### Abstract

It is a long-standing problem to establish the existence of Kähler-Einstein metrics on Fano manifolds since Yau’s solution for the Calabi conjecture in late 70s. It is also one of driving forces in today’s study in Kähler geometry. In this paper, we discuss a program I started more than twenty years ago on this famous problem. It includes some of my results and speculations on the existence of Kähler-Einstein metrics on Fano manifolds, such as, holomorphic invariants, the K-stability, the compactness theorem for Kähler-Einstein manifolds, the partial C^{0}-estimates and their variations. I will also discuss some related problems as well as some recent advances.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 119-159 |

Number of pages | 41 |

DOIs | |

State | Published - Jan 1 2012 |

### Publication series

Name | Progress in Mathematics |
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Volume | 297 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Keywords

- Compactness theorems
- Fano manifolds
- Holomorphic invariants
- K-stability
- Kähler-Einstein metrics

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## Cite this

Tian, G. (2012). Existence of Einstein metrics on Fano manifolds. In

*Progress in Mathematics*(pp. 119-159). (Progress in Mathematics; Vol. 297). Springer Basel. https://doi.org/10.1007/978-3-0348-0257-4_5