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 Doi and Yotsutani (N Y J Math 20:1203–1235, 2014) to construct Calabi–Yau threefolds. We also give examples of Calabi–Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are admissible pairs, which were first dealt with by Kovalev (J Reine Angew Math 565:125–160, 2003). 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 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 $\widehat{A}$-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 any of the toric Fano fourfolds, then the resulting manifold M is a Calabi–Yau fourfold by computing $\widehat{A}(M)=2$.