Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 114 Tropical Geometry and Mirror Symmetry Mark Gross American Mathematical Society with support from the National Science Foundation cbms-114-gross-cov.indd 1 11/16/10 1:58 PM Tropical Geometry and Mirror Symmetry http://dx.doi.org/10.1090/cbms/114 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 114 Tropical Geometry and Mirror Symmetry Mark Gross Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS-Regional Conference in the Tropical Geometry and Mirror Symmetry held at Kansas State University, Manhattan, Kansas December 13–17, 2008 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of the Mathematical Sciences and NSF grant DMS-0735319 2000 Mathematics Subject Classification. Primary 14T05, 14M25, 14N35, 14J32, 14J33, 52B20. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-114 Library of Congress Cataloging-in-Publication Data Gross, Mark, 1965– Tropical geometry and mirror symmetry / Mark Gross. p. cm. — (Conference Board of the Mathematical Sciences regional conference series in mathematics ; no. 114) Includes bibliographical references and index. ISBN 978-0-8218-5232-3 (alk. paper) 1. Tropical geometry. 2. Mirror symmetry. I. Title. QA582.G76 2010 516.35—dc22 2010043384 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. 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Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 “Maybe the tropics,” somebody, probably the General said, “but never the Polar Region, it’s too white, too mathematical up there.” —Thomas Pynchon, Against the Day. Contents Preface ix Introduction xi Part 1. The three worlds 1 Chapter 1. The tropics 3 1.1. Tropical hypersurfaces 3 1.2. Some background on fans 11 1.3. Parameterized tropical curves 13 1.4. Affine manifolds with singularities 19 1.5. The discrete Legendre transform 27 1.6. Tropical curves on tropical surfaces 30 1.7. References and further reading 32 Chapter 2. The A- and B-models 33 2.1. The A-model 33 2.2. The B-model 67 2.3. References and further reading 89 Chapter 3. Log geometry 91 3.1. A brief review of toric geometry 92 3.2. Log schemes 98 3.3. Log derivations and differentials 112 3.4. Log deformation theory 117 3.5. The twisted de Rham complex revisited 126 3.6. References and further reading 129 Part 2. Example: P2. 131 Chapter 4. Mikhalkin’s curve counting formula 133 4.1. The statement and outline of the proof 133 4.2. Log world → tropical world 138 4.3. Tropical world → log world 143 4.4. Classical world → log world 157 4.5. Log world → classical world 165 4.6. The end of the proof 169 4.7. References and further reading 171 Chapter 5. Period integrals 173 5.1. The perturbed Landau-Ginzburg potential 173 vii viii CONTENTS 5.2. Tropical descendent invariants 179 5.3. The main B-model statement 187 5.4. Deforming Q and P1,...,Pk 192 5.5. Evaluation of the period integrals 222 5.6. References and further reading 244 Part 3. The Gross-Siebert program 245 Chapter 6. The program and two-dimensional results 247 6.1. The program 247 6.2. From integral tropical manifolds to degenerations in dimension two 255 6.3. Achieving compatibility: The tropical vertex group 290 6.4. Remarks and generalizations 305 6.5. References and further reading 306 Bibliography 307 Index of Symbols 313 General Index 315 Preface The NSF-CBMS conference on Tropical Geometry and Mirror Symmetry was held at Kansas State University during the period December 13–17, 2008. It was organized by Ricardo Casta˜no-Bernard, Yan Soibelman, and Ilia Zharkov. During this time, I gave ten hours of lectures. In addition, talks were given by M. Abouzaid, K.-W. Chan, C. Doran, K. Fukaya, I. Itenberg, L. Katzarkov, A. Mavlyutov, D. Morrison, Y.