A vanishing-cycle obstruction to smoothing Calabi–Yau threefolds

Abstract

We prove a homological obstruction to smoothing projective Calabi–Yau threefolds with nodes and anticanonical cones over smooth del Pezzo surfaces of degrees six and seven. On a crepant resolution that is small over the nodes and blows up the cone vertices, any smoothing forces a relation among link classes bounding in the local Milnor fibers. The relation is nonzero at every node and degree-seven point, and at each degree-six point smoothed through the one-dimensional component. The proof uses vanishing periods and applies to smoothing arcs of arbitrary contact order, extending the necessity direction of Friedman’s criterion. We apply the obstruction to a Batyrev hypersurface whose thirty singularities are individually smoothable but which admits no smoothing, even formally. On its mirror partner, two nodes cannot be smoothed by any analytic deformation, independently of a locally nonsmoothable cone point.