Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature. (English) Zbl 0711.53056

Nadel, Alan Michael

Proc. Natl. Acad. Sci. USA 86, No. 19, 7299-7300 (1989).

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

Keywords:

positive first Chern class; complex Monge-Ampère equation; Kähler- Einstein metric; analytic sheaf; positive scalar curvature