Smoothing nodal Calabi-Yau n-folds

TL;DR

This paper extends the understanding of smoothing conditions for odd-dimensional Calabi-Yau varieties with nodes, generalizing Friedman's 3-dimensional results and highlighting the role of the Yukawa product in higher dimensions.

Contribution

It provides a topological proof of smoothing conditions for odd-dimensional Calabi-Yau n-folds and extends the results to higher dimensions, emphasizing the Yukawa product's role.

Findings

Extended Friedman's smoothing criteria to higher dimensions

Identified the role of the Yukawa product in higher-dimensional cases

Proved a converse for nodal Calabi-Yau hypersurfaces in projective space

Abstract

Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the "Yukawa product" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.