Polarized Hodge Structures for Clemens Manifolds

TL;DR

This paper studies Clemens manifolds obtained from Calabi-Yau threefolds via conifold transitions, proving they satisfy the $ar{ ext{d}}ar{ ext{d}}$-lemma and have polarized Hodge structures, addressing questions by Friedman.

Contribution

It establishes the $ar{ ext{d}}ar{ ext{d}}$-lemma and polarization of Hodge structures on Clemens manifolds, extending understanding of their complex geometry.

Findings

Any small smoothing of the singular space satisfies the $ar{ ext{d}}ar{ ext{d}}$-lemma.

The Hodge structure on $H^3(Y)$ is polarized by the cup product.

Results answer questions posed by R. Friedman.

Abstract

Let $X$ be a Calabi-Yau threefold. A conifold transition first contracts $X$ along disjoint rational curves with normal bundles of type $(-1,-1)$, and then smooth the resulting singular complex space $\bar{X}$ to a new compact complex manifold $Y$. Such $Y$ is called a Clemens manifold and can be non-K\"{a}hler. We prove that any small smoothing $Y$ of $\bar{X}$ satisfies $\partial\bar{\partial}$-lemma. We also show that the resulting pure Hodge structure of weight three on $H^3(Y)$ is polarized by the cup product. These results answer some questions of R. Friedman.