Generic Vanishing, 1-forms, and Topology of Albanese Maps
Yajnaseni Dutta, Feng Hao, Yongqiang Liu
备注：
Final version to appear in Math Zeitschrift: rewrote the introduction and abstract from a slightly different perspective. Shortened exposition, with some deleted proofs as per referee's comments. Added a special case of the main theorem in positive characteristics (Proposition 3.7). 18 pages, comments are very welcome
Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety $X$; one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This result is related to Kotschick's conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if $X$ has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients.
In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of $H^0(X,Ω_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1.
Non-vanishing of homotopy groups of Manin--Schechtman arrangements
Institute for Chemical Reaction Design and Discovery, Hokkaido University（化学反应用设计发现研究所，北海道大学）；Department of Applied Mathematics, Fukuoka University（福冈大学应用数学系）
Expected topology of random real algebraic submanifolds
设X是一个n维光滑复射影流形，它具有一个充量线丛L和一个秩为k的全纯向量丛E.我们假设0< k <=n，X，E和L定义在实数上，用RX表示X的实轨迹。然后，我们估计从上到下的期望贝蒂数的消失轨迹在RX中的全纯实截面张量为L^d，其中d是一个足够大的整数。此外，给定R^n中任意具有平凡法丛的闭连通余维k子流形S，证明了E中具有L^d张量的实截面在其消失轨迹中有与d无关的正概率包含d^n个与S同构的连通分支的平方根.
