Kähler-Einstein metric on an open algebraic manifold
The author generalises Yau's theorem on compact complex manifolds with ample canonical bundle as follows. Let \(\bar M\) be a complex projective manifold, and \(D\subset \bar M\) a (reduced) normal-crossings divisor with smooth components. If \(K_{\bar M}\otimes D\) is ample, then there is a unique complete Ricci-negative Einstein-Kähler metric of volume one on \(M=\bar M\setminus D.\) From this theorem, the author also deduces an inequality involving the first and second Chern classes of \(\bar M\) and the cohomology classes of D and its components, which reduces to the Miyaoka-Yau inequality when \(D=\emptyset\). [For stronger results when dim \(\bar M=2\), see the author's paper in Math. Ann. 272, 385-398 (1985; Zbl 0556.14019).]
complement of divisor with normal crossings
scientific article; zbMATH DE number 3931381