p . If p a and Li d 1 for İİ | |ă “t ` u İİ “ İİ Pr| |s | |ă p qě p d p prq all i a and d Λ 0 , then D Y A ~ q Y and Cp,s 0 for all r p . Pr s P ´t u “t r` dq u “ Pr| |s

ě0 k r Corollary 4.8. If p Z and max p , 1 νE d for all d Λ 0 , then Pp q p| | qă p q P ´t u p p İ d İ İİ d İİ Zp ~,q x ~ q Y x, ~,q , Zp ~,q x ~ q Y x, ~,q . p q“ ` dq p q p q“ ` dq p q " * " * This follows from Theorem 4.7 and Remark 4.2.

˚ p ˚ s Corollary 4.9. Let gps Q α be homogeneous polynomials such that gpspr1 x pr2 x P r s |p|`|s|ďN´k is the equivariant Poincar´edual to the diagonal in Xτ , where N k is theř complex dimension M ´ 6 Furthermore, gj “0 if bą0 or Dj pdqPt´1, 0u for all dPΛ with νEpdq“0.

44 İ τ of XM . If N k and νE d N k for all d Λ 0 , then the two-point function Z of (4.1) is given by ą p qą ´ P ´t u p s İ İ İİ 1 ˚ d ˚ d Z ~1, ~2,q gpspr1 x ~1 q Y x, ~1,q pr2 x ~2 q Y x, ~2,q . p q“ ~1 ~2 ` dq p q ` dq p q |p|`|s|ďN´k ` ÿ " * " * This follows from Theorem 4.5 and Corollary 4.8. İ İİ Remark 4.10. In the inductive construction of DpY with Y Y or Y Y, the ﬁrst equation in (4.12) may be replaced by “ “

p d péi D Y A, ~,q cp;i Ai ~ q D Y A, ~,q Qα A, ~ Λ , p q” ` dq p qP p qrr ss iPsuppppq ÿ " * r p for any tuple cp;i iPsuppppq of rational numbers with cp;i 1. The power series D Y p q iPsuppppq “ defined by the second equation in (4.12) in terms of theř “new weighted” DpY satisfy (EP1) and (EP2) with Dp correspondingly “weighted” as in Remark 3.6. This follows by the 1 same arguments as in the case when cp;i |suppppq| for all i supp p .r Therefore, (4.14) prq “ P p pqrq continues to define power series Cp,s Y in terms of the “new weighted” Cp,s Y . The resulting p q p q “weighted” power series Yp of (4.25) do not depend on the “weights” cp;i as elements of ˚ τ H N X ~ Λ by the proofr of Theorem 4.7 outlined in Section 5.1. T p M q rr ss ˚ RemarkV 4.11.W We define an equivariant version of Z in (1.10). Let

˚ d e VE Z ~1, ~2,Q Q ev1 ev2 p q , (4.26) ˚ ~ ψ ~ ψ p q” dPΛ´0 p ˆ q 1 1 2 2 ÿ „p ´ q p ´ q İ τ τ ˚ ` ˚ ´ where ev1, ev2 :M0,2 X , d X . Since e VE ev e E e VE ev e E , p M qÝÑ M p q 1 p q“ p q 1 p q İ ˚ ˚ ` ˚ ˚ ´ Z ~1, ~2,Q pr e E Z ~1, ~2,Q pr e E , p q 1 p q“ p q 1 p q İ İ ˚ 0 τ τ τ where Z is obtained from Z by disregarding the Q term and pr1 :XM XM XM is the projection onto the ﬁrst component. This together with Theorem 4.5 expressesˆ ÝÑZ˚ in terms İ İİ ` of Zη and Zη in the E E case. Using an idea from [CoZ],“ we derive a formula for Z˚ in terms of one-point GW generating functions that holds in all cases. Following [CoZ], we then show how to express the latter τ τ τ in terms of explicit power series if b 0. If pri : XM XM XM and gps Q α are as in Corollary 4.9, then ą ˆ ÝÑ P r s

˚ 1 ˚ p ˚ ˚ Z ~1, ~2,Q gps pr1 x pr2 Zs ~2,Q p q“ ~1 ~2 p q ` |p|`|s|ďN´k (4.27) ÿ “ İİ ˚ ˚ ˚ pr Z ~1,Q pr Zs ~2,Q , ` 1 pp q 2 p q where ‰ ˚ p ˚ d e VE ev2 x ˚ τ ´1 Z ~,Q Q ev1 p q H N X ~ , Λ . p ˚ ~ ψ T M p q” dPΛ´0 1 P p qrr ss ÿ „ ´  45 İ İ This follows from Theorem 4.5, using that pr˚ e E` e E´ Z Z Z˚, and 1 p p q{ p qqp ´ Q;0q“ ` İİ ˚ e E ˚ p ˚ s J ˚K p ˚ ˚ pr p q gpspr x pr Zs ~,Q x gpspr x pr Z ~,Q . (4.28) 1 e E´ 1 2 p p q´ q“ 1 2 s p q |p|`|s|ďN´k |p|`|s|ďN´k ˆ p q˙ ÿ ÿ τ n´1 τ n´1 In the XM P case, (4.27) is [CoZ, (2.19)] and the proof of the XM P case of (4.28) in [CoZ] extends“ to the case of an arbitrary toric manifold. “ We give another proof of (4.28), using the Virtual Localization Theorem (5.16) on τ M0,2 XM , d as in Section 5.4. We prove that (4.28) holds when restricted to I J for arbitraryp q I,J V τ . The left-hand side of (4.28) restricted to I J is r sˆr s P M r sˆr s İİ ` ˚ s ˚ e E p d e VE ev2 x ev1 φJ p q gpsx Q p q . (4.29) ´ vir e E rIs rIs M pXτ ,dq ~ ψ1 p qˇ |p|`|s|ďN´k ˇ dPΛ´0 ż 0,2 M ´ ˇ ÿ ˇ ÿ r s The right-handˇ side of (4.28) restrictedˇ to I J is r sˆr s ˚ s ˚ p d e VE ev2 x ev1 φJ gpsx Q p q . (4.30) vir rIs M pXτ ,dq ~ ψ1 |p|`|s|ďN´k ˇ dPΛ´0 ż 0,2 M ´ ÿ ˇ ÿ r s Since φI is the equivariant Poincarédualˇ of I , r s g ˚xp ˚xs ˚φ ˚φ φ φ φ J I J V τ , pspr1 pr2 rIsˆrJs pr1 I pr2 J I J I 0 M “ ∆pXτ q “ Xτ “ p q“ @ ‰ P p,s ż M ż M ÿ ˇ τ ˇτ τ where ∆ XM XM XM denotes the diagonal. Thus, by the Virtual Localization Theo- rem (5.16),p aqĂ graph Γˆ as in Section 5.4 may contribute to (4.29) or (4.30) only if its second marked point is mapped into I . Finally, (4.28) follows from the above since r s e E` İİ V V p ´q e E e E e E rIs p q ZΓ “ p q ZΓ p qˇ ˇ ˇ τ ˇ N ˇ ˇ whenever ZΓ M0,2 XM , d is the Tˇ -pointwiseˇ fixed locusˇ corresponding to a graph Γ whose secondĂ markedp pointq is mapped into I . r s ˚ We next assume that b 0 and νE d 0 for all d Λ and express Zp ~,Q in terms of explicit power series. Alongą with (4.27)p andqě Theorem 4.7,P this will concludep theq computation of Z˚. We define

` ´ a Li pdq k b ´Li pdq k d ` ´ Y A, ~,q q u d;A, ~ ℓriAr s~ ℓriAr s~ . (4.31) p q”dPΛ p q i“1 s“1 ˜r“1 ` ¸ i“1 s“1 ˜r“1 ´ ¸ ÿ ź ź ÿ ź ź ÿ p p prq prq As Y satisfies (4.8), we may define D Y and Cp,s Cp,s Y by (4.12) and (4.14). We define İ ” p İq prq prq Yp A, ~,q by the right-hand side of (4.25) above, with Y replaced by Y and Cp,s by Cp,s. Letpp q p r r p p ` ´ p r r a Li pdq k b ´Li pdq´1 k ˚ d ` ´ Y A, ~,q q u d;A, ~ ℓriAr s~ ℓriAr s~ . (4.32) p q”dPΛ´0 p q i“1 s“1 ˜r“1 ` ¸ i“1 s“1 ˜r“1 ´ ¸ ÿ ź ź ÿ ź ź ÿ r 46 prq Define Ep,s Q α Λ by P r srr ss |p|´b |p|´b´s d p A ~ q Y˚ A, ~,q E prqAr~s mod ~´1. (4.33) ` dq p q– p,s s“0 |r|“0 " * ÿ ÿ r prq It follows that Ep,s is a degree p b s νE d r symmetric homogeneous polynomial r zq;d | |´ ´ ´ p q´| | in α. Then,

p |p|´b |p|´b´s İ İ d ´ prq s Yp x, ~,Q x ~ q Y x, ~,q e E E ~ Yr x, ~,q , (4.34) p q“ ` dq p q´ p q p,s p q s“0 |r|“0 " * ÿ ÿ İ p where Yp is defined by (4.25); see Section 5.1 for a proof of (4.34). Whenever b 2, ě p |p|´b |p|´b´s ˚ ` d ˚ prq s Z ~,q e E x ~ q Y x, ~,q E ~ Yr x, ~,q . (4.35) pp q“ p q » ` dq p q´ p,s p qfi s“0 |r|“0 " * ÿ ÿ – r p fl If b 1, “

´ p epE qf0pqq d Z˚ ~,Q e E` e´ ~ x ~ q Y˚ x, ~,q pp q“ p q ` dq p q « " * (4.36) |p|´b |p|´b´s ` p r 8 ´ n prq s e E x f0 q 1 e E f0 q E ~ Yr x, ~,q p q p q p q p q , ´ p,s p q ´ ~ n 1 ! ´ ~ s“0 |r|“0 ﬀ n“0 ÿ ÿ ÿ p ` q „  p with Q and q related by the mirror map (3.26) and f0 q Q Λ given by (3.35). Equa- İ ˚ epE`q p p q P rr ss tions (4.35) and (4.36) follow from Z ´ Zp x , (4.24), and (4.34). p “ epE q ´ ´ ¯

47 Chapter 5

Proofs

5.1 Outline

In this chapter, we prove Theorems 4.5 and 4.7 and the identity (4.34). The proofs of the two theorems are in the spirit of the proof of mirror symmetry in [Gi1] and [Gi2] but with a twist. Similarly to [Gi1] and [Gi2], our argument revolves around the restrictions on power series imposed by certain recursivity and polynomiality conditions. The concept of τ n´1 C-recursivity was first introduced in [Gi1] in the XM P case, extended to an arbitrary τ “ XM in [Gi2], and re-defined in [Z1]; all these definitions involve an explicit collection C of structure coefficients. Our concept of C-recursivity introduced in Definition 5.2 extends the notion of C-recursivity with an arbitrary collection of structure coefficients from the τ n´1 τ XM P case considered in [PoZ] to an arbitrary XM . The concept of (self-) polynomiality “ τ n´1 τ introduced in [Gi1] in the XM P case and extended to an arbitrary XM in [Gi2] was “ τ n´1 modified into the concept of mutual polynomiality for a pair of power series in the XM P τ “ case in [Z1]; we extend the latter to an arbitrary XM in Definition 5.5. By Proposition 5.6, τ n´1 τ which extends [Z1, Proposition 2.1] from the XM P case to an arbitrary XM , C- recursivity and mutual polynomiality impose severe restri“ ctions on power series, more severe than the restrictions imposed by recursivity and self-polynomiality as discovered in [Gi1]. Analogous to [Z1] and [PoZ], the proof of Theorem 4.7 relies on the one-point mirror theorem of [LLY3]. We begin by stating it. The coefficient of αj ~ for j N in the Laurent İ 1 ´1 { Pr s İ Y ~ expansion of A“0 at 0 is given by I0pqq “

