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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.

##### MSC:
 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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