Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homological Mirror Symmetry for orbifold log Calabi-Yau surfaces

Bogdan Simeonov

Abstract

We construct mirror abstract Lefschetz fibrations associated to a class of surfaces with cyclic quotient singularities which we call effective. These surfaces can be obtained by contracting disjoint chains of smooth rational curves inside the anticanonical cycle $D$ of a smooth log Calabi-Yau surface $(Y,D)$ with maximal boundary and considering the result as an orbifold. The Fukaya-Seidel categories of these abstract Lefschetz fibrations admit semiorthogonal decompositions akin to the ones described via the derived special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. We apply this construction to establish an equivalence at the large volume limit between the derived category of an effective orbifold log Calabi-Yau surface with points of type $\frac{1}{k}(1,1)$ and the Fukaya-Seidel category of its mirror Lefschetz fibration. We also compare the abstract construction to an explicit Landau-Ginzburg model defined by a Laurent polynomial associated to a toric degeneration in the case of the family of hypersurfaces $X_{k+1}\subset \mathbb{P}(1,1,1,k)$. The hypersurfaces $X_{k+1}$ admit a non-trivial moduli of complex structures, which we compare with an open subset of the space of symplectic structures on the total space of the mirror Landau-Ginzburg model via a mirror map built out of intrinsic quantities in a non-exact Fukaya-Seidel category.

Homological Mirror Symmetry for orbifold log Calabi-Yau surfaces

Abstract

We construct mirror abstract Lefschetz fibrations associated to a class of surfaces with cyclic quotient singularities which we call effective. These surfaces can be obtained by contracting disjoint chains of smooth rational curves inside the anticanonical cycle of a smooth log Calabi-Yau surface with maximal boundary and considering the result as an orbifold. The Fukaya-Seidel categories of these abstract Lefschetz fibrations admit semiorthogonal decompositions akin to the ones described via the derived special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. We apply this construction to establish an equivalence at the large volume limit between the derived category of an effective orbifold log Calabi-Yau surface with points of type and the Fukaya-Seidel category of its mirror Lefschetz fibration. We also compare the abstract construction to an explicit Landau-Ginzburg model defined by a Laurent polynomial associated to a toric degeneration in the case of the family of hypersurfaces . The hypersurfaces admit a non-trivial moduli of complex structures, which we compare with an open subset of the space of symplectic structures on the total space of the mirror Landau-Ginzburg model via a mirror map built out of intrinsic quantities in a non-exact Fukaya-Seidel category.
Paper Structure (63 sections, 39 theorems, 150 equations, 24 figures)

This paper contains 63 sections, 39 theorems, 150 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Context: mirror symmetry and orbifold del Pezzo surfaces
  4. The derived category of an orbifold surface
  5. The construction of the mirror abstract Lefschetz fibration
  6. Example
  7. The case of log CY orbifold surfaces with singularities of type $\frac{1}{k}(1,1)$
  8. Strategy of the proof of Theorem \ref{['thm:1/k(1,1)']}
  9. Describing the gluing structure on the B-side by restricting to a categorical resolution of $\mathrm{Perf}\,(D)$
  10. Describing the gluing structure on the A-side by restricting to a partially wrapped Fukaya category of a torus
  11. The family of del Pezzo surfaces $X_{k+1}$
  12. The derived category of $X_{k+1}$
  13. The mirror LG model
  14. Homological mirror symmetry for $X_{k+1}$
  15. Contracting a $-k$ curve and a Lefschetz stabilization procedure
  16. ...and 48 more sections

Key Result

Theorem 1

Suppose $(\mathcal{X},D^{orb})$ is an effective log Calabi-Yau surface whose orbifold points $\{p_i\}_{i=1}^m$ are of type $\frac{1}{k_i}(1,1)$, with $k_i>2$. Assume that $Y$, the minimal resolution of the coarse space of $\mathcal{X}$, is equipped with the distinguished complex structure as in hack

Figures (24)

  • Figure 1: The general fiber of the Lefschetz fibration associated to a surface with a $\frac{1}{5}(1,3)$ orbifold point. The non-special handles are drawn with a dash. The blue curve describes the Lagrangian $\tilde{L}_2$ and the red one describes $\tilde{L}_4$.
  • Figure 2: A reference fiber (green), $k-2$ critical values which go to infinity as $s\rightarrow 0$ (purple), three critical values close to $0$ in blue, as well as a critical value at $-1$ (orange) with multiplicity $k+1$, in the case $k=15, \mathbf{q}_i=1, \tau_1=1, \tau_j=0,j>1$.
  • Figure 3: The case $k=5$: a surface of genus $3$ with two boundary components, which can be viewed as a torus with three boundary components and $k-2=3$ handles attached. The blue curve describes the vanishing cycle $\tilde{L}_{k-1}$ mirror to $e_{k-1}=e_4$ and the red one describes the mirror $\tilde{L}_{k-2}$ to $e_{k-2}=e_3$. There is also another Lagrangian vanishing cycle $\tilde{L}_2$ that is not depicted.
  • Figure 4: The case $\tfrac{1}{7}(1,1)$. Thick arrows are degree 1 and dashed ones are degree 2.
  • Figure 5: The mirror to two $[\mathbb{A}^1/\mu_n]$'s glued at a $\frac{1}{n}(1,1)$ point is given by two cylinders glued together with $n$ handles. The picture above depicts the Lagrangian $\tilde{L}_1$ mirror to the Koszul resolution of $\mathcal{O}_0\otimes \rho_1$ in the case $n=5$.
  • ...and 19 more figures

Theorems & Definitions (95)

  • Theorem 1: Homological Mirror Symmetry at the large complex structure limit
  • Theorem 2
  • Remark 3
  • Remark 5
  • Definition 6
  • Remark 7
  • Remark 8
  • Theorem 10
  • Remark 11
  • Definition 1.1
  • ...and 85 more