ˇ a ˇ ` Li d ! Dj pdq δb,0 d i“1 p q 1 gj q İ » q p q” śN “ ‰ s I0 q dPΛ,ν pdq“0 ˜ s“1 ¸ E Dr d ! p q —Dspdqěÿ0 @ sPrNs ÿ — r“1 r p q s – ś a (5.1) L` d ! i fi d Dj pdq i“1 p q q 1 Dj d 1 ! . ` p´ q r´ p q´ s ś “ Ds d‰ ! dPΛ,ν pdq“0 ffi ÿE sPrNs´tju r p q sffi Dj pdqă´1 ffi Dspdqě0 @ sPrNs´tju ś ffi ffi fl 48 By [LLY3, Theorem 4.7] together with [LLY3, Section 5.2], if νE d 0 for all d Λ, then p qě P k N 1 İ 1 ´ ~ Gpqq` xifipqq` αj gj pqq İ « i“1 j“1 ff Z ~ İ Y ~ 0 ,Q e ř ř x, ,q , (5.2) p q“ I0 q p q p q

with Q and q related by the mirror map (3.26), G, fi, and gj deﬁned by (3.27), (3.25), and (5.1).7

Remark 5.1. By (4.7), (3.27), and (3.25),

İ İ İ İ I0 q G q Y A, ~,q and I0 q fi q Y A, ~,q i k , α“0 Ai ;1 p q p q”s p q A“0{~´1;1 p q p q”r p qz ~ @ Pr s ˇ ˇ ˇ ´1 Ai where ´1 and Ai denote the coefficients of ~ and respectively within the ~ ;1 ~ ;1 ~ Laurent expansion around ~´1 0 of the power series inside of the brackets. Thus, J K J K “ k N İ İ Y A, ~,q I0 q G q Aifi q αjgj q . ~´1;1 r p qz ” p q « p q` i“1 p q` j“1 p qff ÿ ÿ Some of the proofs in this chapter also hold if we replace Q by any field R Q. Given such a field R, let Ě

˚ τ ˚ τ Rα Qα Q R R α1,...,αN P :P Q α 0 and H N X ; R H N X Q R. ” b “ r sx P r s´ y T p M q” T p M qb ˚ τ An element in HTN XM ; R ~ Λ admits a lift to an element in R α,x ~ Λ and an p q rr ss ˚ τ r s rr ss element in R α,x ~ Λ induces an element in HTN XM ; R ~ Λ via Proposition 2.21. r s ˚ rr τ ss V W V τ p q rr ss V W Given Y ~,Q HTN XM ; R ~ Λ and J M , we write p qP V Wp q rr ss P V W Y ~,Q orV WY ~,Q or Y x J , ~,Q R α ~ Λ p q rJs p q J p p q q P r s rr ss ˇ ˇ for the power series obtainedˇ from Y byˇ replacing each coeﬃcient of ~VsQWd in Y by its image via the restriction map J of (2.28). In proving Theorem¨p 4.7,q we follow the steps outlined in [Z1, Section 1.3] and used for proving [Z1, Theorem 1.1]:

˚ τ (M1) if R Q is any field, Y,Z HTN XM ; R ~ Λ , Z ~,Q is C-recursive in the sense of DefinitionĚ 5.2 and satisfiesP the mutualp polynomialityq rr ss p conditq ion (MPC) of Definition 5.5 with respect to Y ~,Q , the transformsV ofWZ ~,Q of Lemma 5.8 are also C-recursive and satisfy the MPCp withq respect to appropriatep q transforms of Y ~,Q ; p q ˚ τ (M2) if R Q is any field, Z HTN XM ; R ~ Λ is recursive in the sense of Definition 5.2 andĚY,Z satisfies theP MPCp for someq Yrr Hss˚ Xτ ; R ~ Λ with Y ~,Q p q P Tp M q rr ss p q I Q;0 P ˚ V τ V W ~´1 q y Rα for all I M , then Z is determined by its ‘mod part’ (see Propositionˇ 5.6); P V W ˇ 7See Appendix A for the correspondence between the relevant notation in [LLY3] and ours and detailed references within [LLY3] indicating how [LLY3, Theorem 4.7] together with [LLY3, Section 5.2] implies (5.2).

49 İ İ İ İ (M3) Yp of (4.25) and Zζ of (4.2) are C-recursive in the sense of Deﬁnition 5.2 with C given İİ İİ İİ İİ by (5.20), while Yp of (4.25) and Zζ of (4.2) are C-recursive with C given by (5.20);

İ İİ İİ İ İ İİ İİ İ (M4) Y, Yp , Y, Yp , Z1, Zζ , and Z1, Zζ satisfy the MPC; p q p q p q p q (M5) the two sides of (4.24) viewed as powers series in ~´1, agree mod ~´1.

The proof of Theorem 4.5 described below follows the same ideas and extends the proof of [Z1, (1.17)]. İ İİ Claims (M3) and (M4) concerning Zζ and Zζ follow from Lemmas 5.12 and 5.13, since by the string equation of [MirSym, Section 26.3] and (5.21),

İ İ İİ İİ Zζ ~,Q ~Zη,β ~,Q and Zζ ~,Q ~Zη,β ~,Q , p q“ p q p q“ p q

if m 3, β2 β3 0, η2 ζ, and η3 1. “ “ “ “ “ İ İ İİ İİ İ İİ By Lemmas 5.16, 5.17, and 5.7, Y is C-recursive and Y is C-recursive, while Y, Y and İİ İ p q Y, Y satisfy the MPC. This together with the admissibility of transforms (a) and (b) of p q İ İİ Lemma 5.8 proves claims (M3) and (M4) for Yp and Yp. Claims (M3) and (M4) together with (5.2), the admissibility of transforms (c) and (d) of Lemma 5.8, and Proposition 5.6, prove that verifying (4.24) amounts to showing that the two sides of each of these equations agree mod ~´1; this is in turn equivalent to (4.14). Lemma 5.7, Lemma 5.8, and Proposition 5.6 are proved in Section 5.3; the preparations for this section and the ones following it are made in Section 5.2. Lemmas 5.12 and 5.13 are proved in Sections 5.5 and 5.6, respectively. Both proofs rely on the Virtual Localization Theorem [GraPa, (1)]. The localization data provided by [Sp] is presented in Section 5.4. Lemmas 5.16 and 5.17 are proved in Section 5.7. Proof of (4.34). Deﬁne E´ Z with p Zě0 k by p P Pp q b k ´ ´ p ℓ Ar E A . ri ” p i“1 ˜r“1 ¸ p Zě0 k ź ÿ Ppÿ q By (4.7) and (4.31),

p d İ e E´ Y x, ~,q E´ x ~ q Y x,h,q . (5.3) p q p q“ p ` dq p q p Zě0 k Ppÿ q " * İ İ p İİ İ Since Y is C-recursive by Lemma 5.16 and Y, Y satisfies the MPC by Lemmas 5.7 and 5.17, İ İİ p q e E´ Y is C-recursive and Y, e E´ Y satisfies the MPC by (5.3) and Lemma 5.8(a). This p q p p q q İ together with Lemma 5.8(a)(b) implies that the right-hand side of (4.34) is C-recursive and İİ İ satisfiesp the MPC with respect to Y. Sincep Yp also satisfies these two properties by (M3) and (M4), the claim follows from (M2) and the fact that both sides of (4.34) are congruent to xp İ ´1 modulo ~ . The latter follows from the fact that Yp x,h,Q and Yp x,h,Q are congruent to xp modulo ~´1 by (4.14) together with (4.7), (4.32),p andq (4.33). p q p 50 Proof of Theorem 4.5. By (4.1), (2.35), and (2.21),

İ İ ˚ ˚ d e VE ev1 φI ev2 φJ ~1 ~2 Z ~1, ~2,Q ~1~2 Q p q (5.4) vir p ` q p q rIsˆrJs “ τ ~1 ψ1 ~2 ψ2 dPΛ M0,3pXM ,dq ˇ ÿ żr s p ´ qp ´ q ˇ İ τ ˇ for all I,J V . Applying Lemmas 5.12 and 5.13 for Zη,β ~1,Q with P M p q

m 3, β2 n, β3 0, η2 φJ , η3 1, “ “ “ “ “ İ ń along with Lemma 5.8(b), we obtain that the coefficient of ~2 in ~1 ~2 Z ~1, ~2,Q is İ İ p ` İİq p q C-recursive with C given by (5.20) and satisfies the MPC with respect to Z1 ~1,Q for all n 0. Using this, Proposition 2.21(b), (M3), (M4), and (M2), it follows thatp in orderq to proveě (4.23) it suffices to show that

s İ İ İİ ´1 ~1 ~2 Z ~1, ~2,Q Zηj ~1,Q Zηj ~2,Q mod ~1 (5.5) p ` q p q rIsˆrJs – p q rIs p q rJs ˇ j“1 ˇ ˇ ˇ ÿ ˇ q ˇ for all I,J V τ . By (5.4) and theˇ string equation, theˇ left-hand sideˇ of (5.5) mod ~´1 is P M 1 İ ˚ ˚ d e VE ev1 φI ev2 φJ φI J ~2 Q p q vir p q` M pXτ ,dq ~2 ψ2 dPΛ´0 ż 0,3 M ´ ÿ r s İ (5.6) ˚ ˚ e VE ev φI ev φJ ∆ 1 Qd p q 1 2 , ˚ vir “ rIsˆrJs ` M pXτ ,dq ~2 ψ2 ˇ dPΛ´0 ż 0,2 M ´ ˇ ÿ r s where ∆:Xτ Xτ Xτ , ∆ˇ z z , z . The right-hand side of (5.5) mod ~´1 is M ÝÑ M ˆ M r s”pr s r sq 1 s İİ

ηj Zηj ~2,Q . (5.7) rIs p q rJs j“1 ˇ ˇ ÿ ˇ q ˇ İİ ˇ ˇ Applying Lemmas 5.12 and 5.13 for Zη,β ~2,Q with p q

m 3, β2 β3 0, η2 φI , η3 1, “ “ “ “ “ İİ along with Lemma 5.8(b), we obtain that (5.6) is the restriction to J of a C-recursive formal İ r s power series which satisfies the MPC with respect to Z1 ~2,Q . Since (5.7) also satisfies these two properties, by Proposition 5.6 the power series (5.6)p andq (5.7) agree if and only if they ´1 ´1 agree mod ~2 . The latter is the case since (5.7) mod ~ is the equivariant Poincarédual to the diagonal in Xτ Xτ restricted to the point I J . M ˆ M r sˆr s 5.2 Notation for fixed points and curves

