Semi-orthogonality in Fukaya-Seidel mirrors to blowups of abelian varieties

TL;DR

The paper investigates homological mirror symmetry for a blow-up X of an abelian surface along a genus 2 curve, using a generalized SYZ mirror (Y^0,W_0) as a symplectic LG model and X as the complex side.It constructs a bipartite semi-orthogonal decomposition on the B-side and identifies two A-side subcategories, A_L and A_K, which mirror the B-side blocks Db Coh(H) and Db Coh(V×C) respectively.The authors develop a wrapped Fukaya category framework for the LG model, including quasi-units, continuation maps, and a homotopy-colimit construction to handle partial and full wrapping in the base, and then compute morphisms and product structures to match Ext groups.The main result asserts an equivalence between the A_L,A_K-generated subcategory and Db_L Coh(X), thus providing categorical HMS evidence for this non-Calabi–Yau, general-type setting and highlighting semi-orthogonality as a structural bridge between the two sides.

Abstract

We prove evidence of Kontsevich's homological mirror symmetry conjecture (HMS) for a blow-up of an abelian surface times the complex plane, on the complex side, and its symplectic Landau-Ginzburg mirror. Specifically, the first author proved evidence of HMS for a 1-parameter family of genus 2 curves on the complex side, as a hypersurface in an abelian surface. The generalized SYZ mirror to the hypersurface is then the SYZ mirror to the Landau-Ginzburg model given by the blow-up of the abelian surface times the complex plane, along the hypersurface times zero, with superpotential given by projection to the complex plane. The mirror to the blow-up - without the superpotential - is obtained by removing a generic smooth fiber from the generalized SYZ mirror superpotential. We prove a categorical HMS result for the latter pair, between categories expected to split-generate. To do so, we equip the punctured superpotential with a Fukaya category which involves both partial and full wrapping in the base of the symplectic Landau-Ginzburg model due to the removal of a generic fiber. Semi-orthogonality appears in the categorical invariants on both sides of HMS.

Figures (9)

• Figure 1: HMS squares for two components of Remark \ref{['rem:glue']}
• Figure 2: Projections of the two types of Lagrangians to the cylinder model for the base of $W_0:Y^0 \to \mathbb C$
• Figure 3: $Y^0$ is a fibration over the $T^\varepsilon$ disc about 0. U-shape curves $W_0(L_j)$ pass through $b_S=-S$. Arcs $W_0(K_j)$ start from, but don't include, 0. The singular fiber of $W_0$ is above $-T^\epsilon$ marked by an $\times$.
• Figure 4: Angles depicted as fractions of $2\pi$ relative the positive horizontal line field
• Figure 5: The differential on $CF(L_i,L_j)$ counts curves covering this bigon depicted in the base, from point $b_1$ of degree $-1$ to point $b_0$ of degree 0. ${}^*$: Add $\tfrac14 \cdot 2\pi$ as the oriented tangent vector rotates counterclockwise $\frac{\pi}{2}$ from $b_0$ to $b_1$

Theorems & Definitions (63)

• Remark 1.1
• Example 1.2
• Remark 1.3
• Theorem 1.4: AAK
• Remark 1.5
• Definition 1.6: notation $D^b_\mathcal{L}$, Ca20
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10