Description:

Title: Fukaya categories, variations of Hodge structure, and integrality of mirror maps

Abstract: I will describe joint work with Perutz-Sheridan and Sheridan relating the Fukaya category of a semi-positive projective variety to the variation of Hodge structure associated to its cohomology and to rational curve counting. From this work we can deduce that homological mirror symmetry in Calabi-Yau cases automatically implies genus-0 enumerative mirror symmetry correspondences between rational curve counts and period integrals. In joint work with Hanlon-Hicks-Pomerleano-Sheridan, we use this framework to prove that the "integrality of Taylor coefficients of mirror maps" conjecture follows from an arithmetic refinement of homological mirror symmetry. Our work also establishes this latter arithmetic HMS statement - and thereby integrality - for Greene-Plesser mirror pairs.