Homological Mirror Symmetry for Affine Log Calabi-Yau Surfaces

=== DRAFT ===
Problem Statement

The paper addresses the Homological Mirror Symmetry (HMS) conjecture for smooth complex affine log Calabi–Yau surfaces. Specifically, given such a surface $U$, the goal is to construct a "mirror" object $U^\vee$—a finite-type quasi-projective $\mathbb{Z}$-scheme—such that the derived category of coherent sheaves on $U^\vee$ is equivalent to the derived wrapped Fukaya category of the Liouville completion $\widehat{U}$ of $U$.

The surface $U$ is defined as the complement $Y \setminus D$ of a reduced, effective anticanonical divisor $D$ in a smooth projective surface $Y$. The paper distinguishes two non-overlapping cases based on the geometry of the compactification $(Y, D)$:

1. Looijenga interior: $D$ contains at least one node in any log Calabi–Yau compactification.
2. Elliptic: $D$ is smooth (an elliptic curve).

While the wrapped Fukaya category $W(\widehat{U}, k)$ is well-defined up to deformation equivalence, explicitly constructing the algebraic mirror $U^\vee$ and proving the equivalence $D^\pi W(\widehat{U}, k) \cong D^b \text{Coh}(U^\vee_k)$ for arbitrary fields $k$ remains a challenge, particularly for the elliptic case where previous algorithms were less uniform.

Methodology

The author's approach relies on translating the symplectic geometry of $U$ into combinatorial data via "rational ray diagrams" and then applying existing mirror symmetry theorems. The methodology proceeds in three main stages:

1. Construction of Generalized Toric Models:
   The paper establishes that every affine log Calabi–Yau surface admits a "generalized toric model." This involves starting with a smooth projective toric surface $(X, D_X)$, performing "nodal smoothing" (deforming chains of boundary components into smooth curves), and then performing a sequence of "non-toric blowups" at specific points (specifically at the point $-1 \in \mathbb{G}_m$ on the boundary divisors).

   • For Looijenga interiors, the model is derived from a toric start where the boundary components are smoothed and blown up.
   • For elliptic cases, the models are derived from $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$. The author utilizes "Symington triangles" and "nodal trades" (local modifications replacing corners with nodes) to encode the non-toric blowups. In the elliptic case, the construction relies on specific Markov triples (e.g., $(29, 5, 2)$) to embed the necessary number of disjoint Symington triangles.

2. Encoding via Rational Ray Diagrams:
   The geometric operations (blowups and nodal trades) are translated into a "rational ray diagram" $R$. This diagram consists of a finite collection of rational slope rays in $\mathbb{R}^2$ starting from the origin, each equipped with a non-negative integer multiplicity representing the number of nodes (or blowups) on that ray.

   • The diagram defines a Liouville manifold $U_R$ (via an almost toric fibration with focus-focus singularities) and a corresponding algebraic scheme $U^\vee_R$ (constructed by performing toric blowups and non-toric blowups at $-1$ on a toric surface defined by the fan of the rays).

3. Symplectic Torelli Argument:
   The core technical contribution is proving that the Liouville manifold $\widehat{U}$ associated with the original surface $U$ is strongly exact symplectomorphic to the manifold $U_R$ constructed from the diagram.

   • The author constructs a "preferred grading structure" on the complements of the boundary divisors using the holomorphic volume form.
   • Using the Symplectic Torelli Theorem for symplectic log Calabi–Yau pairs (specifically for orthogonal pairs), the author demonstrates that the symplectic invariants (intersection forms and symplectic areas) of the algebraic compactification $(Y, D'')$ and the almost toric compactification $(M_{AT}, D_{AT})$ match.
   • This matching implies the existence of a grading-preserving, strongly exact symplectomorphism $\Phi: \widehat{U} \to U_R$.

Key Contributions and Results

• Algorithmic Construction of Mirrors: The paper provides an explicit, algorithmic procedure to construct the mirror $U^\vee$ for any affine log Calabi–Yau surface $U$.
  • Case 1 (Looijenga): The mirror is obtained by taking the fan of a toric model of $U$ and performing non-toric blowups at $-1$ on the boundary divisors, with the number of blowups determined by the non-toric blowups in the model of $U$.
  • Case 2 (Elliptic):
    • If $U = \mathbb{P}^1 \times \mathbb{P}^1 \setminus V_{2,2}$, the mirror is constructed via toric blowups at corners and non-toric blowups at $-1$.
    • For other elliptic cases (characterized by the self-intersection number $d = D^2 \in \{1, \dots, 9\}$), the mirror is defined by a specific fan with rays $(1, 0), (7, 1), (-41, -5)$, and $(2d-17, -2)$, with multiplicities $9-d, 1, 1, 1$ respectively.
• Proof of Mirror Symmetry: The main theorem (Theorem 1.4) states that for any affine log Calabi–Yau surface $U$, the constructed scheme $U^\vee$ is a $k$-mirror for any field $k$. That is:
  $$D^\pi W(\widehat{U}, k) \cong D^b \text{Coh}(U^\vee_k)$$
• Uniformity: The paper offers a more uniform approach to the elliptic case compared to previous algorithms (such as those by Hacking–Keating), unifying the treatment of different elliptic surfaces through the rational ray diagram framework.
• Grading Preservation: A significant technical result is the proof that the constructed symplectomorphism preserves the natural grading structures induced by the log Calabi–Yau condition, which is necessary for the equivalence of the $A_\infty$-categories.

Significance and Claims

The paper establishes the HMS conjecture for the specific class of affine log Calabi–Yau surfaces by reducing the problem to the known results of Hacking and Keating [12]. The author explicitly states that the result is a "special case" of the main theorem in [12], with the novelty lying in the construction of the specific rational ray diagram that models the symplectic geometry of $U$.

The significance of the work is twofold:

1. Algorithmic Accessibility: It provides a concrete method to generate the mirror scheme for a broad class of surfaces, including Painlevé character varieties (mentioned as a specific family covered by Case 1).
2. Theoretical Unification: By utilizing the Symplectic Torelli theorem and the concept of exact eigenray diagrams, the paper bridges the gap between the algebraic construction of the mirror and the symplectic topology of the original space, ensuring the equivalence holds over arbitrary fields $k$.

The author notes that while the computation of the toric model in Case 1 can be computationally expensive, the process is theoretically algorithmic. The paper does not propose new experimental applications but rather solidifies the theoretical framework for mirror symmetry in this specific geometric setting, relying on the structural results of Eliashberg-Gromov regarding Liouville manifolds and the algebraic geometry of log Calabi–Yau pairs.