Determinant bundle, Quillen metric, and Petersson-Weil form on moduli spaces
type: Item
instance of: scholarly article
title: Determinant bundle, Quillen metric, and Petersson-Weil form on moduli spaces
zbMATH Open document ID: 0968.32008
author: Indranil Biswas
author: Georg Schumacher
published in: Geometric and Functional Analysis. GAFA
publication date: 2001-09-20T00:00:00Z
review text: The authors study moduli spaces of pairs \((X,E)\) where \(E\) is a unitary flat bundle over a smooth projective manifold \(X\) with ample canonical line bundle. The first cohomology of the Atiyah holomorphic vector bundle over \(X\) is used to describe the tangent space at a point of the moduli space, and the moduli space carries a Kähler structure given by the Petersson-Weil metric. The 2-form belonging to this metric coincides with the natural Kähler form on the unitary representation space for fixed \(X\), and this form plays a role in a certain foliation. The main property established about the Petersson-Weil form is that it can be realized as a fiber-integral, i.e. a fiberwise integral of a form over the universal space for \(X\). Then the curvature of the Petersson-Weil metric is computed. Finally, the case of a smooth quasi-projective variety \(X\) over \(\mathbb{C}\) is considered. In particular, the Kähler form on the representation space is discussed. Here the determinant bundle and the Quillen metric are introduced.
reviewed by: Alfred Aeppli
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 1309442
zbMATH Keywords: moduli spaces
zbMATH Keywords: first cohomology
zbMATH Keywords: Atiyah holomorphic vector bundle
zbMATH Keywords: Petersson-Weil metric
zbMATH Keywords: determinant bundle
zbMATH Keywords: Quillen metric
MaRDI profile type: Publication
full work available at URL: s000390050086
OpenAlex ID: W2002612103
DOI: 10.1007/S000390050086
description: scientific article; zbMATH DE number 1309442