Corpus ID: 116977053

Geometry of cohomology support loci for local systems I

  title={Geometry of cohomology support loci for local systems I},
  author={D. Arapura},
  journal={arXiv: Algebraic Geometry},
  • D. Arapura
  • Published 1996
  • Mathematics
  • arXiv: Algebraic Geometry
Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $\Sigma^k(X)$ of one dimensional local systems on X with nonvanishing kth cohomology. We show that under certain conditions (X compact, X has a smooth compactification with trivial first Betti number, or k=1) $\Sigma^k(X)$ is a union of translates of sets of the form $f^*H^1(T,C^*)$, where $f:X \to T$ is a holomorphic map to a complex Lie group which is an extension of a compact complex torus by a… Expand
Geometry of cohomology support loci II: integrability of Hitchin's map
In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projectiveExpand
Jumping loci and finiteness properties of groups
Characteristic varieties. Let X be a connected CW-complex with finitely many cells in each dimension, and G its fundamental group. The characteristic varieties of X are the jumping loci forExpand
Characteristic Varieties and Betti Numbers of Free Abelian Covers
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of theExpand
The topology of compact Lie group actions through the lens of finite models
Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smoothExpand
Homological finiteness of abelian covers
We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finiteExpand
Generic vanishing and holomorphic 1-forms
The goal of this article is three-fold; first we establish an interesting link between some sets related to cohomologies of complex local systems and the stratification of the Albanese morphism ofExpand
Geometric and algebraic aspects of 1-formality
Formality is a topological property, defined in terms of Sullivan's model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rationalExpand
On the twisted cohomology of arrangements of lines
Let $\A$ be an affine line arrangement in $\C^2,$ with complement $\M(\A).$ The twisted (co)homology of $\M(\A)$ is an interesting object which has been considered by many authors, strictly relatedExpand
Fibred Kähler and quasi-projective groups.
We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group πι^) of a compact Kahler manifold onto the fundamental group Π^ of aExpand
Topology and geometry of cohomology jump loci
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted groupExpand


Subspaces of moduli spaces of rank one local systems
Suppose X is a smooth projective variety. The moduli space M (X) of rank one local systems on X has three different structures of complex algebraic group (Betti, de Rham, and Dolbeault). A subgroupExpand
Geometric Invariant Theory
“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory toExpand
Mixed hodge modules
Introduction 221 § 1. Relative Monodromy Filtration 227 §2. Mixed Hodge Modules on Complex Spaces (2. a) Vanishing Cycle Functors and Specializations (Divisor Case) 236 (2.b) Extensions over LocallyExpand
Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville
Introduction 389 w Notation and conventions 392 w Deformations of cohomology groups 392 w Generic vanishing criteria for topologically trivial line bundles 397 w A Nakano-type generic vanishingExpand
Higgs bundles and local systems
© Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www.Expand
Varieties of Representations of Finitely Generated Groups