卡拉比--丘退化與過渡的週期幾何研究

Abstract

We study the moduli space of polarized Calabi--Yau manifolds, especially degenerations of Calabi--Yau manifolds. In the first part of the thesis, we give a Hodge theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance over higher dimensional bases up to a set of codimension $geq 2$ and a description on the codimension 2 locus for the moduli space of Calabi--Yau 3-folds. Also, we prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices and the points on the intersection of exact two infinite divisors have infinite distance measured by the metric induced from the dominant terms of the candidates of the Weil--Petersson potential. In the second part of the thesis, we study the degeneration of $mathcal{D}$-modules arsing from conifold transitions. Via the degeneration of Grassmannian manifolds $G(k,n)$ to Gorenstein toric Fano varieties $P(k,n)$, we suggest an approach to study the relation between the tautological systems on $G(k,n)$ and the (generalized) extended GKZ systems on the small resolution $hat{P}(k,n)$. We carry out the first but highly non-trivial case when $(k,n)=(2,4)$ to ensure its validity. To study the period integrals of Calabi--Yau complete intersections in $hat{P}(k,n)$, we also develop a new PDE system, which is a generalization of extended GKZ systems, governing the period integrals for the Calabi--Yau complete intersections in $hat{P}(k,n)$. We also establish a correspondence between the tautological systems on $G(2,5)$ and the generalized extended GKZ systems on $hat{P}(2,5)$. Finally, we also give an explicit description of the automorphism group of $P(k,n)$.