{"title":"Higher Du Bois and higher rational singularities","authors":"R. Friedman, R. Laza","doi":"10.1215/00127094-2023-0051","DOIUrl":null,"url":null,"abstract":". We prove that the higher direct images R q f ∗ Ω p Y /S of the sheaves of relative K¨ahler 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 ≤ k and all q ≥ 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. Independently, we have proved that the latter statement also holds for isolated local complete intersection singularities, and conjecture that it holds more generally for all local complete intersection singularities.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.3000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Duke Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1215/00127094-2023-0051","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 14
Abstract
. We prove that the higher direct images R q f ∗ Ω p Y /S of the sheaves of relative K¨ahler 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 ≤ k and all q ≥ 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. Independently, we have proved that the latter statement also holds for isolated local complete intersection singularities, and conjecture that it holds more generally for all local complete intersection singularities.
.我们证明,在 p ≤ k 和所有 q ≥ 0 的情况下,相对 K ¨ahler 二项性的剪切的高直映像 R q f ∗ Ω p Y /S 是局部自由的,并且与任意基数变化的 flat 适当族相容,这些族的fibers 具有 k -Du Bois 局部完全交集奇点,这推广了 Du Bois 的一个结果(k = 0 的情况)。然后,我们提出了扩展有理奇点定义的 k 有理奇点定义,并证明如果 X 是具有孤立奇点或局部完全交点奇点的 k 有理综,那么 X 就是 k 杜波依斯。作为应用,我们讨论了族中霍奇数的行为和奇异卡拉比优(Calabi-Yau)变体的无碍性。在附录中,Morihiko Saito 证明了在超曲面奇点的情况下,这里提出的 k - 理性定义等同于之前给出的 k - 理性奇点的数值定义。因此,对于超曲面奇点,k -杜波依斯奇点是 ( k - 1)- 理性的。另外,我们还证明了后一种说法对于孤立的局部完全交点奇点也是成立的,并猜想这种说法对于所有局部完全交点奇点都是普遍成立的。