-G. Oh, T. Pantev, B. Siebert, and B. Young. My talks were meant to give a snapshot of a long-term program currently being carried out with Bernd Siebert aimed at achieving a fundamental conceptual understanding of mirror symmetry. Tropical geometry emerges naturally in this program, so in the lectures I took a rather ahistorical point of view. Starting with the tropical semi-ring, I developed tropical geometry and explained Mikhalkin’s tropical curve-counting formulas, outlining the proof given by Nishinou and Siebert. I then explained my recent work in connecting this to the mirror side. Finally, I sketched the ideas behind recent work by myself and Siebert on constructing degenerations of Calabi-Yau manifolds from affine manifolds with singularities. This monograph follows the structure of the lectures closely, filling in many de- tails which were not given there. Like the lectures, this monograph only represents a snapshot of an evolving program, but I hope it will be useful to those who may wish to become involved in this program. NSF grant DMS-0735319 provided support both for the conference and for this book. Mark Gross, La Jolla, 2010 ix Introduction The early history of mirror symmetry has been told many times; we will only summarize it briefly here. The story begins with the introduction of Calabi-Yau compactifications in string theory in 1985 [11]. The idea is that, since superstring theory requires a ten-dimensional space-time, one reconciles this with the observed universe by requiring (at least locally) that space-time take the form R1,3 × X, where R1,3 is usual Minkowski space-time and X is a very small six-dimensional Riemannian manifold. The desire for the theory to preserve the supersymmetry of superstring theory then leads to the requirement that X have SU(3) holonomy, i.e., be a Calabi-Yau manifold. Thus string theory entered the realm of algebraic geometry, as any non-singular projective threefold with trivial canonical bundle carries a metric with SU(3) holonomy, thanks to Yau’s proof of the Calabi conjecture [113]. This generated an industry in the string theory community devoted to produc- ing large lists of examples of Calabi-Yau threefolds and computing their invariants, the most basic of which are the Hodge numbers h1,1 and h1,2. In 1989, a rather surprising observation came out of this work. Candelas, Lynker and Schimmrigk [12] provided a list of Calabi-Yau hypersurfaces in weighted projective space which exhibited an obvious symmetry: if there was a Calabi-Yau threefold with Hodge numbers given by a pair (h1,1,h1,2), then there was often also one with Hodge numbers given by the pair (h1,2,h1,1). Independently, guided by certain observations in conformal field theory, Greene and Plesser [36] studied the 4 quintic threefold and its mirror partner. If we let Xψ be the solution set in P of the equation 5 ··· 5 − x0 + + x4 ψx0x1x2x3x4 =0 for ψ ∈ C,thenformostψ, Xψ is a non-singular quintic threefold, and as such, has Hodge numbers 1,1 1,2 h (Xψ)=1,h(Xψ) = 101. On the other hand, the group {(a ,...,a )|a ∈ μ , 4 a =1} G = 0 4 i 5 i=0 i {(a, a, a, a, a)|a ∈ μ5} acts diagonally on P4,via (x0,...,x4) → (a0x0,...,a4x4). Here μ5 is the group of fifth roots of unity. This action restricts to an action on Xψ, and the quotient Xψ/G is highly singular. However, these singularities can be xi xii INTRODUCTION resolved via a proper birational morphism Xˇψ → Xψ/G with Xˇψ anewCalabi-Yau threefold with Hodge numbers 1,1 1,2 h (Xˇψ) = 101,h(Xˇψ)=1. These examples were already a surprise to mathematicians, since at the time very few examples of Calabi-Yau threefolds with positive Euler characteristic were known (the Euler characteristic coinciding with 2(h1,1 − h1,2)). Much more spectacular were the results of Candelas, de la Ossa, Green and Parkes [10]. Guided by string theory and path integral calculations, Candelas et al. conjectured that certain period calculations on the family Xˇψ parameterized by ψ would yield predictions for numbers of rational curves on the quintic threefold. They carried out these calculations, finding agreement with the known numbers of rational curves up to degree 3. We omit any details of these calculations here, as they have been exposited in many places, see e.g., [43]. This agreement was very surprising to the mathematical community, as these numbers become increasingly difficult to compute as the degree increases. The number of lines, 2875, was known in the 19th century, the number of conics, 609250, was computed only in 1986 by Sheldon Katz [66], and the number of twisted cubics, 317206375, was only computed in 1990 by Ellingsrud and Strømme [22].