Polarized Hodge Structures for Clemens Manifolds

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.