Soliton-type metrics and Kähler-Ricci flow on symplectic quotients

In this paper, we first show an interpretation of the Kähler-Ricci flow on a manifold X as an exact elliptic equation of Einstein type on a manifold M of which X is one of the (Kähler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on X will do). Such an equation is called V-soliton equation, which can be regarded as a generalization of Kähler-Einstein equations or Kähler-Ricci solitons. As in the case of Kähler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kähler manifoldM. One of our motivations is to apply the interpretation to studying finite time singularities of the Kähler-Ricci flow.