Differential geometry of moduli spaces of quiver bundles

Let P be a parabolic subgroup of a semisimple affine algebraic group G defined over C and X a compact Kähler manifold. L. Álvarez-Cónsul and O. García-Prada associated to these a quiver Q and representations of Q into holomorphic vector bundles on X (Álvarez-Cónsul and García-Prada, 2003) [1, 2]. Our aim here is to investigate the differential geometric properties of the moduli spaces of representations of Q into vector bundles on X. In particular, we construct a Hermitian form on these moduli spaces. A fiber integral formula is proved for this Hermitian form; this fiber integral formula implies that the Hermitian form is Kähler. We compute the curvature of this Kähler form. Under an assumption which says that X is a complex projective manifold, this Kähler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric.