Mok, Ngaiming; Zhong, Jia-Qing Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature. (English) Zbl 0682.53065 Ann. Math. (2) 129, No. 3, 427-470 (1989). This paper shows that a complex manifold X which is homotopic to a finite CW-complex and carries a complete Kähler metric g of finite volume and negative Ricci curvature with bounded sectional curvatures is bihomolorphic to a Zariski-dense open subset of some projective manifold. As a consequence of a theorem of Ballman-Gromov-Schroeder, the topological condition can be dropped if g is real-analytic and has nonpositive sectional curvatures. The idea is to embed X birationally as an open subset of some projective manifold Z, using \(L^{\alpha}\) integrable sections of \(K^ p_ x\), for some \(p,\alpha >0\). The topological condition ensures that this map F can be made holomorphic after finitely many blow-ups. The argument to show that F(X) is Zariski-open in Z rests on very refined techniques, and uses in particular Bombieri’s result on sets of non-summability of \(e^{-\phi}\), \(\phi\) plurisubharmonic, as well as the analyticity of sets associated with Lelong numbers. Reviewer: F.Campana Cited in 17 Documents MSC: 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32J05 Compactification of analytic spaces Keywords:projective compactification; Kähler manifold; complex manifold; finite volume; negative Ricci curvature; bounded sectional curvatures PDFBibTeX XMLCite \textit{N. Mok} and \textit{J.-Q. Zhong}, Ann. Math. (2) 129, No. 3, 427--470 (1989; Zbl 0682.53065) Full Text: DOI