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Homological mirror symmetry for curves of higher genus

Alexander I. Efimov

TL;DR

The paper extends homological mirror symmetry to genus $g\ge 3$ curves by proving an equivalence $D^{\pi}(\mathcal{F}(M)) \simeq \overline{D_{sg}}(H)$ between the Fukaya category of a genus-$g$ curve and the category of singularities of a corresponding Landau–Ginzburg model with explicit superpotential $W$ and singular fiber $H$. It leverages Kontsevich formality to classify $A_{\infty}$-structures on $\Lambda(V)$ via formal polyvector fields, showing the $A_{\infty}$-structure on the B-side is governed by $W$, and establishes a reconstruction theorem to recover $W$ from the endomorphism DG algebra $\mathcal B_W$ up to formal changes of variables. A McKay-type argument relates two LG presentations, proving their categories of singularities are equivalent, and together with an explicit construction of generators for the Fukaya category, yields the main HMS equivalence for curves of genus $g\ge 3$. The work also develops a reconstruction framework for hypersurface singularities and extends the McKay correspondence to the LG setting, linking to matrix factorizations and Orlov-type results.

Abstract

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. Seidel \cite{Se} has proved a version of this conjecture in the simplest case of the genus two curve. Basing on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus $g\geq 3,$ relating the Fukaya category of a genus $g$ curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D$(\Z/2)$-G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2.

Homological mirror symmetry for curves of higher genus

TL;DR

The paper extends homological mirror symmetry to genus curves by proving an equivalence between the Fukaya category of a genus- curve and the category of singularities of a corresponding Landau–Ginzburg model with explicit superpotential and singular fiber . It leverages Kontsevich formality to classify -structures on via formal polyvector fields, showing the -structure on the B-side is governed by , and establishes a reconstruction theorem to recover from the endomorphism DG algebra up to formal changes of variables. A McKay-type argument relates two LG presentations, proving their categories of singularities are equivalent, and together with an explicit construction of generators for the Fukaya category, yields the main HMS equivalence for curves of genus . The work also develops a reconstruction framework for hypersurface singularities and extends the McKay correspondence to the LG setting, linking to matrix factorizations and Orlov-type results.

Abstract

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. Seidel \cite{Se} has proved a version of this conjecture in the simplest case of the genus two curve. Basing on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus relating the Fukaya category of a genus curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D-G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2.

Paper Structure

This paper contains 16 sections, 32 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Maurer-Cartan theory for pro-nilpotent DG Lie algebras
  3. $A_{\infty}$-structures and formal polyvector fields
  4. Classification lemma for polyvector fields
  5. Classification theorem on $A_{\infty}$-structures
  6. Categories of singularities and matrix factorizations
  7. Minimal $A_{\infty}$-model for ${\mathcal{B}}_W$
  8. Reconstruction theorem
  9. Equivalence of two LG models
  10. Generalities on Fukaya categories
  11. The definition
  12. Split-generators in Fukaya categories
  13. Additional $\mathbb{Z}$-gradings
  14. Fukaya categories of orbifolds
  15. Fukaya category of a genus $g\geq 3$ curve
  16. ...and 1 more sections

Key Result

Theorem 1.1

The triangulated categories $D^{\pi}(F(M))$ and $\overline{D_{sg}}(H)$ are equivalent.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 3.1
  • ...and 51 more