Doubling construction of Calabi-Yau threefolds

Abstract

We give a differential-geometric construction and examples of Calabi-Yau threefolds, at least one of which is {\it{new}}. Ingredients in our construction are {\it admissible pairs}, which were dealt with by Kovalev in \cite{K03} and further studied by Kovalev and Lee in \cite{KL11}. An admissible pair $(\overline{X},D)$ consists of a three-dimensional compact K\"{a}hler manifold $\overline{X}$ and a smooth anticanonical $K3$ divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ satisfy the {\it gluing condition}, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a Calabi-Yau threefold $M$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we can {\it always} construct a Calabi-Yau threefold from a {\it single} admissible pair $(\overline{X},D)$ by {\it doubling} it. Furthermore, we can compute all Betti and Hodge numbers of the resulting Calabi-Yau threefolds in the doubling construction.