论文标题
较高的杜波伊斯和更高的理性概念
Higher Du Bois and higher rational singularities
论文作者
论文摘要
我们证明，较高的直接图像$ r^qf_*ω^p _ {\ Mathcal y/s} $相对Kähler差异的束带是本地免费的，并且与fr的基础更改且与任意的基本变化兼容，其纤维具有$ k $ k $ k $ k $ -du bois bois bois local interception singulinities $ p \ pe uis $ pe uis $ q $ q $ q $ q $ q $ q \ quis（$ k） $ k = 0 $）。然后，我们提出了$ k $ - 理性的奇异性的定义，以扩展了理性概念的定义，并表明，如果$ x $是$ k $ - 理性的品种，则具有孤立或本地完整的交叉点奇异点，则$ x $为$ k $ -du bois。作为应用，我们讨论了家庭中杂货数的行为以及奇异卡拉比雅的变形的不良结构。 在附录中，Morihiko Saito证明，在高度表面奇点的情况下，此处提出的$ k $ - 理性定义等同于先前给出的$ K $理性奇异性的数值定义。直接的结果，因此，对于超表面奇异性，$ k $ -du bois的奇异性是$（k-1）$ - 理性。最近，Chen-Dirks-Mustaţă证明了所有本地完整的交叉点奇异性。
We prove that the higher direct images $R^qf_*Ω^p_{\mathcal Y/S}$ of the sheaves of relative Kähler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have $k$-Du Bois local complete intersection singularities, for $p\leq k$ and all $q\geq 0$, generalizing a result of Du Bois (the case $k=0$). We then propose a definition of $k$-rational singularities extending the definition of rational singularities, and show that, if $X$ is a $k$-rational variety with either isolated or local complete intersection singularities, then $X$ is $k$-Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties. In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the $k$-rationality definition proposed here is equivalent to a previously given numerical definition for $k$-rational singularities. As an immediate consequence, it follows that for hypersurface singularities, $k$-Du Bois singularities are $(k-1)$-rational. This statement has recently been proved for all local complete intersection singularities by Chen-Dirks-Mustaţă.
