Deformations of Calabi-Yau varieties with $k$-liminal singularities

TL;DR

This paper explores nonlinear topological obstructions to smoothing certain singular Calabi-Yau varieties, extending previous linear conditions to a broader class of singularities called weighted homogeneous $k$-liminal hypersurface singularities.

Contribution

It generalizes nonlinear topological smoothing conditions from nodal Calabi-Yau varieties to those with weighted homogeneous $k$-liminal singularities, unifying earlier results.

Findings

Established nonlinear topological conditions for $k$-liminal singularities.

Extended smoothing criteria to higher-dimensional Calabi-Yau varieties.

Unified previous linear and nonlinear smoothing conditions.

Abstract

The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first order smoothings of mildly singular Calabi-Yau varieties of dimension at least $4$. For nodal Calabi-Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first order smoothing was given by the first author in 1986. Subsequently, Rollenske-Thomas generalized this picture to nodal Calabi-Yau varieties of odd dimension, by finding a necessary nonlinear topological condition for the existence of a first order smoothing. In a complementary direction, in our recent work, the linear necessary and sufficient conditions for nodal Calabi-Yau threefolds were extended to Calabi-Yau varieties in every dimension with $1$-liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of Rollenske-Thomas for Calabi-Yau varieties with weighted homogeneous $k$-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.