Derived Equivalences, Generic Vanishing, and the Structure of Cohomology

导出等价、泛型消失和上同调的结构

• 批准号: 1101323
• 负责人: Mihnea Popa
• 金额: $ 22.38万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101323&HistoricalAwards=false
• 关键词: Derived; Equivalences; Generic; Vanishing; Structure

A central problem originating on the homological side of mirror symmetry and birational geometry is to compare the numerical invariants, cohomology, and geometry of varieties with equivalent bounded derived categories. Continuing work with Schnell on the behavior of the Picard variety under derived equivalence and the invariance of certain Hodge numbers, the PI plans to address problems like the invariance of the canonical cohomology, the invariance of cohomological support loci, and the invariance of all Hodge numbers for Fourier-Mukai partners, or the study of the dimension of derived categories of coherent sheaves. In a different direction, many results on the cohomology of compact Kahler manifolds lie at the intersection of the areas of Generic Vanishing theory, Fourier-Mukai theory, and homological and commutative algebra. The PI will continue work in this area, initiated with Lazarsfeld, on the structure of the canonical cohomology as a module over the exterior algebra. He would like to derive consequences on either the existence of indecomposable bundles of low rank on projective space, or on improved lower bounds on the holomorphic Euler characteristic of compact Kahler manifolds without irregular fibrations, after applying the BGG correspondence to such modules. He will attempt to apply methods of this type to approach a conjecture of Carrell and Hacon-Kovacs on holomorphic one-forms on varieties of general type. He will also continue work towards the Beauville-Debarre-Ran Schottky-type conjecture predicting which principally polarized abelian varieties contain subvarieties representing minimal cohomology classes, and towards the Beauville conjecture on filtrations on the rational Chow groups of abelian varieties induced by the Fourier-Mukai transform.Some of the problems the PI proposes to attack, like the invariance of cohomology groups or Hodge numbers under derived equivalences, restrictions on the total cohomology of a variety, or the study of minimal classes on abelian varieties, are among the most prominent and established problems in their respective areas and will have a high impact as proved statements. Others, like for instance the dimension of derived categories or the exterior structure on cohomology modules, are part of newly emerging theories where a better understanding will surely lead to more applications. All parts of the project will have a broad range of applications, further our knowledge in the fi eld, create interaction with people of diff^erent mathematical backgrounds, and produce problems suitable for Ph.D. students. Outside of Mathematics, the PI will continue to be involved in non-departmental activities, like his membership on the WISEST Committee devoted to creating a better environment for women in science and engineering, and on the UIC Senate. In the international mathematical community, he will be involved in organizing conferences and workshops, and editing volumes with the goal of disseminating knowledge. Funds will help the PI continue to deliver lectures at summer schools and conferences in the US and abroad; in the recent past this has led to working with students outside of the PI's base institution and getting part of their research started. In the UIC department they will be used for assisting the research and travel of graduate students, supporting seminars, and developing and improving the graduate curriculum.

源自镜像对称性和异性几何形状的同源侧的中心问题是比较具有等效界定类别的品种的数值不变性，同时组和几何形状。继续与施内尔（Schnell）继续研究PICARD品种在派生的等效性下的行为和某些杂物数的不变性，PI计划解决诸如规范共同体学的不变性，共同体支持基因座的不变性，以及对傅立叶伴侣的所有杂物数的不变性，或者是对衍生型的fourier-mukai伙伴的所有hodge数字的不变性。在不同的方向上，关于紧凑型卡勒歧管的共同体的许多结果都位于通用消失理论，傅立叶 - 穆凯理论以及同源和交换代数的相交。 PI将继续在Lazarsfeld发起的该领域继续工作，以典型的共同体结构为外部代数上的模块。他想在投射空间上存在低等级的不可塑性捆绑包，或者在将BGG对应于此类模块后，在没有不规则振动的情况下，对塑形欧拉（Holomorthic Euler）的特征的不可分解的捆绑物产生了后果。他将尝试采用这种类型的方法，以在一般类型的各种形式上对carrell和Hacon-kovacs的猜想进行猜想。 He will also continue work towards the Beauville-Debarre-Ran Schottky-type conjecture predicting which principally polarized abelian varieties contain subvarieties representing minimal cohomology classes, and towards the Beauville conjecture on filtrations on the rational Chow groups of abelian varieties induced by the Fourier-Mukai transform.Some of the problems the PI proposes to attack, like the invariance of cohomology在派生的等效性下，对多样性总和的限制或对阿贝尔品种的最小阶层的研究是其各自领域中最突出，最确定的问题之一，并且将对证明的陈述产生很大的影响。其他人，例如衍生类别的维度或同事模块上的外部结构，是新出现的理论的一部分，在这些理论中，更好的理解肯定会导致更多的应用。该项目的所有部分都将具有广泛的应用程序，进一步我们在FI中的知识，与不同的数学背景的人建立互动，并产生适合博士学位的问题。学生。在数学之外，PI将继续参与非部门活动，例如他在最聪明的委员会中的成员资格，致力于为科学和工程领域的女性和UIC参议院创造更好的环境。在国际数学社区中，他将参与组织会议和研讨会，并进行编辑，以传播知识。资金将帮助PI继续在美国和国外的暑期学校和会议上进行讲座；最近，这导致与PI基础机构以外的学生合作，并开始研究他们的一部分研究。在UIC部门，他们将用于协助研究生的研究和旅行，支持研讨会以及开发和改善研究生课程。