Deformations of some local Calabi-Yau manifolds

Friedman, Robert M.; Laza, Radu

doi:10.48550/arxiv.2203.11738

Cited by 1 publication

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“…The deformation theory of generalized Fano and Calabi-Yau threefolds with ordinary double points (or nodes), or more generally isolated canonical hypersurface singularities, has been extensively studied [Fri86], [Nam94], [NS95], [Nam02], [Ste97], [Gro97]. This paper is part of a series [FL22a,FL22b,FL22c,FL23] that aims to revisit and sharpen these results and explore generalizations to higher dimensions. A motivating question throughout has been the problem of understanding the local structure of compactified moduli spaces of Calabi-Yau varieties.…”

Section: Introductionmentioning

confidence: 99%

Deformations of Calabi–Yau varieties with k-liminal singularities

Friedman,

Laza

2024

Forum of Mathematics, Sigma

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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 first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] 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 [FL22a], the linear necessary and sufficient conditions of [Fri86] 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 [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

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