The Ricci curvature of symplectic quotients of Fano manifolds (Q1093938)

Let M be a Kähler manifold on which a compact connected Lie group K acts as a group of holomorphic isometries. Let k be the Lie algebra of K. If \(\mu\) : \(X\to k^*\) is a momentum mapping and K acts freely on \(\mu^{-1}(0)\) then the symplectic quotient (or Marsden-Weinstein reduction) \(M_ K=\mu^{-1}(0)/K\) has an induced Kähler structure. In this paper a formula is given for the Ricci curvature of \(M_ K\) in terms of the Ricci curvature of M. In particular it is shown that if M is a Fano manifold (i.e. a compact complex manifold of positive first Chern class) then so is the quotient \(M_ K\). When M is a compact Kähler- Einstein manifold of positive Ricci curvature a necessary and sufficient condition is given for \(M_ K\) to be Kähler-Einstein.