论文标题
奇异的Fano和Calabi-yau品种的变形
Deformations of singular Fano and Calabi-Yau varieties
论文作者
论文摘要
本文的目的是概括有关Calabi-yau和Fano三倍的变形理论，其孤立的超脸奇异物的变形理论，这是由于第一作者Namikawa和Steenbrink。特别是，在终极奇点的假设下，namikawa在Fano情况下证明了平稳性，也可以在广义的Calabi-yau三倍上，假设满足了一定的拓扑一级条件。在尺寸$ 3 $的情况下，我们通过用规范代替终端来扩展其结果。在较高的维度中，我们确定了我们方法应用的奇异性类别。我们研究的一个令人惊讶的方面是较高的杜波伊斯和更高理性的概念扮演的角色。 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 $3$.
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 $3$.
