zbMATH — the first resource for mathematics

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

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q99 Complex manifolds
Full Text: DOI
[1] T. AUBIN, Equations du type Monge-Ampere sur les variete kaehleriennes compactes, C. R. Acad. Paris. 283 (1976), 119-121. · Zbl 0333.53040
[2] T. AUBIN, ”Nonlinear Analysis on manifolds, Monge-Ampere Equations”, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, Grundlehren der Math. 252. · Zbl 0512.53044
[3] B. Y. CHEN AND K. OGIUE, Some characterization of complex space forms in terms o Chern classes, Quart. J. Math. 26 (1975), 459-464. · Zbl 0315.53034 · doi:10.1093/qmath/26.1.459
[4] S. Y. CHENG AND S. T. YAU, On the existence of a complete Kaehler metric on non compact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1978), 507-544. · Zbl 0506.53031 · doi:10.1002/cpa.3160330404
[5] H. GRAUERT, Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann 146 (1962), 331-368. · Zbl 0178.42702 · doi:10.1007/BF01441136 · eudml:160940
[6] P. A. GRIFFITHS AND J. HARRIS, ”Principles of Algebraic Geometry”, John Wiley & Sons, New York, Chichester, Brisbane, Toronto. · Zbl 0836.14001
[7] F. HIRZEBRUCH, Hubert modular surfaces, Ens. Math. 71 (1973), 183-281 · Zbl 0285.14007
[8] F. HIRZEBRUCH, Chern numbers of algebraic surfaces–an example–, preprint serie 030-83, M. S. R. I. Berkeley. · Zbl 0504.14030 · doi:10.1007/BF01475584 · eudml:163860
[9] R. KOBAYASHI, Kaehler-Einstein metric on open algebraic manifolds, Osaka J. Math 21 (1984), 399-418. · Zbl 0582.32011
[10] R, KOBAYASHI, A remark on the Ricci curvature of algebraic surfaces of general type, Thoku M. J. 36 (1984), 385-399. · Zbl 0536.53064 · doi:10.2748/tmj/1178228805
[11] K. KODAIRA, On compact complex analytic surfaces, I, Ann. of Math. 71 (1960), 111-152 JSTOR: · Zbl 0098.13004 · doi:10.2307/1969881 · links.jstor.org
[12] K. KODAIRA, Pluricanonical system on algebraic surfaces of general type, J. Math. Soc Japan. 20 (1968), 170-192. · Zbl 0157.27704 · doi:10.2969/jmsj/02010170
[13] H. LAUFER, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131-164 · Zbl 0281.32010 · doi:10.1007/BF01350842 · eudml:162485
[14] Y. MIYAOKA, On the Chern numbers of surfaces of general type, Invent. Math. 4 (1977), 225-237. · Zbl 0374.14007 · doi:10.1007/BF01389789 · eudml:142501
[15] Y. MIYAOKA, The maximal number of quotient singularities on surfaces with give numerical invariants, Math. Ann. 268 (1984), 159-171. · Zbl 0521.14013 · doi:10.1007/BF01456083 · eudml:182912
[16] D. MUMFORD, Hirzebruch’s proportionality theorem in the noncompact case, Invent Math. 42 (1977), 230-272. · Zbl 0365.14012 · doi:10.1007/BF01389790 · eudml:142502
[17] U. PERSSON, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 189 (1977), 1-144. · Zbl 0368.14008
[18] F. SAKAI, Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), 89-120. · Zbl 0431.14011 · doi:10.1007/BF01467073 · eudml:163485
[19] I. SATAKE, The Gauss-Bonnet theorem for F-manifolds, J. Math. Soc. Japan. 9 (1957), 464-492. · Zbl 0080.37403 · doi:10.2969/jmsj/00940464
[20] S. T. YAU, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl Math. 28 (1975), 201-228. · Zbl 0291.31002 · doi:10.1002/cpa.3160280203
[21] S. T. YAU, A general Schwarz lemma for Kaehler manifolds, Amer. J. Math. 100 (1978), 197-203. JSTOR: · Zbl 0424.53040 · doi:10.2307/2373880 · links.jstor.org
[22] S. T. YAU, On Calabi’s conjecture and some new results in algebraic geometry, Nat Acad. Sci. U. S. A. 74 (1977), 1789-1799. JSTOR: · Zbl 0355.32028 · doi:10.1073/pnas.74.5.1798 · links.jstor.org
[23] S. T. YAU, On the Ricci curvature of a compact Kaehler manifold and the comple Monge-Ampere equation, I, Comm. Pure Appl. Math. 31 (1978), 339-411. · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.