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Homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$

Mohammed Abouzaid, Denis Auroux

Abstract

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in $(\mathbb{C}^*)^n$ (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a "fiberwise wrapped" version of the Fukaya category of a toric Landau-Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasi-embeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

Homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$

Abstract

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a "fiberwise wrapped" version of the Fukaya category of a toric Landau-Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasi-embeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

Paper Structure

This paper contains 33 sections, 54 theorems, 246 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fiberwise wrapped Fukaya categories
  3. A Floer cohomology calculation
  4. A functorial perspective
  5. Complete intersections and compactifications
  6. Related works
  7. The mirror Landau-Ginzburg model
  8. The main construction
  9. Construction as a Hamiltonian reduction
  10. The Fukaya category of a Landau-Ginzburg model
  11. Landau-Ginzburg models
  12. Maximum principle and energy estimates
  13. Definition of the directed category
  14. Quasi-units and continuation maps
  15. Fiberwise wrapped category via colimits
  16. ...and 18 more sections

Key Result

Theorem 1.2

For a suitable choice of bulk deformation class $\mathfrak{b}\in H^2(Y,\Lambda_{\geq 0})$, the fiberwise wrapped Floer cohomology ring $H\mathcal{W}^*(L_0,L_0)$ is isomorphic to the quotient $\mathbb{K}[x_1^{\pm 1},\dots,x_n^{\pm 1}]/(f)$ of the ring of Laurent polynomials by the ideal generated by

Figures (5)

  • Figure 1: Constructing the mirror: $f(x_1,x_2)=1+x_1+x_2+t^{2\pi}x_1x_2+t^{4\pi}x_1^2$
  • Figure 2: A homotopy between $\mu^2(e_{L_1^j},\cdot)$ and $F(\mu^2(\cdot,e_{L_0^k}))$.
  • Figure 3: A homotopy between $\mu^2(e_{L_1^j},\cdot)$ and $\mu^2(F(\cdot),e_{L_0^k})$.
  • Figure 4: The extremal vectors $\mathbf{v}\in\mathcal{V}$ and the regions $S_{\mathbf{v},\gamma}$, for $f(x_1,x_2)=1+x_1+x_2+t^{2\pi}x_1x_2+t^{4\pi}x_1^2$ (cf. Example \ref{['ex:example_42']}).
  • Figure 5: The Lagrangians $L_0$ and $L_0(t)=\phi^{t}\rho^{t}(L_0)$ ($t<0$).

Theorems & Definitions (110)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • ...and 100 more