Hasse-Witt matrices, unit roots and period integrals

TL;DR

This paper explores the relationship between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces, proving a conjecture and connecting algebraic and analytic aspects of these geometric objects.

Contribution

It proves Vlasenko's conjecture on higher Hasse-Witt matrices for toric hypersurfaces using Katz's local expansion method, linking these matrices to period integrals.

Findings

Proved a conjecture by Vlasenko on higher Hasse-Witt matrices.

Established a connection between Hasse-Witt matrices and period integrals.

Provided a method to relate Katz's and Dwork's congruence relations.

Abstract

Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about periods.