奇异凯勒-爱因斯坦度量族

奇异Kähler-Einstein度量的Psh变化与光滑Kähler族的多亏格不变性

Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families

本文证明光滑Kähler族中多重亏格维数不变性，解决Siu在2002年提出的猜想，方法涉及奇异Kähler-Einstein度量的Psh变化。

On the multiplicities of families of complex hypersurface-germs

We show that the possible drop in multiplicity in an analytic family $F(z,t)$ of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of $t$. We prove equimultiplicity of $μ$-constant families of the form $f + tg + t^2h$ if the singular set of the tangent cone of $\{f = 0}$ is not contained in the tangent cone of $\{h = 0}$.

Extension of the curvature form of the relative canonical line bundle on families of Calabi-Yau manifolds and applications

Given a proper, open, holomorphic map of Kähler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus,…

Families of distributions and Pfaff systems under duality

A singular distribution on a non-singular variety $X$ can be defined either by a subsheaf $D$ of the tangent sheaf, or by the zeros of a subsheaf $D^0$ of the sheaf of 1-forms. Although both definitions are equivalent under mild conditions on $D$, they give rise, in general, to non-equivalent notions of flat families. In this work we investigate conditions under which both notions of flat families are equivalent. In the last sections we focus on the case where the distribution is integrable, and we use our results to generalize a theorem of Cukierman and Pereira.

Positivity of relative canonical bundles for families of canonically polarized manifolds

Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.