Novikov homology, jump loci and Massey products

TL;DR

This paper explores the relationships between Novikov homology, jump loci, and Massey products, providing spectral sequence computations and degeneracy results for Kähler manifolds, and analyzing the behavior of Betti numbers under deformations.

Contribution

It introduces a spectral sequence for cohomology with local coefficients, computes its differentials via Massey products, and studies the invariance of Betti numbers in relation to cohomology classes.

Findings

Spectral sequence degenerates for Kähler manifolds with semi-simple representations.

Differentials in the spectral sequence are expressed through Massey products.

Twisted Novikov Betti numbers are constant outside finitely many hyperplanes.

Abstract

Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms of Massey products. We show that the spectral sequence degenerates in case when X is a Kaehler manifold, and the representation R is semi-simple. If A is a real cohomology class, one associates to the triple (X,R,A) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the above local system. We investigate the dependance of these numbers on A and prove that they are constant in the complement to a finite number of integral hyperplanes in the first cohomology group.