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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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##### References:
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