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. (2) 36, 385–399 (1984; Zbl 0536.53064)], the author proved a much stronger result for the two-dimensional case. On the other hand, let \(p_1,\ldots, p_k\) be distinct points in \(\mathfrak P^1\). If \(k\) is greater than two, \(\mathfrak P^1 - \{p_1,\ldots, p_k\}\) 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 \(\overline M\) be a compact complex surface and \(D\) a reduced divisor with normal crossings. Assume \((\overline M,D)\) satisfies the following conditions: (i) Let \(L=K_{\overline M}\otimes D\) 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 \(M'=\Phi_{mL}(\overline M - D)\) is called the logarithmic canonical model of \(\overline M - D\).
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 \((\overline M,D)\). Then the inequality \(3\bar c_2 - \bar c^2_1\geq k(\overline M - D)\geq 0\) holds, where \(k(\overline M - D)\) is a rational number which is universally determined by the configuration of all \((-2)\)-curves on \(\overline M - D\).
The author obtains a two-dimensional analogue of the fact mentioned above. Let \(\overline M\) be a compact complex surface and \(D\) a reduced divisor with normal crossings. Assume \((\overline M,D)\) satisfies the following conditions: (i) Let \(L=K_{\overline M}\otimes D\) 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 \(M'=\Phi_{mL}(\overline M - D)\) is called the logarithmic canonical model of \(\overline M - D\).
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 \((\overline M,D)\). Then the inequality \(3\bar c_2 - \bar c^2_1\geq k(\overline M - D)\geq 0\) holds, where \(k(\overline M - D)\) is a rational number which is universally determined by the configuration of all \((-2)\)-curves on \(\overline M - D\).
Reviewer: Toru Ishihara (Tokushima)
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 32Q99 | Complex manifolds |
Keywords:
algebraic surface; Kähler metric; Gaussian curvature; logarithmic canonical model; V-metric; Chern classReferences:
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