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

Abstract

For any positive integer $g$, we introduce the moduli space ${\mathcal{A}}_g^F=[{\mathcal{H}}_g/P_g(\mathbb{Z})]$ 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 ${\mathcal{A}}_g^F$, 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 $H{F}^{\ast }(\hat{\ell },{\hat{\ell }}^{\prime })$ form coherent sheaves over a complex manifold parametrizing triples $(\tau ,\hat{\ell },{\hat{\ell }}^{\prime })$ where $\tau \in {\mathcal{H}}_g$ defines a complexified symplectic form ${\omega }_{\tau }$ on ${\mathbb{T}}^{2g}$ and $\hat{\ell }$, ${\hat{\ell }}^{\prime }$ are affine Lagrangian branes in $({\mathbb{T}}^{2g},{\omega }_{\tau })$.