With V τ as in (2.4) and for all I,J V τ with I J k 1, we denote by M P M | X |“ ´ IJ Xτ I J Xτ IJ IJ M M and deg rXτ s Λ (5.8) ” p Y qĎ ” M P “ ‰ 51 the P1 passing through the points I and J and its homology class, respectively; see Corollary 2.20. Given I V τ and j rNs I, wer s denote by P M Pr s´ Ij Xτ I j Ij Ij M and deg rXτ s Λ ” p Yt uq ” M P τ“ ‰ the compact one-dimensional complex submanifold of XM deﬁned by Remark 2.10 and its τ homology class, respectively. Since XM admits a K¨ahler form,

deg IJ, deg Ij Λ 0 P ´t u by [GriH, Chapter 0, Section 7]. By the last part of Remark 2.10, there exists a unique element v I, j of V τ such that p q M v I, j I and v I, j I j . p q‰ p qĂ Yt u Since v I, j I j I, j v I, j and Ij Iv I, j . Let j I v I, j . Applyingp qY the“t LocalizationuY P p Theoremq (2.22)“ p to theq integralt u” of´ 1 poverqIj P1 and using – (2.34) and Corollary 2.20, we ﬁnd that p

τ uj I u v I, j 0 I V , j N I. (5.9) p q` j p p qq“ @ P M Pr s´ p ˘ Applying the Localization Theorem (2.22) to the integrals of xi, λi , and us over Ij and using Corollary 2.20, (2.34), and (5.9), we ﬁnd that

τ xi I xi v I, j Hi, deg Ij uj I I V , j N I,i k , (5.10) p q´ p p qq “ x y p q @ P M Pr s´ Pr s ˘ ˘ ˘ τ λ I λ v I, j L Ij uj I I V , j N I,i a i b , (5.11) i p q´ i p p qq “ i p q @ P M Pr s´ Pr s p Pr sq τ us I us v I, j Ds Ij uj I I V , j N I,s N . (5.12) p q´ p p qq “ ` ˘ p q @ P M Pr s´ Pr s By (5.12), (5.9), (2.30), and (2.33),` ˘

Dj Ij D Ij 1,Ds Ij 0 s I v I, j . (5.13) “ j “ “ @ P X p q The last ﬁve identities` are˘ statedp ` in˘ [Gi2]. ` ˘

5.3 Recursivity, polynomiality, and admissible transforms

As in [Gi2], we introduce a partial order on Λ: if s, d Λ, we deﬁne sĺd if d s Λ. By Proposition 2.16, P ´ P d Λ s Λ: s ĺ d is ﬁnite. (5.14) P ùñ t P u This implies that for every non-empty subset S of Λ, there exists d S such that P s Λ, s ă d s S. P ùñ R

52 dě1 Definition 5.2. Let R Q be any field and C CI,j d V τ any collection of IP M ,jPrNsÍ Ě ˚ τ ” p p qq elements of Rα. A power series Z HTN XM ; R ~ Λ is C-recursive if the following holds: if d˚ Λ is such that P p q rr ss P V W τ Z x v I, j , ~,Q ˚ Rα ~ I V , j N I,d 1, p p p qq q Q;d ´d¨deg Ij P p q @ P M Pr s´ ě V τ and Z x v I,J j , ~,Q Q;d˚´d¨degK Ij is regular at ~ uj I d for all I M , j N I, and d 1, thenp p p qq q “´ p q{ P Pr s´ ě J K CI,j d ´1 Z x I , ~,Q ˚ p q Z x v I, j , ~,Q ˚ uj pIq Rα ~, ~ , Q;d uj pIq Q;d ´d¨deg Ij ~“´ p p q q ´ p p p qq q d P r s dě1 jPrNsÍ ~ d ÿ ÿĺ ˚ ` ˇ J K d¨deg Ij d J K ˇ V τ ˚ τ for all I M . A power series Z HTN XM ; R ~ Λ is called recursive if it is C-recursive P Pdě1 p q rr ss for some collection C CI,j d IPV τ ,jPrNsÍ of elements of Rα. ”p p qq M V W ˚ τ dě1 By Remark 5.3 below, if Z HTN XM ; R ~ Λ is CI,j d IPV τ ,jPrNsÍ -recursive, then P p q rr ss p p qq M for each I V τ P M V W

Nd prq ´r d Z x I , ~,Q ZI;d~ Q p p q q“ dPΛ r“Ń ÿ ÿ d 8 d¨deg Ij CI,j d Q uj I p q Z x v I, j , p q,Q ` uj pIq p p qq ´ d d“1 jPrNsÍ ~ ÿ ÿ ` d ˆ ˙ prq for some integers Nd and some Z Rα. I;d P ˚ τ Remark 5.3. Let R Q be any field. If Z H N X ; R ~ Λ is recursive, then Z Ě P T p M q rr ss I P R ~ Λ and Z x v I, j , ~,Q is regular at ~ új pIq for all I V τ , d Λ, j N I, α Q;d d V W M ˇ andp dqrr 1;ss this followsp p p byqq inductionq on d Λ. The regularity“ claimP also usesP RemarkPr s´ˇ 5.4 below.ě J K P The C-recursivity is an Rα-linear property (that is, if Z1 and Z2 are C-recursive, then so is f1Z1 f2Z2 for any f1,f2 Rα). By Lemma 5.8(b), C-recursivity is actually an Rα ~ Λ - linear property.` P r srr ss

Remark 5.4. For all I V τ , j N I, all d Q 1 , and all s N , P M Pr s´ P ´t u Pr s

uj I d us v I, j 0. p q` ¨ p p qq ‰ Proof. Assume that uj I d us v I, j 0 (5.15) p q` ¨ p p qq “ τ for some I V , j N I, d Q 1 , and s N . If d 0 or s v I, j , then uj I 0 by P M Pr s´ P ´t u Pr s “ P p q p q“ (2.30) which contradicts (2.33). If d 0 and s I v I, j , then uj I 1 d 0 by (5.15) ‰ P p ´ p qq p qp ´ q“ and (5.9), which again contradicts (2.33). If d 0 and s I v I, j , then setting αi 0 for ‰ Rp Y p qq “ all i I v I, j in (5.15) and using (2.33), we ﬁnd that dαs 0, which is false. Pp Y p qq ´ “

53 For the purposes of Definition 5.5 and the transforms (a) and (d) in Lemma 5.8 below τ k as well as all statements involving them, we identify H2 XM ; Z with Z via the dual basis k p q to H1,..., Hk so that Λ Z . t u Ă ˚ τ Definition 5.5. For any Y Y ~,Q ,Z Z ~,Q H N X ; R ~ Λ , define ΦY,Z ” p q ” p q P T p M q rr ss P Rα ~ z, Λ by rr ss V W V W expIq¨z Φ ~,z,Q Y x I , ~,Qe~z Z x I , ~,Q , Y,Z u I p q” IPV τ j p q p p q ´ q ÿM jPrNsÍ p q ` ˘ ś k ~z ~z1 ~zk where z z1,...,zk , x I z xi I zi, and Qe Q1e ,...,Qke . ”p q p q ¨ ” i“1 p q ” ˚ τ If Y,Z H N XM ; R ~ Λ ř, the pair Y,Z satisfies` the mutual polynomiality˘ condition P T p q rr ss p q (MPC) if ΦY,Z Rα ~ z, Λ . P r srr V Wss ˚ τ Proposition 5.6. Let R Q be a field. Assume that Z HTN XM ; R ~ Λ is recursive Ě ˚ τ P p q rr ss and that Y,Z satisfies the MPC for some Y HTN XM ; R ~ Λ with p q P p q rr ss V W Y ~,Q R˚ I VVτ .W p q I Q;0 P α @ P M q y ´1 ˇ Then, Z ~,Q 0 mod ~ if and onlyˇ if Z ~,Q 0. p q– p q“ Proof. By the second statement in Proposition 2.21(b),

Z ~,Q 0 Z ~,Q 0 I V τ . p q“ ðñ p q I “ @ P M 1 Set fI Y ~,Q and assume that Z ~,Qˇ 0 for all 0 ĺ d ă d and all ” p´ q I Q;0 p ˇ q I Q;d1 “ V τ q y ~ q ~´y1 I M . Since Z is recursiveˇ and Z ,Q 0 modulo ˇ , P ˇ p q– ˇ Nd Z ~,Q Zprq ~´r p q I Q;d “ I;d q y r“1 ˇ ÿ prq ˇ for some Nd 0 and some Z Rα. Thus, ě I;d P expIq¨z Nd Φ ~,z,Q f Zprq ~´r R ~ z . Y,Z Q;d u I I I;d α p´ q “IPV τ j ˜r“1 ¸P r srr ss M p q J K ÿ jPrNs´I ÿ ś This implies that

m Nd x I z prq ´r p p q ¨ q fI Z ~ Rα ~,z m 0. u I I;d IPV τ j ˜r“1 ¸P r s @ ě ÿM jPrNsÍ p q ÿ ś In particular, m x I z prq τ p p q ¨ q fI Z 0 0 m V 1, r Nd . u I I;d M IPV τ j “ @ ď ď| |´ @ Pr s ÿM jPrNsÍ p q ś 54 prq V τ For each r Nd , this is a linear system in the ‘unknowns’ fI ZI;d uj I with I M . Pr s jPrNsÍ p q P Its coefficient matrix has a non-zero Vandermonde determinant, sinceM ś x I x J I J V τ p q‰ p q @ ‰ P M prq τ by Proposition 2.21(a). It follows that Z 0 for all I V and all r Nd . I;d “ P M Pr s τ n´1 Lemmas 5.7 and 5.8 below extend [Z1, Lemmas 2.2, 2.3] from the XM P case to an τ “ arbitrary XM . Our proof of the former is completely different from and much simpler than the one in [Z1]. For the latter, the arguments in [Z1] go through with only two significant changes required.

