### 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-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.

Original language | English (US) |
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Pages (from-to) | 139-166 |

Number of pages | 28 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 2016 |

Issue number | 711 |

DOIs | |

State | Published - Feb 1 2016 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal fur die Reine und Angewandte Mathematik*,

*2016*(711), 139-166. https://doi.org/10.1515/crelle-2013-0114