Determinant bundle, Quillen metric, and Petersson-Weil form on moduli spaces
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.
scientific article; zbMATH DE number 1309442