  Let P be a parabolic subgroup of a simple affine algebraic group G defined
over C and X a compact connected K\"ahler manifold. L. \'Alvarez-C\'onsul and
O. Garc\'ia-Prada associated to these a quiver Q and representations of Q into
holomorphic vector bundles on X. Our aim here is to investigate the
differential geometric properties of the moduli spaces 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 K\"ahler. We compute
the curvature of this K\"ahler form. Under an assumption which says that X is a
complex projective manifold, this K\"ahler form is realized as the curvature of
a certain determinant line bundle equipped with a Quillen metric.

Complex Variables;Algebraic Geometry