˚ τ Lemma 5.7. Let R Q be a ﬁeld and Y,Z H N X ; R ~ Λ . Then, Ě P T p M q rr ss ΦY,Z Rα ~ z, Λ ΦZ,Y RVα W~ z, Λ . P r srr ss ðñ P r srr ss

Proof. Let Yd ~ Y ~,Q Q;d and Zd ~ Z ~,Q Q;d. It follows that ΦY,Z ~,z,Q Q;d is p q” p q p q” p q p q J K J K J K xpIq¨z xpIq¨z e ~zd1 e ~zpd´d1q Yd1 ~ e Zd´d1 ~ Yd´d1 ~ e Zd1 ~ , uj I p q I p´ q I “ uj I p q I p´ q I 0ĺd1ĺd 0ĺd1ĺd I V τ jPrNsÍ p q ˇ ˇ I V τ jPrNsÍ p q ˇ ˇ ÿP M ˇ ˇ ÿP M ˇ ˇ ś ˇ ˇ ś ˇ ˇ ~z ~z1 ~zk ~zd where e e ,..., e . The right-hand side is e times ΦZ,Y ~,z,Q . ”p q p´ q Q;d dě1 Lemma 5.8. Let R Q be any field and C CI,j d V τ J any collectionK of elements IP M ,jPrNsÍ Ě ˚ τ ”p p qq of Rα. Let Y1,Y2,Y3 HTN XM ; R ~ Λ . If Y1 is C-recursive and Y2,Y3 satisfies the MPC, then P p q rr ss p q V W d ( a) if Yi xs ~ Qs Yi for all i and s k , then Y1 is C-recursive and Φ ” ` dQs P r s Y2,Y3 P Rα ~ z,!Λ ; ) r srr ss ( b) if f Rα ~ Λ , then fY1 is C-recursive and ΦY ,fY Rα ~ z, Λ ; P r srr ss 2 3 P r srr ss f{~ ( c) if f Rα Λ 0 and Yi e Yi for all i, then Y1 is C-recursive and Φ Rα ~ z, Λ ; P rr ´ ss ” Y2,Y3 P r srr ss f¨x{~ f ( d) if fr Rα Λ 0 for all r k and Yi ~,Q e Yi ~,Qe for all i, where f x k P rr ´ ss Pr s p q” p q ¨ ” f f1 fk ~ frxr and Qe Q1e ,...,Qke , then Y1 is C-recursive and ΦY2,Y3 Rα z, Λ . r“1 ”p q P r srr ss Proof.řFor all I V τ , P M d C d u I x I ~ Q I,j Qd¨deg IjY x v I, j , j ,Q s s uppIqq 1 p q p q` dQs ˜~ j p p qq ´ d ¸“ " * ` d ˆ ˙ CI,j d d¨deg Ij uj I CI,j d d¨deg Ij p q Q Y1 x v I, j , p q,Q p q Q ~ uj pIq p p qq ´ d ` ~ uj pIq ` d ˆ ˙ ` d uj I d uj I ~ p q Qs ~ d deg Ij xs I xs v I, j Y1 x v I, j , p q,Q . ˆ ` d dQ ` ¨ s ` p q´ p p qq p p qq ´ d ˆˆ ˙ s ˙ ˆ ˙

55 The ﬁrst claim in (a) now follows from Remark 5.3 and (5.10). The second claim in (a) and τ n´1 the claims in (b)-(d) follow similarly to the proof of [Z1, Lemma 2.3] for the XM P case, using Lemma 5.7, Remark 5.3, (5.14), and (5.10). Equation (5.10) and property“ (5.14) are used in the proof of the recursivity claim in (d) when showing that

1 fpQqxpIq ´fpQqxpvpI,jqqd dfpQq¨deg Ij` ~ u pIq ´1 e e j Rα ~, ~ Λ . ~ uj pIq ´ P r srr ss ` d ˆ ˙ ˚ τ ´1 Property (5.14) is also used to show that transforms (c)and(d) preserve HTN XM ; Rα ~, ~ Λ , that p qr srr ss f{~ ´df{u pIq ~ e e j fpQe zq´fpQq ´1 ~ ´ Rα ~, ~ Λ , e Rα ~ z,Q , ~ uj pIq P r srr ss P r srr ss ` d in the case of (c), and that ~z fr Qe fr Q zr p q´ p q Rα ~,z Λ r k ` ~ P r srr ss @ Pr s in the case of (d).

5.4 Torus action on the moduli space of stable maps

N An action of T on a smooth projective variety X induces an action on M0,m X, d as in Chapter 4 and an integration along the ﬁber homomorphism as in Section 2.3. Thep Virtualq Localization Theorem [GraPa, (1)] implies that

η ˚ η Q α η H N M0,m X, d , (5.16) vir vir T vir N M0,mpX,dq “ TN rF s e F {X P r s @ P p q ż F ĎM0,mpX,dq ż r s ÿ p q ` ˘ where the sum runs over the components of the TN pointwise fixed locus TN M0,m X, d M0,m X, d . p q Ď p q τ TN vir This section describes M0,m XM ,d , the equivariant Euler class e NF {X of the virtual p q τ TN p q normal bundle to each component F of M0,m XM ,d , and the restriction of e VE to F . We follow [Sp] where the corresponding statementsp areq formulated in the languagep q of fans rather than toric pairs. τ N If f : Σ,z1,...,zm XM is a T -fixed stable map, then the images of its marked points, nodes,p contractedq ÝÑ components, and ramification points are TN-fixed points and so V τ points of the form I for some I M by Corollary 2.20(a). Each non-contracted component r s N P V τ Σe of Σ maps to a closed T -fixed curve which is of the form IJ for some I,J M with I J k 1 by Corollary 2.20(b). Since all such curves IJ are biholomorphicP to P1 by |CorollaryX |“ 2.20(´ b), the map

f : Σe IJ Σe ÝÑ ˇ is a degree d e covering map ramified onlyˇ over I and J . To each such map we associate a decorated graphp q as in Definition 5.9 below;ˇ the verticesr s ofr ths is graph correspond to the nodes and contracted components of Σ or the ramification points of f; the edges e correspond to non-contracted components Σe of Σ, and d e describes the degree of f . p q Σe ˇ 56 ˇ Definition 5.9. A genus 0 m-point decorated graph Γ is a collection of vertices Ver Γ , edges Edg Γ , and maps p q p q d : Edg Γ Zą0, p : Ver Γ V τ , dec : m Ver Γ p q ÝÑ p q ÝÑ M r s ÝÑ p q satisfying the following properties: 1. the underlying graph Ver Γ , Edg Γ has no loops; p p q p qq 2. if two vertices v and v1 are connected by an edge, then p v p v1 k 1. | p qX p q|“ ´ Such a decorated graph is said to be of degree d Λ if P d e deg p v p v1 d, p q p q p q “ ePEdgpΓq Be“tÿv,v1u ´ ¯ where e v, v1 for an edge e joining vertices v and v1. B ”t u For a decorated graph Γ as in Definition 5.9, we denote by Aut Γ the group of automor- p q phisms of Ver Γ , Edg Γ . It acts naturally on Zdpeq; let p p q p qq ePEdgpΓq ś AΓ Zd e Aut Γ ” p q ¸ p q ePEdgpΓq ź denote the corresponding semidirect product. For any v Ver Γ , let P p q Edg v e Edg Γ :v e and val v dec´1 v Edg v p q”|t P p q P B u| p q ” | p q| ` p q denote the number of edges to which the vertex v belongs and its valence, repectively. A flag F in Γ is a pair v,e , where e is an edge and v is a vertex of e. For a flag F v,e , p q 1 1 “ p 1 q let val F val v . For a flag F v,e , let ωF e T ´1 v P , where f :P p v p v is p q” p q “p q ” p f ppp qq q ÝÑ p q p q the degree d e cover of p v p v1 corresponding to e, e v, v1 , and the TN-action on P1 is induced fromp q the actionp onq pXτq via f. If j p v1 Bp“tv , u M t u” p q´ p q uj p v ωF p p qq (5.17) “ d e p q by (2.34). If v is a vertex that belongs to exactly 2 edges e1 and e2, then we write Fi v p q” v,ei . p Givenq a decorated graph Γ as above, let

MΓ M0,val v , ” p q vPVerpΓq ź 2 where M0,m point, whenever m 2. For a ﬂag F v,e , let ψF H N MΓ denote the ” ď “ p q P T p q equivariant Euler class of the universal cotangent line bundle on MΓ corresponding to F (that is, the pull-back of the ψ class on M0,valpvq corresponding to e).

57 τ Proposition 5.10 ([Sp, Lemma 6.9]). There is a morphism γ :MΓ M0,m XM ; d whose τ TN ÝÑ p q image is a component of M0,m X ; d and every such component occurs as the image of p M q such a morphism corresponding to some degree d decorated graph. With Zdpeq acting ePEdgpΓq trivially on MΓ, the induced map ś τ γ AΓ : MΓ AΓ M0,m X , d { { ÝÑ p M q τ TN identifies MΓ AΓ with the corresponding component of M0,m X , d . { p M q Proposition 5.11 ([Sp, Theorem 7.8]). Let Γ be a degree d genus 0 m-point decorated graph vir τ and N the virtual normal bundle to γ :MΓ M0,m X , d . Then, Γ ÝÑ p M q 1 1 N vir ω ψ ω ω e Γ F F Edgpvq´1 F1pvq F2pvq “ p ´ q ` ωF flags F of Γ φppvq p v vPVerpΓq flags F of Γ valźpF qě3 vPVerpΓq p p qq valźpvq“2 ` ˘ ` ˘ valpF q“1 ś “ ‰ dec´1pvq“H ś

dpeqDrpIjq s dpeq 2 2dpeq ur I dpeq uj I 1 d e ! uj I p q´ p q ¨ s“0 ˛ . p´ q p p q q2dpeq p qq ´1 ´ ¯ I“ppvq ˆ d e ś 1 e Edg Γ r N I j s ˇtju“ppv q´I P p q˚ p p qq Pr s´p Yt uq ur I uj I ‹ ˇ Be“tźv,v1u˚ ź p q´ dpeq p q ‹ ˇ ˚ s“dpeqDrpIjq`1 ‹ ˇ ˚ ś ´ ¯‹ ˇ By (5.9)˝ and (5.12), ‚

dpeq 2dpeq dpeq dpeq 1 1 uj I I“ppvq uppv1q´ppvq p v uppvq´ppv1q p v , p´ q p p qq ˇtju“ppv1qÍ “ p p qq p p qq ˇ dpeqD ˇ ppvqppv1q ´s u pppvqq`su pppv1qq rˇ r r 1 s r p q s 1 if Dr p v p v 0, ˇ dpeqDr ppvqppv q p q p q ‰ ur I uj I I“ppvq p q p q´ d e p q ˇtju“ppv1qÍ “ $u p v u p v1 if D ´p v p v1 ¯ ,s 0; p q ˇ & r r r ˇ p p qq “ p p qq p q p q “ ˇ vir ´ ¯ so the edge contributionsˇ to e N%Γ in Proposition 5.11 are indeed symmetric in the vertices of each edge. p q 1 N Let f : P ,z1,...,zm IJ be a T -fixed stable map. Thus, f is a degree d cover of IJ for some dp Zą0. By (1.1),qÝÑ P 0 1 ˚ ` 1 1 ˚ ´ VE 1 H P ,f E H P ,f E . rP ,z1,...,zm,fs “ ‘ By [MirSym, Exercise 27.2.3]ˇ together with` (5.9) and˘ (5.11)`, and with˘ j J I, ˇ t u” ´ ` a dLi pIJq b ´1 ` s ´ s e VE 1 λi I uj I λi I uj I . (5.18) p q rP ,z1,...,zm,fs “ p q´ d p q p q´ d p q i“1 s“0 i“1 s dL´ IJ 1 ˇ ź ź ” ı ź “ źi p q` ” ı By (5.11),ˇ

dL`pIJq´s λ`pIq`sλ`pJq s i i i if L` IJ 0, ` r dLs`pIJq i λi I uJ´I I i p q‰ p q´ d p q“ $λ` I λ` J if L` IJ s 0, & i p q“ i p q i p q“ “ ´ ´ ´ ´ s dLi IJ s λi I sλi J λ I uJ´I I % p q´ p q` p q. i p q´ d p q“ dL´ IJ “ ‰i p q

