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Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature. (English) Zbl 0711.53056
Let M be a compact complex manifold with $$C_ 1(M)>0$$ and $$g_{\alpha {\bar \beta}}$$ be a Kähler metric whose Kähler form lies in the anticanonical class. A Kähler-Einstein metric exists on M if a certain zeroth order a priori estimate holds for any sequence $$S=(\phi_ k)$$ of smooth real-valued functions on M satisfying $$\sup_ M\phi_ k=0$$ and $$\partial_{{\bar \alpha}}\phi_ k+g_{\alpha {\bar \beta}}>0$$ for each k. The author constructs a coherent analytic sheaf $${\mathcal I}$$ of ideals on M for each such S so that if $${\mathcal I}={\mathcal O}_ M$$, then the a priori estimate holds. Using $$L^ 2$$ estimates of $${\bar \partial}$$, the author finds significant global algebro-geometric properties of $${\mathcal I}$$ and the complex subspace V cut out by $${\mathcal I}$$. In particular, $$H^ r(M,{\mathcal I})=0$$ for $$r\geq 1$$. Now suppose a compact subgroup G of biholomorphisms of M is also given. Then a Kähler- Einstein metric exists on M, provided M does not admit a proper G- invariant coherent sheaf of ideals with the algebro-geometric properties obtained above. This method yields various new examples of Kähler- Einstein metrics of positive scalar curvature.
Reviewer: H.-S.Luk

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32C35 Analytic sheaves and cohomology groups 35B45 A priori estimates in context of PDEs
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