Deformations of singular Fano and Calabi-Yau varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866918137703694336 |
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| author | Friedman, Robert Laza, Radu |
| author_facet | Friedman, Robert Laza, Radu |
| contents | The goal of this paper is to generalize results concerning the deformation theory of Calabi-Yau and Fano threefolds with isolated hypersurface singularites, due to the first author, Namikawa and Steenbrink. In particular, under the assumption of terminal singularities, Namikawa proved smoothability in the Fano case and also for generalized Calabi-Yau threefolds assuming that a certain topological first order condition is satisfied. In the case of dimension $3$, we extend their results by, among other things, replacing terminal with canonical. In higher dimensions, we identify a class of singularities to which our method applies. A surprising aspect of our study is the role played by the higher Du Bois and higher rational singularities. Among other deformation theoretic results in higher dimensions, we obtain smoothing results for generalized Fano varieties whose singularities are not $1$-rational, and for generalized Calabi-Yau varieties whose singularities are not $1$-rational but are $1$-Du Bois under a topological condition on the links which is similar to the first order obstruction in dimension $3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_04823 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformations of singular Fano and Calabi-Yau varieties Friedman, Robert Laza, Radu Algebraic Geometry Differential Geometry 14B07, 14J32, 14J45, 32Q25, 32S30 The goal of this paper is to generalize results concerning the deformation theory of Calabi-Yau and Fano threefolds with isolated hypersurface singularites, due to the first author, Namikawa and Steenbrink. In particular, under the assumption of terminal singularities, Namikawa proved smoothability in the Fano case and also for generalized Calabi-Yau threefolds assuming that a certain topological first order condition is satisfied. In the case of dimension $3$, we extend their results by, among other things, replacing terminal with canonical. In higher dimensions, we identify a class of singularities to which our method applies. A surprising aspect of our study is the role played by the higher Du Bois and higher rational singularities. Among other deformation theoretic results in higher dimensions, we obtain smoothing results for generalized Fano varieties whose singularities are not $1$-rational, and for generalized Calabi-Yau varieties whose singularities are not $1$-rational but are $1$-Du Bois under a topological condition on the links which is similar to the first order obstruction in dimension $3$. |
| title | Deformations of singular Fano and Calabi-Yau varieties |
| topic | Algebraic Geometry Differential Geometry 14B07, 14J32, 14J45, 32Q25, 32S30 |
| url | https://arxiv.org/abs/2203.04823 |