58 5.5 Recursivity for the GW power series

For all d Zą0,I V τ , j N I, let P P M Pr s´ 0 s ur I uj I d 2d´1 p q´ d p q 1 d 1 s“dDrpIjq`1 Q CI,j d p´ q 2 2d´1 ś “ ‰ α, (5.19) p q” d dDrpIjq P ! uj I rPrNs´pIYtjuq s p q r p qs ź ur I d uj I r s“1 p q´ p q

` ´ś “ ‰ a dL pIjq b ´dL pIjq´1 İ i s i s C d C d λ` I u I λ´ I u I Q , I,j I,j i d j i d j α p q” p qi“1 s“1 p q´ p q i“1 s“0 p q` p q P ź ź ” ı ź ź ” ı (5.20) ` 1 ´ r a dLi pIjq´ b ´dLi pIjq İİ s s C d C d λ` I u I λ´ I u I Q . I,j I,j i d j i d j α p q” p q i“1 s“0 p q´ p q i“1 s“1 p q` p q P ź ź ” ı ź ź ” ı r τ τ Lemma 5.12. If m 3, evj :M0,m XM , d XM is the evaluation map at the j-th marked ˚ τ ě ě0 p qÝÑ point, ηj H N X and βj Z for j 2,...,m, then the power series P T p M q P “ İ m İ e VE d βj ˚ ˚ τ Z ~,Q Q ev ψ ev η H N X ~ Λ and η,β 1˚ » ~´ ψ¯ j j j fi T M p q” dPΛ 1 j“2 P p q rr ss ÿ ´ ź ´ ¯ İİ V W (5.21) – m fl İİ e VE d βj ˚ ˚ τ Zη,β ~,Q Q ev1˚ ψj evj ηj HTN XM ~ Λ p q” » ~´ ψ1¯ fiP p q rr ss dPΛ ´ j“2 ´ ¯ ÿ ź V W İ İİ – İ İİ fl are C- and C-recursive, respectively, with C and C given by (5.20). τ Proof. This is obtained by applying the Virtual Localization Theorem (5.16) on M0,m XM , d , using Section 5.4, and extending the proof of [Z1, Lemma 1.1] from the case of a positivep lineq bundle over Pn´1 to that of a split vector bundle E E` E´ as in (1.5) over an arbitrary symplectic toric manifold Xτ . By (2.35), (2.21), (5.16),“ and‘ the second equation in (2.34), M İ İİ a decorated graph may contribute to Zη,β ~,Q I and Zη,β ~,Q I only if p dec 1 I. p qp q p qp q p p qq “ τ There are thus two types of contributing graphs: the AI and the BI graphs, where I V . P M In an AI graph the first marked point is attached to a vertex v0 of valence 2, while in a BI graph the first marked point is attached to a vertex v0 of valence at least 3. If Γ is a BI τ TN graph and ZΓ the corresponding component of M0,m X , d , then p M q n ψ 0 n val v0 3. 1 “ @ ą p q´ İ ´1 d Thus, Γ contributes a polynomial in ~ to the coefficient of Q in Zη,β ~,Q I and İİ p qp q Zη,β ~,Q I . p qp q In an AI graph there is a unique vertex v joined to v0 by an edge. Let A I,j d0 be the p qp q set of all AI graphs such that p v v I, j and the edge having v0 as a vertex is labeled d0. Thus, p q“ p q 8

AI ApI,jq d0 . “ d “1 jRI p q ď0 ď 59 1 1 2 1

d0 d0 I vpI,jq 2 I vpI,jq vpI,jq 2 Γ Γ Γ 0 c 3 3

Figure 5.1: A graph of type A I,j d0 and its two subgraphs p qp q

We ﬁx Γ A I,j d0 and denote by Γ0 and Γc the two graphs obtained by breaking Γ at v, P p qp q adding a second marked point to the vertex v in Γ0 and a ﬁrst marked point to v in Γc, and 8 requiring that marked points 2,...,m are in Γc; see Figure 5.1. Thus, Γ0 consists only of the vertices v0 and v and the marked points 1 and 2 attached to v0 and v, respectively. With τ TN ZΓ denoting the component in M0,m X , d corresponding to Γ, p M q

ZΓ ZΓ ZΓ ; – 0 ˆ c

we denote by π0 and πc the two projections. Thus,

İ İ İ İİ İİ İİ ˚ ˚ ˚ ˚ VE π VE π VE and VE π VE π VE. (5.22) “ 0 ‘ c “ 0 ‘ c These identities are obtained by considering the short exact sequence of sheaves

0 f ˚E˘ f ˚E˘ f ˚E˘ E˘ 0, ÝÑ ÝÑ 0 ‘ c ÝÑ p ÝÑ τ N ˇ where f : Σ X is a T -ﬁxed stable map whose correspondingˇ graph is Γ, while f0 and ÝÑ M fc are its restrictions to the components of Σ corresponding to the edge leaving v0 and the rest of Γ. Let m β βj ˚ η ψj evj ηj . ” j“2 ź ´ ¯ By (5.22),

İ İ β e VE η e VE İ ˚ ˚ β π0 πc e VE η , ´~ ψ¯1 ZΓ “ ¨ ~´ ψ1¯˛ ´ ˇ ´ ´ ´ ¯ ¯ İİ ˇ İİ (5.23) β ˇ ˝ ‚ e VE η e VE İİ ˚ ˚ β π0 πc e VE η . ´~ ψ¯1 ZΓ “ ¨ ~´ ψ1¯˛ ´ ˇ ´ ´ ´ ¯ ¯ ˇ ˇ ˝ ‚ By Proposition 5.11, (5.17), and (5.9),

˚ ev φI ˚ ˚ 1 ZΓ ˚ ev1 φI ˚ ev1 φvpI,jq 1 π0 πc . (5.24) e N vir e N vir e N vir uj pIq ˚ Γˇ “ Γ0 Γc π ψ1 p ˇ q ˆ p q˙ ˆ p q ˙ ´ d0 ´ c 8Figure 5.1 is [Z1, Figure 2] adapted to the toric setting.

60 By (5.18) and (5.11), on ZΓ0

` ´ a d0L pIjq b ´d0L pIjq´1 İ i i ` s ´ s e VE λi I uj I λi I uj I , “ p q´ d0 p q p q` d0 p q i“1 s“1 „  i“1 s“0 „  ´ ¯ ź ź` ź ź´ (5.25) a d0L pIjq´1 b ´d0L pIjq İİ i s i s e V λ` I u I λ´ I u I . E i d j i d j “ i“1 s“0 p q´ 0 p q i“1 s“1 p q` 0 p q ´ ¯ ź ź „  ź ź „  By Proposition 5.11,

d0DrpIjq u I s u I d0 2 r d0 j vir 1 d0! 2d0 s“0 p q´ p q e N p´ q p q uj I . (5.26) p Γ0 q“ 2d0 r p qs ś´1 ” ı d0 r N I j s Pr s´p Yt uq ur I uj I ź d0 s“d0DrpIjq`1 p q´ p q ś ” ı By (5.25), (5.26), (5.17), and (5.20),

İ İİ ˚ İ ˚ İİ e VE ev φI e VE ev φI 1 CI,j d0 1 CI,j d0 p q and p q. (5.27) ´ ¯ vir uj pIq ´ ¯ vir uj pIq ZΓ ~ ψ1 e NΓ “ ~ ZΓ ~ ψ1 e NΓ “ ~ ż 0 p ´ q 0 ` d0 ż 0 p ´ q 0 ` d0 ` ˘ ` ˘ By (5.23), (5.24), and (5.27),

İ İ İ ˚ β ˚ β e VE ev1 φI η 1 CI,j d0 e VE ev1 φvpI,jqη 1 p q vir p q p q vir u pIq , ~ Z uj pIq ~ ~“´ j ZΓ ψ1 Γ e NΓ “ ~ ZΓ ψ1 e NΓc d ż ´ p q d0 ż c ´ p q 0 İİ ˇ İİ` İİ ˇ (5.28) ˚ β ˇ ˚ β ˇ e VE ev1 φI η ˇ 1 CI,j d0 e VE ev1 φvpI,jqη 1 ˇ p q vir p q p q vir u pIq . ~ Z uj pIq ~ ~“´ j ZΓ ψ1 Γ e NΓ “ ~ ZΓ ψ1 e NΓc d ż ´ ˇ p q ` d0 ż c ´ p qˇ 0 ˇ ˇ By the first equationˇ in (5.28) and the Virtual Localization Theorem (5.16),ˇ the contribution İ d Z of the AI graphs to the coefficient of Q in η,β I is İ ˇ CI,j d0 İ ˇ p q Zη,β x v I, j , ~,Q uj pIq uj pIq ~“´ ~ p p p qq q Q;d´d0¨deg Ij d0 d0ě1 jPrNsÍ d0 r z ÿ ÿ ĺ ` ˇ d0¨deg Ij d ˇ whenever d d˚ satisfies the two properties in Definition 5.2 (which make evaluation at ” ~ uj pIq meaningful). An analogous statement holds when summing in the second equation d0 in“´ (5.28).

5.6 MPC for the GW power series

İ İİ İ İİ Let Z1 and Z1 be as in (4.2) and Zη,β and Zη,β be as in (5.21).

61 K 2 2 J K 1 2 I I 1 3 L J L 3 5 J 1 3

τ τ Figure 5.2: A graph representing a ﬁxed locus in Xd X ; I,J,K,L V , I J,K,L. p M q P M ‰

İİ İ ˚ τ ě0 m´2 Lemma 5.13. For all m 3,ηj H N XM , βj Z , the pairs Z1 ~,Q , ~ Zη,β ~,Q İ İİ T m´2 ě P p q P p p q p qq and Z1 ~,Q , ~ Zη,β ~,Q satisfy the MPC. p p q p qq Lemma 5.13 extends [Z1, Lemma 1.2] from the case of a positive line bundle over Pn´1 to that of a split vector bundle E E` E´ as in (1.5) over an arbitrary symplectic toric τ “ ‘ manifold XM . While [Z1, Lemma 1.2] follows from [Z1, Lemma 3.1], Lemma 5.13 follows from Lemma 5.15 below, which extends [Z1, Lemma 3.1] to the general toric case. The proof of Lemma 5.15 uses the Virtual Localization Theorem (5.16) instead of the classical one used τ n´1 in the XM P case and Lemma 5.14, which is a general toric version of the first displayed formula in“ [Z1] after [Z1, (3.32)]. 1 2 ´1 As in [Gi1] and [Z1], we consider the action of T on V C given by ξ z0,z1 z0,ξ z1 and the induced action on PV . Let ~ be the weight of the” standard action¨p of q”pT1 on C. Forq any d Λ, let P τ τ τ τ Xd X f M0,m PV X , 1, d : ev1 f 1, 0 X , ev2 f 0, 1 X . p M q” P p ˆ M p qq p qPr sˆ M p qPr sˆ M τ T1ˆTN (1 N By Proposition 5.10, the components of the fixed locus Xd XM of the T T - τ p q ˆ action on Xd XM are indexed by decorated graphs Γ of the following form. Such a graph Γ has a uniquep edgeq of positive PV -degree; this special edge corresponds to a degree-one 1 V τ map f : P PV I for some I M . Edges to the left (respectively right) of this edge ÝÑ ˆr s τ P τ are mapped into 1, 0 XM (respectively 0, 1 XM ); see Figure 5.2, where we dropped the PV -label of the vertices.r sˆ 9 Thus, the first markedr sˆ point is attached to some vertex to the left of the special edge, while the second marked point is attached to some vertex to the right of the special edge. Let dL dL Γ , dR dR Γ Λ ” p q ” p qP τ denote the XM -degrees of the left- and right-hand side (with respect to the special edge) τ T1ˆTN sub-graphs, respectively; thus, d dL dR. Let ZΓ be the component of Xd XM corresponding to Γ. “ ` p q Lemma 5.14. For every i k and d Λ, there exists Pr s P 2 τ Ωi HT1 TN Xd XM such that Ωi xi I dL Γ ~ P ˆ p p qq ZΓ “ p q`p p qqi ˇ τ T1ˆTN for all graphs Γ corresponding to components of Xd Xˇ M , with dL Γ and I depending on Γ as above. p q p q 9Figure 5.2 is [Z1, Figure 3] adapted to the toric setting.

