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

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.