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Closed-String Mirror Symmetry for Log Calabi-Yau Surfaces

Hyunbin Kim

TL;DR

The paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, showing $ ext{Jac}(W)\\cong QH^{*}(X)$ where $W$ is the Landau-Ginzburg potential arising from SYZ mirror symmetry. It analyzes how blowups and, crucially, blowdowns of $(-1)$-divisors affect the critical locus of $W$, proving that each blowdown reduces the number of geometric critical points by exactly one and preserves Morse non-degeneracy. A key technical advance is the use of the star-shaped Newton polytope to accurately count geometric critical points and to track potential changes under blowdown, together with energy considerations of basic and broken disks. The authors then prove that $QH^{*}(X)$ is semisimple by linking distinct critical values of $W$ to eigenvalues of the quantum product by $c_{1}(X)$, solidifying closed-string mirror symmetry for this class and highlighting implications for the spectral theory of quantum cohomology in non-Fano settings.

Abstract

This paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, where the exceptional divisor are sufficiently small. We demonstrate that blowing down a $(-1)$-divisor removes a single geometric critical point, ensuring that the resulting potential remains a Morse function. Additionally, we show that the critical values are distinct, which implies that the quantum cohomology $QH^{\ast}(X)$ is semi-simple.

Closed-String Mirror Symmetry for Log Calabi-Yau Surfaces

TL;DR

The paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, showing where is the Landau-Ginzburg potential arising from SYZ mirror symmetry. It analyzes how blowups and, crucially, blowdowns of -divisors affect the critical locus of , proving that each blowdown reduces the number of geometric critical points by exactly one and preserves Morse non-degeneracy. A key technical advance is the use of the star-shaped Newton polytope to accurately count geometric critical points and to track potential changes under blowdown, together with energy considerations of basic and broken disks. The authors then prove that is semisimple by linking distinct critical values of to eigenvalues of the quantum product by , solidifying closed-string mirror symmetry for this class and highlighting implications for the spectral theory of quantum cohomology in non-Fano settings.

Abstract

This paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, where the exceptional divisor are sufficiently small. We demonstrate that blowing down a -divisor removes a single geometric critical point, ensuring that the resulting potential remains a Morse function. Additionally, we show that the critical values are distinct, which implies that the quantum cohomology is semi-simple.
Paper Structure (13 sections, 8 theorems, 55 equations, 4 figures)

This paper contains 13 sections, 8 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Critical points of Mirror Landau-Ginzburg Potentials
  3. SYZ Mirror Construction and Lagrangian Floer Theory
  4. Blowups of Surfaces and Wall-Crossing
  5. The Toric Model
  6. Non-toric Blowups of $\left( X_{\Sigma},D_{\Sigma} \right)$
  7. Critical points of the Landau-Ginzburg Model
  8. Blowdown of Log Calabi-Yau Surfaces
  9. Notations and Geometric Setup
  10. The blowdown $(X, D)$ of $( \widetilde{X}, \widetilde{D} )$
  11. Weakly unobstructedness of $L_{u}$
  12. Number of geometric critical points
  13. Mirror symmetry for log Calabi-Yau surfaces

Key Result

Theorem 2.1

Kush Let $\mathbb{K}$ be an algebraically closed field with characteristic 0. If a convenient Laurent polynomial $W \in \mathbb{K}[z_{1}^{\pm}, \dots, z_{n}^{\pm}]$ is non-degenerate, then where $\rvert \mathrm{Crit}(W) \rvert$ is the number of critical points of $W$ counted with multiplicity, and $V_{n} (\Delta_W)$ is the $n$-dimensional volume of the Newton polytope $\Delta_W$.

Figures (4)

  • Figure 1: (Case I.)
  • Figure 2:
  • Figure 3: $(X', D' )$ and $(\underline{X}, \underline{D})$. The moment polytope of $(\underline{X}, \underline{D})$ is shown in red. The tropicalization of $W'$ is shown in blue.
  • Figure 4: $(X", D" )$ and $(\underline{X}, \underline{D})$. The moment polytope of $(\underline{X}, \underline{D})$ is shown in red. The tropicalization of $W"$ is shown in blue.

Theorems & Definitions (20)

  • Remark 2.1
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.2: HK23 Proposition 4.2
  • Remark 2.3
  • Proposition 2.3: HK23
  • Lemma 3.1
  • proof
  • Remark 3.1
  • ...and 10 more