62 Proof. We follow the proof in [Gi1, Section 11] and [Gi2, Section 2]. Given s Zě0 and n 1, let P ě n ‘n Poly P P C z0,z1 : P homogeneous of degree s . s ” t P r s u We next deﬁne a morphism` ˘

n´1 n θ0 : M0,0 PV P , 1,s Poly . p ˆ p qq ÝÑ s n´1 1 If Σ,f is an element of M0,0 PV P , 1,s , Σ Σ0 Σ1 ... Σr, where Σ0 is a P , r s p ˆ p qq “ Y Y Y f has degree 1,s0 , Σi is connected for all i r , and f has degree 0,si for all i r . Σ0 p q Pr s Σi p q Pr s Thus,ˇ ˇ ˇ n´1 ˇ ˇ f Σi Ai,Bi P for some Ai,Bˇi PV i r . p qĎtr suˆ r sP @ Pr s Let θ0 Σ,f P1g,...,Png , where r s”r s r ´1 n si f f1,f2 , f2 f P1,...,Pn Poly , g Aiz1 Biz0 . Σ0 1 s0 ” p q ˝ ”r sP ” i“1p ´ q ˇ ź ˇ Let θ θ0 fgt, where ” ˝ n´1 n´1 fgt : M0,m PV P , 1,s M0,0 PV P , 1,s p ˆ p qq ÝÑ p ˆ p qq is the forgetful morphism. By [Gi1, Section 11, Main Lemma], θ n´1 is continuous. XspP q T1 Tn n The torus acts on Polys by ˇ ˆ ˇ ξ,t1,...,tn P1 z0,z1 ,...,Pn z0,z1 t1P1 z0,ξz1 ,...,tnPn z0,ξz1 . p q ¨ p r s r sq”p r s r sq n 1 n This action naturally lifts to the hyperplane line bundle over Polys . The map θ0 is T T - equivariant and hence so is θ. ˆ τ Let L XM be any very ample line bundle. For any d Λ, let L d c1 L , d . ÝÑ N τ P p q” p q Consider the canonical lift of the T -action on XM to L given by Proposition 2.17 together N @ n D with (2.23). Thus, there exists n, an injective group homomorphism ιT : T T , and τ n´1 ˚ ÝÑ an ιT-equivariant embedding ι : XM P such that ι OPn´1 1 L. We consider the N n´1 ÝÑ 1 p Nq“ T -action on P induced by ιT. The embedding ι induces a T T -equivariant embedding ˆ τ F n´1 Xd X XL d P . p M q ÝÑ p qp q The composition τ F n´1 θ n Xd X XL d P Poly p M q ÝÑ p qp q ÝÑ Lpdq LpdRq LpdLq LpdRq LpdLq maps ZΓ onto z0 z1 a1,...,z0 z1 an , where a1,...,an ι I . 2 r n s r s” pr sq Let Ω H 1 N Poly be the equivariant Euler class of the hyperplane line bundle and P T ˆT p Lpdqq ˚ ˚ 2 τ Ω L F θ Ω H 1 N Xd X . p q” P T ˆT p p M qq It follows that

L L d L d L d L d L L Ω Z Ω p Rq p Lq p Rq p Lq e I c1 , dL ~, (5.29) p q Γ “ rz0 z1 a1,...,z0 z1 ans “ p qp q` p q ˇ ˇ @ D ˇ ˇ 63 where e L is the TN-equivariant Euler class of L. p q By Proposition 2.16, there exist very ample line bundles Li for all i k such that 2 τ N P r s c1 Li : i k is a basis for H XM ; so, using the T -action on each Li deﬁned by (2.23), twep ﬁndq thatPr su p q

Span e Li : i k Span xi : i k . Q t p q Pr su “ Q t Pr su 2 τ Via Proposition 2.21(b), this shows that e Li ,αj : i k , j N is a basis for HTN XM . t p 2 q τ Pr s 2 Pr su τ p Lq As in [Gi2], we deﬁne a Q-linear map from HTN XM to HT1ˆTN Xd XM by sending e i p q 2 p p τqq p q to Ω Li for all i k and αj to αj for all j N . Let Ωi H 1 N Xd X be the image of p q Pr s Pr s P T ˆT p p M qq xi under this map. The claim now follows from (5.29).

m β βj ˚ ˚ τ Lemma 5.15. Let η ψj evj ηj in HTN M0,m XM , d and let ” j“2 p p qq ś ´ ¯ τ τ π : M0,m PV X , 1, d M0,m X , d p ˆ M p qq ÝÑ p M q

denote the natural projection. With Φ as in Deﬁnition 5.5 and Ωi as in Lemma 5.14,

k m Ω z İ m´2 d i i ˚ β ˚ ~ Φ İİ İ ~,z,Q Q ei“1 π e V η ev e OP 1 , Z ,Z E j V p´ q 1 η,β p q“ τ vir ř p p qq dPΛ rXdpXM qs j“3 ÿ ż ” ´ ¯ ı ź (5.30) k m Ω z İİ m´2 d i i ˚ β ˚ ~ Φ İ İİ ~,z,Q Q ei“1 π e V η ev e OP 1 . Z ,Z E j V p´ q 1 η,β p q“ τ vir ř p p qq dPΛ rXdpXM qs j“3 ÿ ż ” ´ ¯ ı ź Proof. We apply the Virtual Localization Theorem (5.16) to the right-hand side of each of the two equations in (5.30), using Section 5.4 and extending the proof of [Z1, Lemma 3.1] from the case of a positive line bundle over Pn´1 to that of a split vector bundle E E` E´ τ “ ‘ as in (1.5) over an arbitrary symplectic toric manifold XM . The possible contributing fixed vir loci graphs are described above. Given such a fixed locus graph Γ, we denote by NΓ the virtual normal bundle to the corresponding component of the fixed locus inside the moduli 1 N space. We denote by AI the set of all T T -fixed loci graphs whose unique edge of positive 1 ˆ V τ PV -degree corresponds to a map P PV I , where I M . A graph Γ AI breaks into ÝÑ ˆr s P10 P 3 graphs - ΓL, ΓR, and Γ0 - as follows; see also Figure 5.3. The graph ΓL is obtained by considering all vertices and edges of Γ to the left of the special edge (of positive PV -degree) and adding a marked point labeled 2 at the vertex belonging to the special edge. Given that V τ all vertices in this “left-hand side graph” are labeled 1, 0 ,I for some I M , it defines a τ TN pr s q P component of M0,2 XM , dL . The graph ΓR is obtained by considering all vertices of Γ to the right of the specialp edgeq and adding a marked point labeled 1 at the vertex belonging to the special edge. Given that all vertices in this “right-hand side graph” are labeled 0, 1 ,I V τ τ TN pr s q for some I M , it defines a component of M0,m XM , dR . Finally, Γ0 is the special edge with 2 markedP points added. They are labeled 1 inp the left-hanq d side and 2 in the right-hand side. Thus, ZΓ ZΓ ZΓ ZΓ ; – L ˆ 0 ˆ R 10Figure 5.3 is [Z1, Figure 4] adapted to the toric setting.

64 K 2 2 1 2 1 2 J K 1 2 I I I I 1 3 L J L 3 5 J 1 3

Figure 5.3: The three sub-graphs of the graph in Figure 5.2

we denote by πL, π0, and πR the corresponding projections. It follows that İ İ İ İİ İİ İİ ˚ ˚ ˚ ˚ ˚ ˚ π VE π VE π VE, π VE π VE π VE, “ L ‘ R “ L ‘ R vir N vir N vir (5.31) NΓ ˚ ΓL ˚ ΓR ˚ ˚ ˚ ˚ π π π L2 π L1 π L2 π L1, T Xτ “ L T Xτ ‘ R T Xτ ‘ L b 0 ‘ 0 b R rIs M ˆ rIs M ˙ ˆ rIs M ˙

where L2 ZΓL ,L1,L2 ZΓ0 , and L1 ZΓR are the tautological tangent line bundles. The ﬁrst twoÝÑ equations inÝÑ (5.31) follow similarlyÝÑ to (5.22). By (5.31) and (2.34),

m İ İ İ ˚ β ˚ ˚ ˚ β m´2 π e VE η ev e OPV 1 π e VE π e VE η ~ , j ZΓ L R j“3 r p p qqs “ p´ q ” ´ ¯ ı ź ˇ ” ´ ¯ı ” ´ ¯ ı (5.32) τ ˚ ˇ ˚ e TrIsXM ˚ ev2 φI ˚ ev1 φI 1 p vir q πL vir πR vir ˚ ˚ , e N “ e N e N ~ π ψ2 ~ π ψ1 p Γ q « ΓL ff « ΓR ff p ´ L q pp´ q´ R q ` İ ˘ ` İİ ˘ and the first equation in (5.32) with VE replaced by VE also holds. By (5.32) and Lemma 5.14,

k İ m Ωizi k i“1 ˚ β ˚ e π e VE η ev e OPV 1 xipIqzi ř j ZΓ j“3 p p qq ei“1 ” ´ ¯ ı ~ m´2 ř e N virś ˇ “ p´ q e T Xτ ZΓ Γ ˇ rIs M (5.33) ż İ p q İ p q k ˚ β ˚ e VE ev φI e VE η ev φI pdLqizi~ 2 1 1 1 i“1 e Z vir vir ; ˆ $ ř Z ´ ~ ¯ψ2 ΓL e N , $ Z ´ ~¯ ψ1 e N , ż ΓL ΓL ż ΓR ΓR & ´ ˇ . & p´ q´ . İ İİ ˇ ` ˘ ` ˘ (5.33)% with VE replaced by VE also holds. In the- %dL 0 case, the ﬁrst curly bracket- on the right-hand side of (5.33) is deﬁned to be 1. By the Virtual“ Localization Theorem (5.16) and (2.21),

