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Journal of the European Mathematical Society

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Volume 14, Issue 6, 2012, pp. 2001–2038
DOI: 10.4171/JEMS/353

Published online: 2012-10-10

The Kähler Ricci flow on Fano manifolds (I)

Xiuxiong Chen[1] and Bing Wang[2]

(1) University of Wisconsin-Madison, Madison, USA
(2) Princeton University, USA

We study the evolution of pluri-anticanonical line bundles $K_M^{-\nu}$ along the Kähler Ricci flow on a Fano manifold $M$. Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of $M$. For example, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface satisfying $c_1^2(M)=1$ or $c_1^2(M)=3$. Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian in [Tian90].

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Chen Xiuxiong, Wang Bing: The Kähler Ricci flow on Fano manifolds (I). J. Eur. Math. Soc. 14 (2012), 2001-2038. doi: 10.4171/JEMS/353