Differential forms on singular spaces in arbitrary characteristic

任意特征奇异空间上的微分形式

• 批准号: 364017874
• 负责人: Dr. Patrick Graf
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/364017874?language=en
• 关键词: Differential; forms; singular; spaces; arbitrary

An algebraic variety is given locally as the zero set of finitely many polynomials with coefficients in a field. It makes a substantial difference whether that field has characteristic zero (e.g. the complex numbers) or positive characteristic (i.e. 1+1+...+1=0).A variety is called regular if locally it looks like affine space. In classification theory, also singular varieties are interesting. We investigate reflexive differential forms on varieties with mild singularities, i.e. differential forms on the smooth locus. More precisely, we consider the question of their extendability to a resolution of singularities.In positive characteristic, F-pure and strongly F-regular singularities are relevant here. In general, extension does not hold, but maybe extension with logarithmic poles does hold. This would contribute to a better understanding of the failure of the Lipman-Zariski conjecture in positive characteristic: there are singular varieties with locally free tangent sheaf.In characteristic zero we investigate extendability of reflexive pluri-differential forms, i.e. sections of tensor powers of the cotangent sheaf. Here extendability with logarithmic poles should also hold. This would yield a geometric criterion for a rationally connected variety not to carry any reflexive pluri-differentials.

局部给出了一个代数变量，它是一个域中有限多个带系数多项式的零集。该域是特征为零（例如复数）还是正特征（例如1+1+…+1=0），这是有本质区别的。如果一个变种局部看起来像仿射空间，那么它就被称为正则变种。在分类理论中，奇异变量也很有趣。研究了具有轻微奇异点的变异上的自反微分形式，即光滑轨迹上的微分形式。更确切地说，我们考虑它们的可扩展性问题，以解决奇点。在正特征中，f纯奇点和强f正则奇点是相关的。一般来说，扩展是成立的，但也许对数极点的扩展是成立的。这将有助于更好地理解Lipman-Zariski猜想在正特征下的失效：存在具有局部自由切线束的奇异变体。在特征零点，我们研究了自反多微分形式的可拓性，即余切束的张量幂的部分。这里对数极点的可拓性也应该成立。这将产生一个几何标准，合理连接的品种不携带任何反身多元微分。

项目成果

A note on Flenner’s extension theorem

关于弗伦纳可拓定理的注记

• DOI: 10.1007/s00229-020-01233-y
• 期刊: manuscripta mathematica
• 影响因子: 0.6
• 作者: P. Graf

THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER

高等一属中的利普马纳扎里斯基猜想

• DOI: 10.1017/fms.2020.19
• 期刊: Forum of Mathematics, Sigma
• 作者: H. Bergner;P. Graf
• 通讯作者: P. Graf