We show that the \partial\bar{\partial}-lemma holds for the non-Kahler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi-Yau threefolds contracted along smooth rational curves with normal bundle of type (-1, -1), at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.

# Algebraic Geometry Seminar

For more information about this seminar, contact Gabriele Di Cerbo or János Kollár.

**Please click on seminar title for complete abstract.**

**There is no seminar on September 26.**

##### The Hodge decomposition for some non-Kahler threefolds with trivial canonical bundle.

Columbia University

##### Transcendence of period maps

Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group. In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the André-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which...

University of Georgia