turkmath.org

A theorem of Bell, Satriano and Sierra state that for a given smooth complex surface $X$ with an automorphism $\phi$ the set of natural numbers $n$ such that $ \mathrm{Ext}^i (\mathscr{F} , (\phi^*)^n (\mathscr{F})' )\neq 0$ is a union of finitely many arithmetic progressions and finitely many other numbers. Due to homological mirror symmetry conjecture, one can expect a symplectic version of this statement. In this talk, we will present such a theorem for a class of symplectic manifolds and symplectomorphisms isotopic to identity. The technique is analogous to its algebro-geometric counterpart: namely we construct $p$-adic analytic action on a version of the Fukaya category, interpolating the action of the iterates of the symplectomorphism.