Tautological systems developed in [8, 9] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to zero loci of global sections of vector bundles. In particular, we obtain similar criterion as in [8, 9] about holonomicity and regularity of the systems. We also prove solution rank formulas and geometric realizations of solutions following the work on hypersurfaces in homogeneous varieties .
Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent
progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.
We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G^r_d (C) that parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d ≤ g(C)−1.
In this paper we realize the moduli spaces of singular sextic curves with specified symmetry type as arithmetic quotients of complex hyperbolic balls or type IV domains. We also identify their GIT compactifications with the Looijenga compactifications of the corresponding period domains, most of which are actually Baily-Borel compactifications.
We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.