Some families of local systems over smooth projective varieties
type: Item
instance of: scholarly article
title: Some families of local systems over smooth projective varieties
zbMATH Open document ID: 0813.14014
DOI: 10.2307/2946614
published in: Annals of Mathematics. Second Series
publication date: 1995-05-30T00:00:00Z
review text: One can try to study the fundamental group of a smooth, complex projective variety \(S\) by looking at spaces of representations of \(\pi_ 1 (S)\). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties \(S\) where there exist nontrivial continuous families of representations (or, equivalently, local systems on \(S) \). It is not too hard to see that if \(S\) is a projective algebraic curve of genus \(g \geq 2\), then the moduli space of representations of rank \(r \geq 1\) has dimension \((2g - 2) r^ 2 + 2\); or, for example, if \(S\) is a variety with \(\dim H^ 1(S, \mathbb{C}) = a\), then the space of representations of rank 1 has dimension \(a\). Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist? The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that \(f:X \to S\) is a smooth projective morphism and \(\{W_ t\}\) is a family of local systems on \(X\), chosen in a simple way. Let \(V_ t = R^ i f_ * (W_ t)\). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that \(\{V_ t\}\) will be an interesting family of local systems on \(S\). The principal question that needs to be addressed is whether, if the family \(\{W_ t\}\) varies nontrivially, the family of direct images \(\{V_ t\}\) varies nontrivially. We are also interested in constructing examples where we can show that the family \(\{V_ t\}\) does not, by some miracle, arise from a simpler construction such as the one described before. This article consists essentially of two parts. The first (sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we devel…
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14D07
zbMATH DE Number: 490545
zbMATH Keywords: representations of fundamental group
zbMATH Keywords: local systems on complex projective variety
zbMATH Keywords: moduli space of algebraic curve
zbMATH Keywords: families of representations
zbMATH Keywords: direct image of a harmonic bundle
zbMATH Keywords: Higgs bundle
zbMATH Keywords: spectral varieties of the Higgs bundles
author: Carlos Simpson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946614
OpenAlex ID: W2328808682
description: scientific article; zbMATH DE number 490545