Derived Equivalences, Generic Vanishing, and the Structure of Cohomology

一个起源于镜像对称和双曲几何同调面的中心问题是比较具有等价有界导出范畴的簇的数值不变量、上同调和几何。继续与Schnell一起研究Picard簇在导出等价下的行为和某些Hodge数的不变性，PI计划解决一些问题，如正则上同调的不变性，上同调支点的不变性，以及傅立叶-Mukai对偶的所有Hodge数的不变性，或者研究凝聚层的派生范畴的维度。另一方面，许多关于紧致Kahler流形上同调的结果位于一般消失论、傅立叶-Mukai理论、同调和交换代数的交集上。PI将继续这一领域的工作，从Lazarsfeld开始，研究作为外代数上的模的典型上同调的结构。在将BGG对应应用于紧致Kahler流形上的BGG对应之后，他想要得到关于射影空间上存在低阶不可分解丛的结果，或者关于没有不规则纤颤的紧致Kahler流形的全纯Euler特征的改进的下界的结果。他将尝试应用这类方法来逼近Carrell和Hacon-Kovacs关于一般类型簇上的全纯One-形式的猜想。他还将继续研究Beauville-Debarre-Ran肖特基型猜想，预测主要是极化的阿贝尔变种包含代表最小上同调类的子簇，以及关于由傅立叶-穆凯变换诱导的阿贝尔变种的有理Chow群上的滤子的Beauville猜想。PI建议解决的一些问题，如上同调群或Hodge数在导出等价下的不变性，对一个变种的全上同调的限制，或关于阿贝尔变种的极小类的研究，是各自领域中最突出和公认的问题之一，并将产生很大的影响。其他的，如派生范畴的维度或上同调模的外部结构，是新出现的理论的一部分，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有广泛的应用，进一步加深我们在该领域的知识，创造与不同数学背景的人的互动，并产生适合博士生的问题。在数学课之外，他将继续参与非部门活动，例如他是致力于为科学和工程领域的女性创造更好环境的最明智委员会的成员，以及UIC参议院的成员。在国际数学界，他将参与组织会议和讲习班，并以传播知识为目标编辑卷册。资金将帮助PI继续在美国和海外的暑期学校和会议上讲课；在最近的过去，这导致了与PI基础机构以外的学生合作，并开始了他们的部分研究。在UIC系，它们将用于帮助研究生的研究和旅行，支持研讨会，以及开发和改进研究生课程。

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.