谱曲线的镜像计数
Mirror Counts of Spectral Curves
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- California Institute of Technology（加州理工学院）
- Northwestern University（西北大学）
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中文总结 AI 辅助
本文证明凸格多边形定义的环面簇上结点曲线计数与勒让德链环的rulings计数等价，并猜想ruling分解恢复精细热带不变量。
中文摘要 AI 辅助
给定一个凸格多边形 $\Delta\subset \mathbb R^2$，设 $N_\Delta$ 为 $\Delta$ 定义的环面簇 $\mathbb P_\Delta$ 上丰沛线丛 $L_\Delta$ 的线性系统中的有理结点曲线计数，这些曲线与环面边界有固定交点。我们证明该线性系统的相对雅可比矩阵定义了一个可积系统，该可积系统在二维环面上微支撑于勒让德链环的可构造层语言中具有镜像对偶。在此框架下，我们利用三维空间中勒让德链环的rulings类比来定义对偶计数问题，然后证明其与 $N_\Delta$ 等价。我们使用可构造方法、热带曲线计数以及对数Gromov-Witten理论中的局部化对若干例子进行 $N_\Delta$ 的计算，以展示不同方法的一致性。此外，rulings给出了可构造层模空间的一个分层。我们猜想该ruling分解恢复了Block-Göttsche的精细热带不变量，从而根据Bousseau的一个定理编码了更高亏格的对数Gromov-Witten不变量。
英文摘要
Given a convex lattice polygon $Δ\subset \mathbb R^2$, let $N_Δ$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_Δ$ on the toric variety $\mathbb P_Δ$ defined by $Δ$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_Δ$. We perform calculations of $N_Δ$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.
