Derived Equivalences, Generic Vanishing, and the Structure of Cohomology

起源于镜像对称和双民族几何同调方面的一个中心问题是比较具有等价有界派生范畴的变量的数值不变量、上同调和几何。继续与Schnell在派生等价和某些Hodge数的不变性下对Picard变的行为的研究，PI计划解决诸如正则上同调的不变性，上同调支持座的不变性，以及Fourier-Mukai伙伴的所有Hodge数的不变性等问题，或者研究相干束的派生范畴的维数。在另一个方向上，紧Kahler流形的上同调的许多结果位于一般消失理论、Fourier-Mukai理论、同调和交换代数领域的交叉点。PI将继续在这个领域工作，由Lazarsfeld发起，研究正则上同调作为外部代数上的一个模块的结构。他将BGG对应应用于射影空间上低秩不可分解束的存在性，或无不规则纤颤的紧Kahler流形的全纯欧拉特征的改进下界。他将尝试应用这种类型的方法来逼近Carrell和Hacon-Kovacs关于一般类型变种上全纯一型的猜想。他还将继续致力于预测哪些主要极化阿贝尔变体包含代表最小上同类的子变体的Beauville- debarre - ran schottky型猜想，以及关于由Fourier-Mukai变换引起的阿贝尔变体的理性Chow群的过滤的Beauville猜想。PI提出要解决的一些问题，如上同群或在派生等价下的Hodge数的不变性，对品种的全上同的限制，或对阿贝尔品种的最小类的研究，都是各自领域中最突出和最成熟的问题之一，并且将作为证明命题具有很高的影响。其他的，例如派生范畴的维度或上同模的外部结构，是新兴理论的一部分，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具有广泛的应用范围，进一步扩大我们在该领域的知识，与不同数学背景的人建立互动，并产生适合博士生的问题。在数学之外，PI将继续参与非学院活动，比如他是致力于为女性在科学和工程领域创造更好环境的WISEST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研讨会，并以传播知识为目标编辑书籍。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部门，它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.