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The Kähler-Ricci flow on surfaces of positive Kodaira dimension. (English) Zbl 1134.53040

The authors propose a program of finding canonical metrics on canonical models of algebraic varieties of positive Kodaira dimension. Also they carry out this program for minimal Kähler surfaces. This is done by a study of the Kähler-Ricci flow starting from any Kähler metric, and describing its limiting behaviour as time goes to infinity. To develop the program the authors start with the Kähler-Ricci flow
\[ \begin{cases} \frac{\partial}{\partial t}\omega(t,\cdot) = -\text{Ric}(\omega(t,\cdot))- \omega(t,\cdot)\\ \omega(0,\cdot) = \omega_0,\end{cases}\tag{\(*\)} \]
where \(\omega(t,\cdot)\) is a family of Kähler metrics on an \(n\)-dimensional compact Kähler manifold \(X\), \(\omega_0\) is a given Kähler metric and \(\text{Ric}(\omega(t,\cdot))\) denotes the Ricci curvature of \(\omega(t,\cdot)\). Now, let \(f:X\Rightarrow \Sigma \) be a minimal elliptic surface of codimension 1 with singular fibres. Then it is proved that for any initial Kähler metric, the Kähler-Ricci flow \((*)\) has a global solution \(\omega(t,\cdot)\) for all time \(t\geq 0\) satisfying some conditions. Also, a metric classification for Kähler surfaces with a nef canonical line bundle bu the Kähler-Ricci flow is given.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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