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

Abstract

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-Ampere type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kähler manifold $M$. One of our motivations is to apply the interpretation to studying finite time singularities of the Kähler-Ricci flow.

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