Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. I
The authors construct and analyse certain surprising Calabi--Yau threefolds in characteristic three. The construction starts with two rational quasielliptic Jacobian surfaces \(Y_1\) and \(Y_2\), and the Calabi--Yau threefold \(X\) is obtained as a crepant resolution of the fiber product of \(Y_1\), \(Y_2\) over the projective line. This idea is a variant of a construction \textit{C. Schoen} [Math. Z. 197, 177--199 (1988; Zbl 0631.14032)]. Here the crucial point is to ensure that the singularities on the fiber product admit a crepant resolution. These singularities are closely related to the structure of the degenerate fibers of the two quasielliptic fibrations. Using the local equations for the fibrations obtained in \textit{H. Ito} [Math. Z. 211, No. 1, 1--40 (1992; Zbl 0739.14011)], the authors make a careful analysis of these singularities, which behave very different from the case of characteristic zero, and compute the Betti numbers of the resulting Calabi--Yau threefolds \(X\). Some of them have \(b_3=0\), whence are nonliftable to characteristic zero.
scientific article; zbMATH DE number 5319822
scientific article