Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme

Abstract

A Calabi–Yau manifold is a compact Kahler simply connected manifold with a nowhere vanishing global n-form but no global i-form with 0 < i < n = dimX. By Kodaira’s criterion, it is projective if the dimension n ≥ 3. As well known, Calabi–Yau manifolds, hyperkahler manifolds and complex tori form the building blocks of compact Kahler manifolds with vanishing first Chern class [2, 6]. A famous theorem of Huybrechts states that two bimeromorphic hyperkahler manifolds are equivalent under smooth deformation [8,9]. In particular, they are homeomorphic to each other, having the same Betti numbers and Hodge numbers. Clearly, the same holds true for complex tori. Another theorem, originally due to Batyrev and Kontsevich, says that two birational Calabi–Yau manifolds have the same Betti numbers and Hodge numbers [1, 4, 10, 19, 21]. However, there are rigid birational non-isomorphic Calabi–Yau manifolds (cf. Theorem 0.1). Obviously, they are not equivalent under any smooth deformation. The aim of this paper is to remark that there nevertheless exist birational Calabi–Yau threefolds which are rigid, non-homeomorphic, but are connected by (necessarily non-smooth) projective flat deformation: