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Homological Mirror Symmetry for the Quartic Surface Paul Seidel EMOIRS M of the American Mathematical Society Volume 236 • Number 1116 (sixth of 6 numbers) • July 2015 Homological Mirror Symmetry for the Quartic Surface Paul Seidel ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society EMOIRS M of the American Mathematical Society Volume 236 • Number 1116 (sixth of 6 numbers) • July 2015 Homological Mirror Symmetry for the Quartic Surface Paul Seidel ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society Providence, Rhode Island Library of Congress Cataloging-in-Publication Data Seidel, P. Homological mirror symmetry for the quartic surface / Paul Seidel. pages cm. – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; volume 236, number 1116) Includes bibliographical references. ISBN 978-1-4704-1097-1 (alk. paper) 1. Mirror symmetry. 2. Quartic surfaces. I. Title. QC174.17.S9S45 2015 516.1–dc23 2015007757 DOI: http://dx.doi.org/10.1090/memo/1116 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2015 subscription begins with volume 233 and consists of six mailings, each containing one or more numbers. Subscription prices for 2015 are as follows: for paper delivery, US$860 list, US$688.00 institutional member; for electronic delivery, US$757 list, US$605.60 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, add US$10 for delivery within the United States; US$69 for outside the United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. 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Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. c 2014 by the American Mathematical Society. All rights reserved. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. This publication is indexed in Mathematical ReviewsR , Zentralblatt MATH, Science Citation IndexR , Science Citation IndexTM-Expanded, ISI Alerting ServicesSM , SciSearchR , Research AlertR , CompuMath Citation IndexR , Current ContentsR /Physical, Chemical & Earth Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Contents Chapter 1. Introduction 1 Chapter 2. A∞-categories 7 Chapter 3. Deformation theory 15 Chapter 4. Group actions 23 Chapter 5. Coherent sheaves 29 Chapter 6. Symplectic terminology 33 Chapter 7. Monodromy and negativity 45 Chapter 8. Fukaya categories 61 Chapter 9. Computations in Fukaya categories 81 Chapter 10. The algebras Q4 and Q64 91 Chapter 11. Counting polygons 101 References 127 iii Abstract We prove Kontsevich’s form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in CP 3. Received by the editor November 1, 2011 and, in revised form, June 7, 2013. Article electronically published on December 29, 2014. DOI: http://dx.doi.org/10.1090/memo/1116 2010 Mathematics Subject Classification. Primary 53D37; Secondary 14D05, 18E30. Key words and phrases. Homological mirror symmetry, Floer cohomology, derived category, Lefschetz fibration. Partially supported by NSF grant DMS-1005288 and a Simons Investigator Grant from the Simons Foundation. c 2014 American Mathematical Society v CHAPTER 1 Introduction This paper deals with a special case of Kontsevich’s “homological mirror sym- metry” conjecture [38]. Before formulating the statement, we need to introduce the relevant coefficient rings. Notation 1.1. Let ΛN = C[[q]] be the ring of formal power series in one variable. We denote by ΛZ its quotient field, obtained by formally inverting q. By adjoining 1/d roots q of all orders to ΛZ, one obtains its algebraic closure ΛQ.Elementsofthis field are formal series m (1.1) f(q)= amq m where m runs over all numbers in (1/d)Z ⊂ Q for some d ≥ 1 which depends on f,andam ∈ C vanishes for all sufficiently negative m.Geometrically,one should think of ΛN as functions on a small (actually infinitesimally small) disc. Then ΛZ corresponds to the punctured disc (with the origin removed), and ΛQ to its universal cover. Consider the semigroup End(ΛN) of q-adically continuous C- algebra endomorphisms of ΛN. More concretely, these are substitutions (1.2) ψ∗ : q −→ ψ(q) with ψ ∈ qC[[q]]. For each nonzero ψ, ψ∗ extends to an endomorphism of the quotient field. The q-adically continuous Galois group of ΛZ/C is the group of × invertibles End(ΛN) , consisting of those ψ with ψ (0) =0 . Moreover, the Ga- ∗ lois group of ΛQ/ΛZ is the profinite group Zˆ = Hom(Q/Z, C ), whose topological generator 1ˆ takes qm to e2πimqm. On the symplectic side of mirror symmetry, take any smooth quartic surface 3 X0 ⊂ CP , with its standard symplectic structure. To this we associate a triangu- π lated category linear over ΛQ, the split-closed derived Fukaya category D F(X0), defined using Lagrangian submanifolds of X0 and pseudo-holomorphic curves with boundary on them (as well as a formal closure process). The use of formal power se- ries to take into account the area of pseudo-holomorphic curves is a familiar device (the relevant coefficient rings usually go under the name “Novikov rings”, see e.g. [30]; what we are using here is a particularly simple instance). ΛQ appears because we allow only a certain class of Lagrangian submanifolds, namely the rational ones (there is also another restriction, vanishing of Maslov classes, which is responsible for making the Fukaya category Z-graded). P3 On the complex side, we start with the quartic surface in ΛQ defined by 4 4 4 4 (1.3) y0y1y2y3 + q(y0 + y1 + y2 + y3)=0. 1 2PAULSEIDEL Note that here q is a “constant” (an element of the ground field). The group Γ16 = { 4 }⊂ ∼ Z × Z [diag(α0,α1,α2,α3)] : αk =1,α0α1α2α3 =1 PSL(V ), Γ16 = /4 /4, ∗ acts on this surface in the obvious way. We denote by Zq the unique minimal resolution of the quotient orbifold. Note that ΛQ is an algebraically closed field of characteristic zero, so the standard theory of algebraic surfaces applies, including minimal resolutions. The associated category is the bounded derived category of b ∗ coherent sheaves, D Coh(Zq ), which is also a triangulated category linear over ΛQ. Remark . ∗ 1.2 In fact, Zq can be defined over the smaller field ΛZ. This is easy to see if one argues “by hand” as follows. The action of Γ16 on (1.3) has 24 points with nontrivial isotropy; their isotropy subgroups are isomorphic to Z/4. Each of these points is defined over ΛZ, hence can be blown up over that field. Take the blowup, and divide it by the action of the subgroup Z/2 × Z/2=Γ16 ∩{(±1, ±1, ±1, ±1)}. The quotient is a regular algebraic variety over ΛZ, and carries an action of the ∼ group Γ16/(Z2 × Z2) = Z2 × Z2,with24 points that have nontrivial isotropy Z/2. These points are again defined over ΛZ, and one repeats the previous process to ∗ get an algebraic surface which is a ΛZ-model of Zq . (There are also other possible P3 strategies, such as using a toric resolution of ΛZ /Γ16 and realizing the desired algebraic surface inside that). × Theorem 1.3. There is a ψ ∈ End(ΛN) and an equivalence of triangulated cate- gories, πF ∼ ˆ∗ b ∗ (1.4) D (X0) = ψ D Coh(Zq ). Here ψˆ is a lift of ψ to an automorphism of ΛQ, which we use to change the b ∗ ˆ∗ ΛQ-module structure of D Coh(Zq ), by letting ψ f act instead of f.Thetheorem π says that the outcome of this “reparametrization” is equivalent to D F(X0).
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