|Sfb 288 Differential Geometry and Quantum Physics|
Abstract for Sfb Preprint No. 565
On the compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel
Marinescu G., Dinh T.C.
We show that the `pseudoconcave holes' of some naturally arising class of manifolds, called hyperconcave ends, can be filled in, including the case of complex dimension 2. As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2. From a differential-geometric point of view, arithmetic quotients are of bounded symmetric domains are Kaehler-Einstein manifolds of finite volume and bounded curvature. Siu and Yau generalized the compactification of arithmetic quotients of rank 1 by compactifying complete Kaehler manifolds X of finite volume and sectional curvature pinched between two negative constants. The first step in their proof is to show that X is a hyperconcave end, using the Busemann function and the lemmas of Margulis and Gromov. Then X can be compactified to a projective manifold by the embedding theorem of Andreotti-Tomassini. For the compactification of general hyperconcave ends we show the existence of a lot of holomorphic functions using the twisting trick of Berndtsson and Siu, using a Kaehler metric satisfying the Donnelly-Fefferman condition. Siu-Yau also prove, using the Ahlfors-Schwarz lemma of Yau and Mok-Yau, that X can be compactified adding finitely many points. We give a new analytic proof of this step in Section4. Nadel and Tsuji generalized the compactification of arithmetic quotients of any rank, by showing that certain pseudoconcave manifolds are quasiprojective. In dimension 2 their condition coincides with hyperconcavity. Proposition 5.3 yields, in dimension 2, a stronger version of their theorem together with a completely complex-analytic proof of the compactification of arithmetic quotients, cf. Remark 5.5.
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