  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.
