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Einstein-Kähler metrics on open algebraic surfaces of general type. (English) Zbl 0582.53046
A compact Riemann surface S admits a Kähler metric with negative constant Gaussian curvature if the genus of S is greater than one. This fact is generalized by T. Aubin [C. R. Acad. Sci., Paris, Sér. I 283, 119-121 (1976; Zbl 0333.53040)] and S. T. Yau [Commun. Pure Appl. Math. 31, 339-411 (1978; Zbl 0362.53049)] to higher dimensional complex manifolds. In the previous paper [Tôhoku Math. J., II. Ser. 36, 385-399 (1984; Zbl 0536.53064)], the author proved a much stronger result for the two-dimensional case. On the other hand, let \(p_ 1,...,p_ k\) be distinct points in \({\mathfrak P}^ 1\). If k is greater than two, \({\mathfrak P}^ 1-\{p_ 1,...,p_ k\}\) admits a complete Kähler metric with negative constant Gaussian curvature with finite volume.
The author obtains a two-dimensional analogue of the fact mentioned above. Let \(\bar M\) be a compact complex surface and D a reduced divisor with normal crossings. Assume \((\bar M,D)\) satisfies the following conditions: (i) Let \(L=K_{\bar M}\otimes D\) then \(L^ 2>0\) and \(L\cdot C\geq 0\) for all irreducible curves C on M, (ii) the divisor determined by curves C such that \(C\subset D\) and \(L\cdot C>0\) has only simple normal crossings as its singularities. Let \(\Phi_{mL}\) be the logarithmic m- canonical map. Then \(M'=\Phi_{mL}(\bar M-D)\) is called the logarithmic canonical model of \(\bar M-D\). The author proves the following: The logarithmic canonical model M’ admits a complete Einstein-Kähler V- metric with negative Ricci curvature, which is unique up to multiplication by positive numbers. Moreover, the total volume is finite and equal to \(L^ 2\) if the Ricci tensor is \(-(2\pi)^{-1}\) times the metric. Write \(\bar c_ i\) for the i-th logarithmic Chern class of \((\bar M,D)\). Then the inequality \(3\bar c_ 2-\bar c^ 2_ 1\geq k(\bar M-D)\geq 0\) holds, where \(k(\bar M-D)\) is a rational number which is universally determined by the configuration of all \((-2)\)-curves on \(\bar M-D\).
Reviewer: T.Ishihara

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q99 Complex manifolds
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