İ k ˚ pd q z ~ e VE ev2 φI İİ d L i i 1 ~z L i“1 Z ~ Q e Z vir 1 ,Qe I , ř ´ ~ ¯ ΓL $ ZΓ ψ2 e NΓ , “ p q ΓL & ż L ´ L . ÿ İ ˇ ˇ (5.34) V β ˇ˚ ` ˘ ˇ % e E η ev1 φI 1 - İ QdR Z ~,Q , ´ ~¯ vir η,β I $ ZΓ ψ1 e NΓ , “ p´ q ΓR ż R R ÿ & p´ q´ . ˇ ` ˘ ˇ % 65 - τ TN where the ﬁrst sum is taken after all graphs ΓL corresponding to components of M0,2 XM , dL and with second marked point mapping to I , while the second is taken after all graphsp cor-q τ r Ts N responding to components of M0,m XM , dR such that the ﬁrst marked point is mapped p q İ İİ İİ İ İ İİ to I . The equation obtained from (5.34) by replacing VE by VE, Z1 by Z1, and Zη,β by Zη,β r s İİ also holds. The claims follow from (5.33) and (5.34) (and their VE analogues), by summing τ the left-hand side of (5.33) over all graphs Γ AI and all I V . P P M 5.7 Recursivity and MPC for the explicit power series

As in [Gi2], for each I V τ we deﬁne P M ˚ ∆ d Λ: Dj d 0 j I . (5.35) I ”t P p qě @ P u By (4.6), (4.7), and (2.30), İ İİ ˚ ˚ Y x I , ~,q q;d 0 d ∆ and Y x I , ~,q q;d 0 d ∆ . (5.36) p p q q ‰ ùñ P I p p q q ‰ ùñ P I

J K İ J İ K İ Lemma 5.16. The power series Y x, ~,q of (4.7) is C-recursive with C given by (5.20). İİ p İİ q İİ The power series Y x, ~,q of (4.7) is C-recursive with C given by (5.20). p qİ Proof. The recursivity of Y in the E E` case is [Gi2, Proposition 6.3]. The proof of the İ “ İİ recursivity of Y in the general case is similar and so is the proof of the recursivity of Y. We İİ prove below the recursivity of Y extending the proof of (a) in [Z1, Section 2.3] and the proof V τ of [Gi2, Proposition 6.3]. Let I M , j N I, J v I, j , j I J. By (5.36), (4.7), Remark 5.4, and (5.9), P Pr s´ ” p q t u” ´ 0 p s ur J d uj I 1 rPrNs s“Drpd q`1 p q´ p q İİ 1 u I 1 Drpd qă0 Y j d ś ś “ ‰ x J , p q,q q 1 p q ´ d “ Drpd q d1P∆˚ s ˆ ˙ J ur J uj I D pÿd1qě´d d (5.37) j1 rPrNs s“1 p q´ p q 1 Drpśd qě0 ś “ ‰ L`pd1q´1 ´L´pd1q a i s b i s λ` J u I λ´ J u I . i d j i d j ˆ i“1 s“0 p q´ p q i“1 s“1 p q` p q ź ź ” ı ź ź ” ı By (5.37), (5.12), and (5.11),

dDrpIjq s ur I d uj I 1 rPrNs s“Drpd q`1`dDrpIjq p q´ p q İİ 1 u I 1 Drpd qă0 Y x J , j ,q qd ś ś “ ‰ p q 1 Drpd q`dDrpIjq p q ´ d “ 1 ˚ ˆ ˙ d P∆J s ÿ1 ur I d uj I Dj pd qě´d (5.38) rPrNs s“1`dDrpIjq p q´ p q D pd1qě0 p r ś ś “ ‰ ` 1 ` ´ 1 ´ a Li pd q´1`dLi pIjq b ´Li pd q´dLi pIjq ` s ´ s λ I uj I λ I uj I . ˆ i p q´ d p q i p q` d p q i“1 s dL` Ij i“1 s 1 dL´ Ij ź “ źi p q ” ı ź “ ´źi p q ” ı 66 By (5.13) and (5.19),

0 s ur I uj I d 2d´1 p q´ d p q 1 d 1 s“1`dDrpIjq CI,j d p´ q . (5.39) 2 2d´1 dDśpIjq “ ‰ p q“ d! uj I r rPrNs´tj,ju s p q r p qs ź ur I d uj I r s“1 p q´ p q p ś “ ‰ If d 1, d˚ Λ, d1 d˚ d deg Ij Λ, then, ě P ” ´ ¨ P ˚ ˚ ˚ 1 ˚ 1 d ∆ and Dj d d d ∆ and D d d (5.40) P I p qě ðñ P J jp qě´ ” ı ” ı by (5.13). By (5.36), (4.7), (5.38), (5.39), and (5.20), p

İİ 1 İİ CI,j d İİ uj I Res uj pIq Y x I ,z,q p q Y x J , p q,q uj pIq z“´ d ~ q;d˚ z r p p q qz “ ~ s p q ´ d {q;d˚´d¨deg Ij " ´ * ` d ˆ ˙ İİ ˚ 1 Y ~ for all d Λ. Finally, viewing ~´z x I ,z,q as a rational function in ,z, and αj P r p p q qzq;d˚ and using the Residue Theorem on P1, we obtain

İİ CI,j d İİ uj I İİ p q Y x J , p q,q Y x I , ~,q uj pIq ˚ ~ s p q ´ d { ˚ “ p p q q q;d dě1jPrNs´I d ˆ ˙ q;d ´d¨deg Ij r z ÿd¨degÿIjĺd˚ ` 1 İİ Resz“0,8 Y x I ,z,q , ´ ~ z p p q q q;d˚ " ´ r z *

where Resz 0, F Resz 0F Resz F. Since “ 8 ” “ ` “8

İİ 1 ´1 Resz“0,8 Y x I ,z,q Qα ~, ~ , ~ z p p q q q;d˚ P r s " ´ r z * this concludes the proof.

İ İİ İ İİ Lemma 5.17. With Y and Y deﬁned by (4.7), Y, Y satisﬁes the MPC. p q We follow the idea of the proof of [Gi2, Proposition 6.2] and begin with some preparations. ˚ Let d ∆I , V τ PIP M Ť

J J d j N : Dj d 0 , S J Dj d . ” p q”t Pr s p qě u ”| |`jPJ p q ÿ Let A be the J S matrix giving PDj pdq as in (2.37). Denote the coordinates of a point | |ˆ jPJ y CS by ś P y ,y ,...,y . j;0 j;1 j;Dj pdq jPJ ` ˘ 67 The pair MJ A, τ is toric in the sense of Definition 2.1. It satisfies (ii) in Definition 2.1, since p q V τ i p ,..., i p MJ A 1; 1 k; k : “ pp q p τ qq (5.41) i1,...,ik V , i1,...,ik J, 0 pr Di d r k t uP M t uĎ ď ď r p q @ Pr s S 0 1 ( by the second statement in Lemma 2.4(b). We identify C with H P , OP1 Dj d via jPJ p p p qqq À Dj pdq Dj pdq´r r yj;0,yj;1,...,yj;D d yj;rz z j p q jPJ ÝÑ 0 1 ˜ r“0 ¸jPJ ` ˘ ÿ τ 1 |J| 0 1 X X T T H P , O 1 D and set d MJ A. The torus acts on P j d by ” ˆ jPJ p p p qq À

ξ, tj Pj z0,z1 tjPj z0,ξz1 , (5.42) p qjPJ ¨p p qqjPJ ” p p qqjPJ ´ ¯ |J| 0 1 while the torus T acts on H P , OP1 Dj d by restricting this action via jPJ p p p qqq À T|J| t ã 1,t T1 T|J|; Q Ñ p qP ˆ these actions descend to actions on Xd.

1 |J| Lemma 5.18. ( a) The ﬁxed points of the T T -action on Xd are ˆ

I, p Pj z0,z1 , (5.43) r s” p p qqjPJ ” ı τ k where I V , I J, p pi Z , 0 pi Di d for all i I, and P M Ď “p qiPI P ď ď p q P

Dj pdq´pj pj z0 z1 , if j I; Pj z0,z1 P p q”#0, otherwise.

τ k ( b) Let I V and p pi i I Z . Then P M “p q P P ´1 ´1 ˚ 0 pi Di d i I pM , d pM ∆ . ď ď p q @ P ðñ I ´ I P I 1 |J| 2 Proof. Let Pj z0,z1 j J be any ﬁxed point of the T T -action on Xd and ξ0,ξ1 C be rp p q P qs ˆ p qP P ξ ,ξ P P ξ ,ξ Xτ such that j 0 1 0 whenever j 0. By Lemma 2.4(i) and (5.41), j 0 1 jPJ MJ . p q‰ |J| ‰ τ τ p p qq P Since Pj ξ0,ξ1 j J is a T -ﬁxed point in X , there exists I V with I J such that rp p qq P s MJ P M Ď Pj 0 if and only if j I; see Corollary 2.20(a). This concludes the proof of (a). Partr (b) ‰ P ´1 follows from (5.35) and the identity Di r iPI rMI for r pMI ; see the second equation in (2.17). p p qq ” “

68 1 N 1 N We consider the T T -action on Xd obtained by composing the projection T T 1 |J| ˆ 1 |J| ˆ ÝÑ T T induced by J ã N with the action (5.42) of T T on Xd. We denote by ˆ 1 N Ñ r s ˆ e T I,p Xd the T T -equivariant Euler class of T I,p Xd and by p r s q ˆ r s ˚ ˚ I, p : H 1 N Xd H 1 N r ¨p q T ˆT p qÝÑ T ˆT 1 N the restriction map induced by the inclusion I, p ã Xd, where I, p is the T T -ﬁxed point deﬁned by (5.43). Let ~ denote the weightr ofs theÑ standard actionr s of T1 on Cˆ.

