Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties

Abstract

For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_\tau$ together with a Strominger-Yau-Zalsow (SYZ) fibration, where $\tau \in \mathcal{H}_g$ is the genus-$g$ Seigel upper half space and $P_g(\mathbb{Z}) \subset \mathrm{Sp}(2g,\mathbb{Z})$ is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli $\mathcal{H}_g$ and $\mathcal{A}^F_g$, relating the B-model on $V_\tau$ and the A-model on its mirror, a compact $2g$-dimensional torus $\mathbb{T}^{2g}$ equipped with a complexified symplectic form. For each $V_\tau$, we establish a homological mirror symmetry (HMS) result at the cohomological level over $\mathbb{C}$. This implies core HMS at the cohomological level over $\mathbb{C}$ and a graded $\mathbb{C}$-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups $HF^*(\hat{\ell}, \hat{\ell}')$ form coherent sheaves over a complex manifold parametrizing triples $(\tau, \hat{\ell}, \hat{\ell}')$ where $\tau \in \mathcal{H}_g$ defines a complexified symplectic form $\omega_\tau$ on $\mathbb{T}^{2g}$ and $\hat{\ell}$, $\hat{\ell} '$ are affine Lagrangian branes in $(\mathbb{T}^{2g}, \omega_\tau)$.