Calabi-Yau threefolds with Picard number three

Abstract

In this paper, we continue the study of boundedness questions for (simply connected) smooth Calabi-Yau threefolds commenced in arXiv:1706.01268. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second chern class. The question addressed in both papers is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowing these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. We showed in the previous paper that if there are no such surfaces, then the answer to the above question is yes. Moreover if the Picard number is 2, the answer was shown to be yes without any further assumptions on the Calabi-Yau threefold. The main results of this paper are for Picard number 3, where we prove boundedness in the case where there is at most one rigid non-movable surface, assuming the cubic form is smooth (thereby defining a real elliptic curve); in the two cases where the Hessian curve is singular, we also assume that the line defined by the second chern class does not intersect this curve at an inflexion point. In addition to the methods described in the previous paper, a further crucial tool in the proofs will be the classical Steinian involution on the Hessian of an elliptic curve.