Geometry of anti-self-dual connections and Kuranishi map (Q1089624)

A Riemannian structure is investigated on the moduli space of anti-self- dual Yang-Mills connections. The moduli space is a finite dimensional subspace of the orbit space of all connections. The gauge invariant \(L_ 2\)-inner product produces a Riemannian metric there. By the crucial aid of the Kuranishi map, a map linearizing the space and the Hodge theory on the Atiyah-Hitchin-Singer complex the Riemannian curvature can be expressed in terms of the \(L_ 2\)-inner product together with the Green operators, the inverses of the Laplacians. As a consequence, over a Kähler surface the moduli space carries a Kähler structure compatible with the canonically defined complex structure. Moreover it is verified that the moduli space of Einstein- Hermitian connections over a Riemannian surface is endowed with a Kähler structure of nonnegative holomorphic curvature and hence of positive scalar curvature. This positivity is consistent with the unirationality of the moduli spaces.