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Homological mirror symmetry for the quartic surface. (English) Zbl 1334.53091
Mem. Am. Math. Soc. 1116, v, 129 p. (2015).
Let \(X_0\) be a smooth quartic surface in \(\mathbb C P^3\). We can regard \(X_0\) as a symplectic manifold on the one hand, and as an algebraic variety on the other hand. Using these arguments, we can associate two kinds of triangulated categories \(D^\pi{\mathcal F}(X_0)\) and \(D^b\mathrm {Coh}(Z^\ast_q)\), both linear over \(\Lambda_{\mathbb Q}\). Here \(\Lambda_{\mathbb Q}\) is the algebraic closure of the ring of formal power series \(\Lambda_{\Lambda_{\mathbb N}}= \mathbb C[[q]]\) obtained by adjoining roots \(q^{1/d}\) of all orders to \(\Lambda_{\mathbb N}\).
\(D^\pi{\mathcal F}(X_0)\) is the split-closed derived Fukaya category, defined using Lagrangian submanifolds of \(X_0\) and pseudo-holomorphic curves with boundary on them (the Fukaya category is explained in Chapter 8). As for the use of \(\Lambda_{\mathbb Q}\), the author refers to [H. Hofer and D. A. Salamon, The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 483–524 (1995; Zbl 0842.58029)].
The group \(\Gamma_{16}= \{[\mathrm {diag}(\alpha_0,\alpha_1, \alpha_2,\alpha_3)] : \alpha^4_k= 1,\,\alpha_0\alpha_1\alpha_2\alpha_3= 1\}\subset \mathrm {PSL}(V)\), \(\Gamma_{16}\cong\mathbb Z/4\times \mathbb Z/4\), acts on the quadratic surface in \(\mathbb P_{\Lambda_{\mathbb Q}}\) defined by \[ y_0 y_1 y_2 y_3+ q(y^4_0+ y^4_1+ y^4_2+ y^4_3)= 0. \] \(Z^\ast_q\) is the unique minimal resolution of the quotient orbifold. Since \(\Lambda_{\mathbb Q}\) is an algebraically closed field of characteristic \(0\), we can apply the standard theory of algebraic surfaces in this case. \(D^b\mathrm {Coh}(Z^\ast_q)\) is the bounded derived category of coherent sheaves (Coherent sheaves are explained in Chapter 5). The object of this book is to prove the following.
Theorem 1.3. There is a \(\Psi\in\mathrm {End}(\Lambda_{\mathbb N})^\times\) and an equivalence of triangulated categories \[ D^\pi{\mathcal F}(X_0)\cong \widehat\psi^\ast D^b\mathrm {Coh}(Z^\ast_q). \] Here \(\widehat\psi\) is a lift of \(\psi\) to an automorphism of \(\Lambda_0\). This proves M. Kontsevich’s form of the mirror symmetry conjecture [Homological algebra of mirror symmetry. Proceedings of the international congress of mathematicians, ICM, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120–139 (1995; Zbl 0846.53021)], for a quartic surface. \(\psi\) is expected to agree with the standard “mirror map” [M. Nagura and K. Sugiyama, Int. J. Mod. Phys. A 10, No. 2, 233-252 (1995; Zbl 1044.14509)], but the author remarks that this is not known.
A consequence is the existence of a homomorphism \[ \pi_1({\mathcal M}^\ast)\to \operatorname{Aut}(D^b\mathrm {Coh}(Z^\ast_q))/\Lambda_{\mathbb Q}. \] (The author says that the kernel and cokernel of this homomorphism are not known.) An isomorphism \[ K_0(D^\pi{\mathcal F}(X_0))\cong K_0(D^b\mathrm {Coh}(Z^\ast_q))= K_0(Z^\ast_q) \] is shown. Here, \({\mathcal M}^\ast\) is the classifying space (or moduli stack) for some K3 surfaces equipped with an ample cohomology class \(A\), with square \(A\cdot A=4\), and with a choice of a nonzero holomorphic two-form. It carries a fiber bundle with structure group \(\operatorname{Aut}(X_0)\) and the presence of the holomorphic two-form on the fibers provides a natural lift of this to \(\widetilde{\operatorname{Aut}}(X_0)\), the central extension of \(\operatorname{Aut}(X_0)\) [the author, Bull. Soc. Math. Fr. 128, No. 1, 103–149 (2000; Zbl 0992.53059)].
The proof of Theorem 1.3 is done on the basis of tools from category theory (Chapters 2–5), using the general theory of symplectic geometry (Chapter 6) and the geometric structure of the quartic surface \(X_0\) known via Picard-Lefschetz theory (Chapter 7). The main part of the proof is given in Chapters 8 to 11. Precisely, it relies on the following three steps:
1.
To reconstruct \(D^\pi{\mathcal F}(X_0)\) entirely from the full \(A_\infty\)-subcategory of \({\mathcal F}(X_0)\) consisting of a particular set of 64 Lagrangian two-spheres, which are vanishing cycles for the standard pencil (Explanations on \(A_\infty\)-categories are given in Chapter 2). The algebraic part of this construction is given in Chapter 9 by refining the author’s previous work [Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS), vi, 326 p. (2008; Zbl 1159.53001)], together with the geometric part prepared in Chapter 7. The author remarks the meanings of this process well understood from physical discussions by K. Hori et al. [“\(D\)-branes and mirror symmetry”, Preprint, arxiv:hepth/0005247].
2.
Following a proposal by the author [Fukaya categories and deformations. Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 351–360 (2002; Zbl 1014.53052)] to formulate the relation between the Fukaya categories \({\mathcal F}(M_0)\) and \({\mathcal F}(X_0)\) in terms of the deformation theory of \(A_\infty\)-structures. Here, \(M_0\) is an affine Zariski-open subset of \(X_0\) in which all 64 vanishing cycles lie. This is done in Chapter 8 with the aid of the explanations on deformation theory given in Chapter 3.
3.
After the above processes are done, the problem is reduced to a finite collection of Lagrangian two-spheres in the affine four-manifold \(M_0\). To compute the relevant full \(A_\infty\)-subcategory of \({\mathcal F}(M_0)\), a general dimensional induction argument [the author, Fukaya categories and Picard-Lefschetz theory, loc. cit.] is applied. The algebraic part of this process is given in Chapter 10, and a subtle geometric part is done in Chapter 11 with the aid of computers (the code is available on the author’s homepage).
This book is written not only to present a proof of Kontsevich’s form of the mirror symmetry conjecture for quartic surfaces, but also to present explanations on necessary advanced knowledges from symplectic geometry and algebraic geometry. Therefore this book is readable for those who have basic knowledges in differential geometry and algebraic geometry. We can recommend it not only to learn the proof of Theorem 1.3, but also to learn a new area of geometry developing by the cooperation of symplectic geometry and algebraic geometry with the influence of quantum physics.

MSC:
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
18E30 Derived categories, triangulated categories (MSC2010)
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