### Abstract

In this paper, I give a brief tour on a program of studying the Kähler-Ricci flow with surgery and its interaction with the classification of projective manifolds. The Kähler-Ricci flow may develops singularity at finite time. It is important to understand how to extend the Kähler-Ricci flow across the singular time, that is, construct solution of the Kähler-Ricci flow with surgery. The first task of this paper is to describe a procedure of constructing global solutions for the Kähler-Ricci flow with surgery. This procedure is rather canonical. I will discuss properties of such solutions with surgery and their geometric implications. I will also discuss their asymptotic limits at time infinity. The results discussed here were mainly from my joint works with Z. Zhang, J. Song et al. Some open problems will be also discussed. The paper is mostly expository.

Original language | English (US) |
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Title of host publication | Differential Geometry, Mathematical Physics, Mathematics and Society Part 2 |

Pages | 71-92 |

Number of pages | 22 |

Edition | 322 |

State | Published - Dec 1 2008 |

### Publication series

Name | Asterisque |
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Number | 322 |

ISSN (Print) | 0303-1179 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Kähler-Einstein metrics
- Kähler-Ricci flow

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## Cite this

*Differential Geometry, Mathematical Physics, Mathematics and Society Part 2*(322 ed., pp. 71-92). (Asterisque; No. 322).