Cohomology of vector bundles and non-pluriharmonic loci

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension $n\geq 2$. We prove a vanishing result for the Bott-Chern cohomology group of type $(1, 1)$ with compact support in $X$, which combined with the well-known technique of Ehrenpreis implies a Hartogs type extension theorem for pluriharmonic functions on $X$.

Cohomology support loci of local systems

The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander modules of an algebraic knot, and of certain cohomology support loci. Moreover, it equals conjecturally the image under the exponential map of the zero locus of the Bernstein-Sato ideal. Sabbah showed that S is contained in a union of translated subtori of codimension one in a complex affine torus. Budur-Wang showed recently that S is a union of torsion-translated subtori. We show here that S is always a…

局部系统的霍奇理论与上同调支撑轨迹

Hodge theory for local systems and cohomological support loci

本文以 $\partial\bar{\partial}$-引理为统一框架，重新建立一般消失定理与上同调跳跃轨迹理论，并发展拟紧凯勒流形上一阶局部系统的 $L^2$-霍奇理论。

Orbits of real semisimple Lie groups on real loci of complex symmetric spaces

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$ with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.

Local systems on analytic germ complements

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.