Doubling construction of Calabi-Yau fourfolds from toric Fano fourfolds

Abstract

We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor $D$ on $\overline{X}$. If two admissible pairs $({\overline{X}}_1,D_1)$ and $({\overline{X}}_2,D_2)$ with ${\dim }_{\mathbb{C}}{\overline{X}}_i=4$ satisfy the \emph{gluing condition}, we can glue ${\overline{X}}_1\setminus D_1$ and ${\overline{X}}_2\setminus D_2$ together to obtain a compact Riemannian $8$-manifold $(M,g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\hat{A}$-genus of $M$ equals $2$, then $M$ is a Calabi-Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy $\mathrm{SU}(4)$. 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 obtain a compact Riemannian $8$-manifold $M$ with holonomy contained in $\mathrm{Spin}(7)$. Moreover, we show that if the admissible pair is obtained from \emph{any} of the toric Fano fourfolds, then the resulting manifold $M$ is a Calabi-Yau fourfold by computing $\hat{A}(M)=2$.

17 pages; minor collections; arXiv admin note: text overlap with arXiv:1305.0074