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Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities. (English) Zbl 0556.14019
The existence of a Ricci-negative Einstein-Kähler V-metric on certain Satake V-surfaces is proved, and some consequences are derived. Let X be a compact complex surface with at worst isolated quotient singularities and C a divisor on X with normal crossings which lies in the regular part of X. Let \(\bar X\to X\) be the minimal resolution of X and D the exceptional divisor. We can determine the non-negative rational numbers \(\mu_ i\) by requiring \(K_{\bar X}+\sum_ i\mu_ iD_ i\) to be trivial near D. Assume the following conditions are satisfied: \((1)\quad \kappa (K_{\bar X}+D+C)=2;\) \((2)\quad K_{\bar X}+C\) is numerically effective on C; \((3)\quad \{C_ i;\quad C_ i\quad is\quad an\quad irreducible\quad component\quad of\quad C\quad with\quad (K_{\bar X}+C)\cdot C_ i>0\}\) has only simple normal crossings if any; (4) there are no (-1)-curves which intersect supp(C) at most one point; (5) there are no (-2)-curves \(E\not\in \sup p(D)\) such that if \(D_ i\) intersects E then \(\mu_ i=0.\) Then there exists a Ricci-negative complete Einstein-Kähler V-metric with finite volume. - As a corollary, we derive an inequality \((K_{\bar X}+\sum_ i\mu_ iD_ i+C)^ 2\leq 3(e(\bar X)-e(C)-e(D)+\sum_ p| G_ p|^{-1}),\) where p runs over the singularities of X and \(G_ p\) the finite group corresponding to p. This is also proved by Y. Miyaoka [Math. Ann. 268, 159-171 (1984; Zbl 0521.14013)]. The advantage of our method, whose original is Yau’s argument in the proof of Calabi’s conjecture, is that we can determine the case of the equality. Namely, the equality occurs iff the universal covering of \(\bar X-\)D-C is biholomorphic to the open 2-ball in \({\mathbb{C}}^ 2\) minus a discrete set of points.

14J25 Special surfaces
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14C20 Divisors, linear systems, invertible sheaves
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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[1] [A] Aubin, T.: Équations du type Monge-Ampère sur les variétés kählériennes compactes. C. R. Acad. Paris283, 119-121 (1976) · Zbl 0333.53040
[2] [B] Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math.4, 336-358 (1968) · Zbl 0219.14003
[3] [C, Y] Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure Appl. Math.26, 507-544 (1978) · Zbl 0506.53031
[4] [H1] Hirzebruch, F.: Hilbert modular surfaces. Ens. Math.71, 183-281 (1973) · Zbl 0285.14007
[5] [H2] Hirzebruch, F.: Chern numbers of algebraic surfaces ? an example Preprint series 030-83, M.S.R.I., Berkeley
[6] [Hö] Höfer, T.: Dissertation, Bonn 1984
[7] [K1] Kobayashi, R.: A remark on the Ricci curvature of algebraic surfaces of general type. Tohoku Math. J.36, 385-399 (1984) · Zbl 0536.53064
[8] [K2] Kobayashi, R.: Einstein-Kähler metrics on open algebraic surfaces of general type. Tohoku Math. J. (to appear)
[9] [M] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann.268, 159-171 (1984) · Zbl 0534.14019
[10] [P] Parsad, G.: Strong rigidity of ?-rank 1 lattices. Invent. Math.21, 255-286 (1973) · Zbl 0264.22009
[11] [S] Sakai, F.: Semi-stable curves on algebraicc surfaces and logarithmic pluricanonical maps. Math. Ann.254, 89-120 (1980) · Zbl 0437.14017
[12] [Y1] Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equations. I. Comm. Pure Appl. Math.31, 339-411 (1978) · Zbl 0369.53059
[13] [Y2] Yau, S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA74, 1789-1799 (1977) · Zbl 0355.32028
[14] [Z] Zariski, O.: The theorem of Riemann-Roch for high multiple of an effective divisor on surfaces. Ann. Math.76, 506-615 (1962) · Zbl 0124.37001
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