Soliton-type metrics and K\"ahler-Ricci flow on symplectic quotients

In this paper, we first show an interpretation of the K\"ahler-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 (K\"ahler) 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 K\"ahler-Einstein equations or K\"ahler-Ricci solitons. As in the case of K\"ahler-Einstein metrics, we can also reduce the $V$-soliton equation to a scalar equation on K\"ahler potentials, which is of Monge-Ampere type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact K\"ahler manifold $M$. One of our motivations is to apply the interpretation to studying finite time singularities of the K\"ahler-Ricci flow.