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

For any positive integer g, we introduce the moduli space AFg =[Hg/Pg(Z)] parametrizing g-dimensional principally polarized abelian varieties V_τ together with a Strominger-Yau-Zalsow (SYZ) fibration, where τ ∈ Hg is the genus-g Seigel upper half space and Pg(Z) ⊂ Sp(2g,Z) is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli Hg and AFg, relating the B-model on V_τ and the A-model on its mirror, a compact $2g$-dimensional torus T²ᵍ equipped with a complexified symplectic form. For each V_τ, we establish a homological mirror symmetry (HMS) result at the cohomological level over C. This implies core HMS at the cohomological level over C and a graded C-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups HF^*(̂, ̂') form coherent sheaves over a complex manifold parametrizing triples (τ, ̂, ̂') where τ ∈ Hg defines a complexified symplectic form ω_τ on T²ᵍ and ̂, ̂ ' are affine Lagrangian branes in (T²ᵍ, ω_τ).