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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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