Equivariant split generation and mirror symmetry of special isogenous tori

Abstract

We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group $G$. Combining this with some generalizations of Seidel's algebraic frameworks from Seidel's book, we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call \textit{special isogenous tori}. This extends the work of Abouzaid-Smith. We also show that derived Fukaya categories are complete invariants of special isogenous tori.