Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature
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.
scientific article; zbMATH DE number 4117411