Picard-Fuchs Systems Arising From Toric and Flag Varieties

Abstract

This thesis studies the Picard-Fuchs systems for families arising as vector bundles zero loci in toric or partial flag varieties, including Riemann-Hilbert type theorems and arith- metic properties of the differential systems. The theory of tautological systems is proposed by Lian, Song and Yau in [51, 52]. These Picard-Fuchs type differential systems arise from the variation of Hodge structures of complete intersections in variety X with large symmetries, generalizing Gel’fand-Kapranov- Zelevinski systems for toric varieties [25]. The form of tautological systems makes it natural to introduces the powerful tool of D-modules into the study of hypersurface family in X. Following this direction, Riemann-Hilbert type theorems are obtained by Bloch, Huang, Lian, Srinivas, Yau and Zhu in [6, 34, 35], including solution rank formula, completeness and geometric interpretation of solution sheaves. In the first part of this thesis, we generalize these results in two aspects, one is the construction of tautological systems for vector bundles, the other is Riemann-Hilbert type theorems in this case. In the second part of the thesis, we examine an explicit description of Jacobian rings for homogenous vector bundles. This can be viewed as a description of the graded quotients of tautological systems with respect to the natural filtered D-module structure. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [16] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties. The last part is devoted to the arithmetric properties of fundamental periods near large complex structure limit. Motivated by the work of Candelas, de la Ossa and Rodriguez- Villegas [14], 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 [67] on higher Hasse-Witt matrices for toric hypersurfaces following Katz’s method of local expansion [38, 39]. 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 [39] to Dwork’s congruence relations [21] about periods.