2015/02/01 by Doi, Mamoru, Yotsutani, Naoto
#14J32 #53C25 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1502.00208
We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \citeDY14 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 in \citeK03. Here in this paper an admissible pair (X,D) consists of a compact Kähler manifold X and a smooth anticanonical divisor D on X. If two admissible pairs (X1,D1) and (X2,D2) with dimℂXi=4 satisfy the gluing condition, we can glue X1∖ D1 and X2∖ D2 together to obtain a compact Riemannian 8-manifold (M,g) whose holonomy group Hol(g) is contained in Spin(7). Furthermore, if the \widehatA-genus of M equals 2, then M is a Calabi-Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy SU(4). In particular, if (X1,D1) and (X2,D2) are identical to an admissible pair (X,D), then the gluing condition holds automatically, so that we obtain a compact Riemannian 8-manifold M with holonomy contained in 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 \widehatA(M)=2.