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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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