## Contracting exceptional divisors by the Kähler-Ricci flow.(English)Zbl 1266.53063

The authors study the Kähler-Ricci-flow contracting exceptional divisors in the sense of blow down. Precisely, they assume that there exists a holomorphic map $$\pi : X \rightarrow Y$$ blowing down disjoint exceptional divisors $$E_1,...,E_k$$ which are submanifolds of $$X$$ of codimension $$1$$, biholomorphic to $$P^{n-1}$$ and with normal bundle $$O(-1)$$. Moreover, $$\pi$$ contracts $$E_1,\dots,E_k$$ to distinct points $$y_1,\dots,y_k$$. Then, under a certain cohomology assumption, the Kähler-Ricci flow blows down the exceptional divisors in the sense of Gromov-Hausdorff and smoothly away from the divisors and continues on the new manifold. Moreover if $$X$$ is a projective algebraic surface and the Kähler class is rational then a maximal flow always exists and gives information on the original manifold $$X$$. In particular in this case the Kähler-Ricci flow performs a sequence of canonical contractions until, in finite time, either the minimal model is obtained or the manifold collapses.

### MSC:

 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 32Q20 Kähler-Einstein manifolds 14E30 Minimal model program (Mori theory, extremal rays)
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### References:

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