Moduli of Hermite-Einstein vector bundles (Q1081855)

On a compact Kähler manifold M, it is shown that the moduli space of Hermite-Einstein vector bundles is a complex manifold under a suitable cohomological assumption, namely, \(H^ 2(M, sl \epsilon)=0\), where sl \(\epsilon\) is the bundle of trace-free endomorphisms of a holomorphic vector bundle \(\epsilon\) over M. The tangent space of the moduli space at the point [\(\epsilon\) ] is isomorphic to the sheaf-cohomology group \(H^ 1(M, End \epsilon)\). Similar argument applies to show that the moduli space of simple holomorphic vector bundles is a (non-Hausdorff) complex manifold under the same cohomological assumption. As a corollary, we obtain that the moduli space of stable bundles on a nonsingular irreducible projective algebraic surface with nonnegative total scalar curvature is a Kähler manifold.