{"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":2,"journal":{"name":"ACS Applied Bio Materials","volume":" 52","pages":""},"PeriodicalIF":4.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1215/00127094-2023-0051","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","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。
期刊介绍:
ACS Applied Bio Materials is an interdisciplinary journal publishing original research covering all aspects of biomaterials and biointerfaces including and beyond the traditional biosensing, biomedical and therapeutic applications.
The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrates knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important bio applications. The journal is specifically interested in work that addresses the relationship between structure and function and assesses the stability and degradation of materials under relevant environmental and biological conditions.