` ´ ˚ Lemma 5.19. There exist classes xi iPrks, ur rPrNs, λi iPras, λi iPrbs HT1ˆTN Xd such that p q p q p q p q P p q

ur mirxi αr r N , (5.44) “ i“1 ´ @ Pr s ÿ and such that for all I, d1 with I V τ , I J, d1, d d1 ∆˚, and all I, p as in (5.43), p q P M Ď ´ P I r s 1 1 1 x1 I, d MI ,..., xk I, d MI x1 I ,...,xk I ~ d, (5.45) p p q p qq “ p p q p qq` e T I,p Xd uj I, p s ~ (5.46) p r s q“ r p q´ s jPJ´I 0ďsďD pdq ź źj r uj I, p s ~ , ˆ r p q´ s jPI 0ďsďDj pdq ź sź‰pj ˘ 1 ˘ ˘ 1 λ I, d MI λ I ~ L d i a i b . (5.47) i p q“ i p q` i p q @ Pr s p@ Pr sq ˚ Proof. We deﬁne the classes x1,..., xk and uj;s in H S Xd with j J and 0 s Dj d by T p q P ď ď p q (2.27) with M,τ replaced by MJ A, τ . By (2.29), p q p q r r k

uj;s mijxi αj;s, (5.48) “ i“1 ´ ÿ ˚ 8 S 8r where αj;s πj;sc1 OP8 1 and πj;s : P P is the projection onto the j; s component. ” p p qq p q ÝÑ S p q By Corollary 2.20(a), (5.41), and (2.34), the T -ﬁxed points in Xd are the points I, p and r s TS e T I,p Xd uj;s uj;s , (5.49) p r s q“ rI,ps ˆ rI,ps jPJ´I 0ďsďDj pdq jPI 0ďsďDj pdq ź ź ” ˇ ı ź sź‰pj ” ˇ ı ˇ ˇ TS S where e T I,p Xd denotes the T -equivariant Euler class of T I,p Xd and p r s q r s ˚ ˚ : H S Xd H S rI,ps T p q ÝÑ T ˇ the restriction homomorphism inducedˇ by I, p ã Xd. The map r s Ñ

8 N`1 8 S 2 Dj pdq F : C 0 C 0 , F e0,e1,...,eN ej,ej e0,ej e0,...,ej e0 , p ´t uq ÝÑ p ´t uq p q” ¨ ¨ ¨ jPJ ´ ¯ 69 where d d d 8 z1,z2,... z1 ,z2 ,... d 1, z1,z2,... C 0 , and p q ” p q @ ě p qP ´t u 8 z1,z2,... y1,y2,... ziyj Zą0 Zą0 z1,z2,... , y1,y2,... C 0 p q¨p q ” p qpi,jqP ˆ @ p q p qP ´t u is equivariant with respect to the homomorphism

1 N S 2 Dj pdq f :T T T , f ξ,t1,...,tN tj,tjξ,tjξ ,...,tjξ . ˆ ÝÑ p q” jPJ 8 N`1 ` 8 S ˘ It induces a map F : C 0 T1 TN Xd C 0 TS Xd, p ´t uq ˆ ˆ ÝÑp ´t uq ˆ

F e0,e1,...,eN , Pj j J F e0,e1,...,eN , Pj j J r rp q P ss ”r p q rp q P ss 8 N`1 e0,e1,...,eN C 0 , Pj j J Xd, @ p qPp ´ q rp q P sP ˚ ˚ ˚ and thus a homomorphism F :H S Xd H 1 N Xd . It follows that T p qÝÑ T ˆT p q ˚ F αj;s αj s~ j; s with j J, 0 s Dj d . (5.50) “ ` @ p q P ď ď p q 1 N We deﬁne xi and ur as the T T -equivariant Euler classes of the line bundles ˆ τ τ X C i Xd and X C r Xd, MJ A ˆ {„ ÝÑ MJ A ˆ {„ ÝÑ where r r Mj Pj jPJ ,c i t Pj jPJ ,tic k τ pp q q„ p q t T , Pj jPJ ,c X C (5.51) Mj Mr MJ A Pj j J ,c r t Pj j J ,t c @ P pp q qP ˆ pp q P q„ `p q P ˘ 1 N r with respect to the lifts of the` T T -action˘ on Xd given by ˆ ξ,t1,...,tN Pj z0,z1 jPJ ,c tjPj z0,ξz1 jPJ ,c and p q¨rp p qq s ” rp p qq s (5.52) ξ,t1,...,tN Pj z0,z1 j J ,c tjPj z0,ξz1 j J ,trc p q¨rp p qq P s ” rp p qq P s respectively. It follows that ˚ xi F xi (5.53) “ and uj satisfy (5.44). The latter follows similarly to the proof of (2.29) using equations analogous to (2.25) and (2.26) with TN replacedr by T1 TN . Equation (5.45) follows from (5.53), Proposition 2.21(a), and (5.50). Equation (5.55)ˆ follows from (5.44), (5.45), and (2.29). Equation (5.46) follows from (5.49) together with (5.48), (5.50), (5.53), and (5.44). Finally, deﬁne k k ` ` ´ ´ λi ℓrixr and λi ℓrixr, (5.54) ” r“1 ” r“1 ` ´ ÿ ÿ with ℓri,ℓri as in (3.5). Equations (5.47) then follow from (5.54), (5.45), and (4.4). 1 With uj and I, d as in Lemma 5.19, p q 1 1 uj I, d MI uj I ~ Dj d j N , (5.55) p q“ p q` p q @ Pr s by (5.44), (5.45), and (2.29).

70 1 N Lemma 5.20. There exists a vector bundle Vd Xd and a lift of the T T -action to Vd 1 N ÝÑ ˆ such that the T T -equivariant Euler class e Vd satisﬁes ˆ p q N ´Dj pdq´1

e Vd I, p r uj I, p s ~ p qp q“ j“1 s“1 r p q` s ź ź 1 N ˚ for all T T -ﬁxed points rI, p deﬁned by (5.43) and with uj H 1 N Xd as in Lemma 5.19. ˆ r s P T ˆT p q Proof. Let

0 1 O 1 Vd Pj jPrNs´J H P , P Dj d 1 : Pj 1, 0 0 j N J , ” #p q PjPrNs´J p´ p q´ q p q“ @ Pr s´ + τ à ` ˘ r X Vd MJ A Mj Mj k Vd ˆ Xd, Pj , Pj t Pj , t Pj t T . ” ÝÑ p qjPJ p qjPrNs´J „ jPJ jPrNs´J @ P r „ r ´ ¯ ´ ¯ Xτ X V X ` ˘ ` ˘ Since MJ A d is a principal bundle, d d is a holomorphic vector bundle. The 1 N ÝÑ ÝÑ T T -action on Xd lifts to Vd via ˆ r ξ,t1,...,tN Pj z0,z1 , Pj z0,z1 tjPj z0,ξz1 , tjPj z0,ξz1 . p q¨ p p qqjPJ p p qqjPrNs´J ” p p qqjPJ p p qqjPrNs´J ” ı ” ı The lemma now follows from the deﬁnition of uj in (5.51) and (5.52). By the Localization Theorem (2.22), Lemma 5.18, Lemma 5.20, and (5.46),

N ´1 1 1 f I, d MI uj I, d MI s~ p q j“1 s“D pdq`1 r p q´ s f e V j , (5.56) d u I, dś1M śs~ u I, d1M s~ Xd p q“ IPV τ j I j I ż M j J I r p q´ sj I r p q´ s 1 ÿ1 ˚ P ´ 0ďsďDj pdq P 0ďsďDj pdq d ,d´d P∆I 1 r ś ś ś s‰Dśj pd q ˚ for all f H 1 N Xd . P T ˆT p q Proof of Lemma 5.17. By Deﬁnition 5.5, (5.36), (4.7), (5.56), (5.45), (5.55), and (5.47),

2 ´Dj pd q´1 uj I uj I s~ jPrNs p q jPrNs s“D pd1q`1 r p q` s pxpIq`~d1q¨z j d e Dj pdqă0 Dj pdqă0 Φ İ İİ ~,z,Q Q ś ś ś Y,Y p q“ uj I uj I s~ IPV τ dP∆˚ # d1,d2P∆˚ M I I jPrNs´I p q D pdqě0 ´D pd2qďsďD pd1qr p q` s ÿ ÿ d1`ÿd2“d j j j ś ś sś‰0 ` 1 ´ 2 a Li pd q b ´Li pd q λ` I s~ λ´ I s~ ˆ i p q` i p q` i“1s“´L`pd2q`1 i“1s“L´pd1q`1 + ź źi “ ‰ ź źi “ ‰ ´ a 0 b ´Li pdq d x¨z ` ´ Q e Vd e λ s~ λ s~ . “ p q i ` i ` d ∆˚ Xd i“1 ` i“1 s“1 P I ż s“´Li pdq`1 I V τ PÿM ź ź “ ‰ ź ź “ ‰ Ť r The last expression is in Q α, ~ z, Λ . r srr ss

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74 Appendix A

Derivation of (5.2) from [LLY3]

LLY3 our notation r s m k

tj e qj

R C α1,...,αN ~ ˚ p qr s C T Q α1,...,αN r s r s α ~ T TN

eT e 2 τ c1 Ld ψ1 H M0,1 X , d p q ´ P p M q ρ forgetful morphism M` 0,1 X, d ˘M0,0 X, d X p q ÝÑ p q ed ev1 : M0,1 X, d X p q ÝÑvir LT0,1 d,X M0,1 X, d p q p q V Vd E“ M0,0 X,‰ d ˚ ÝÑ p q Ud ρ Vd VE M0,1 X, d “ ÝÑ p q Da a N uj j N p q Pr s p q Pr s τ In [LLY3, Section 3.2], we take bT eT (that is, e), X XM , and V E. By [LLY3, Section 3.2], ” “ ” E` ρ˚b V LT d,X V,bT ´H¨t{α e d¨t X T d 0,1 A t A t e p q Ade ,Ad e p qX p q , e E´ ˚ e F M X p q“ p q“ « `dPΛ´0 ﬀ “ G 0 d p q ÿ ˆ p { p qq ˙ 2 τ 2 τ where Ha HTN XM is a basis whose restriction to H XM is a basis of ﬁrst Chern classest of ampleuĂ linep bundles;q see [LLY3, Section 3,viii]. By [LLY3,p q Lemma 3.5],

eG F0 Md X α α c1 Ld . p { p qq “ p ´ p qq Thus, in our notation,

` ´H¨t{~ e E d¨t e VE A t e p q e ev1 p q , (A.1) e E´ ˚ ~ ~ ψ p q“ # `dPΛ´0 1 + p q ÿ „ p ` q 75 τ τ where ev1 :M0,1 XM , d XM is the evaluation map at the marked point. By (4.2), (A.1), and the string relationp qÝÑ [MirSym, Section 26.3],

` İ ´H¨t{~ e E t A t e p qZ1 ~,e . (A.2) p q“ e E´ p´ q p q By (4.7), Remark 5.1, and (4.4), (5.2) is independent of the choice of a Q α -basis for 2 τ r 2s τ H N X and so it is not necessary to assume that the restrictions of xi to H X are T p M q p M q Chern classes of ample line bundles. Thus, we may take H x1,...,xk in [LLY3]. By [LLY3, (5.2)] and [LLY3, Theorem 4.9], “ p q

e E` İ B t e´H¨t{~ p qY x, ~, et (A.3) p q“ e E´ ´ p q ` ˘ in [LLY3, Theorem 4.7]. In the notation of the proof of [LLY3, Theorem 4.7] correlated with Remark 5.1,

N İ İ İ 1 2 C I0 q , C I0 q G q αjgj q , C I0 q f1 q ,...,fk q , “ p q “´ p q ˜ p q`j“1 p q¸ “´ p q ¨ p p q p qq ÿ N (A.4) 1 Gpqq` α g pqq 2 C1 1 ~ j j C f{α ´ log C´ Cα « j“1 ff e e İ e ř , g f1 q ,...,fk q . “ “ I0 q “´ C “p p q p qq p q Finally, by [LLY3, Section 5.2] and [LLY3, Corollary 4.11], the hypothesis of [LLY3, Theorem 4.7] are satisfied with A t and B t as in (A.2) and (A.3) if νE d 0 for all d Λ, since e E` and e E´ arep non-zeroq wheneverp q restricted to any TN -fixedp qě point; see PropositionP 2.21(p aq). Thus,p (5.2)q follows from [LLY3, Theorem 4.7], (A.2), (A.3), and (A.4).