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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 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 homogeneousk-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.
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Deformations of Calabi–Yau Varieties With Isolated Log Canonical Singularities
Abstract Recent progress in the deformation theory of Calabi–Yau varieties $$Y$$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $$k$$-liminal singularities for $$k\ge 1$$. The goal of this paper is to show that certain aspects of this study extend naturally to the $$0$$-liminal case as well, that is, to Calabi–Yau varieties $$Y$$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $$Y$$ in the case of isolated $$0$$-liminal hypersurface singularities, and extend Namikawa’s unobstructedness theorem for deformations of singular Calabi–Yau three-folds $$Y$$ with canonical singularities to the case where $$Y$$ has an isolated $$0$$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples.
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- Award ID(s):
- 2101640
- PAR ID:
- 10589759
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 10
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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