×

Holomorphic conformal structures and uniformization of complex surfaces. (English) Zbl 0611.32023

We introduce the notion of generalized holomorphic conformal structure (GHCS) on a smooth compact complex surface and discuss the uniformization problem of surfaces admitting a GHCS. A GHCS defined by a holomorphic line bundle L on a compact complex surface X is a primitive section of \(L\otimes T^ *X\). For example, by taking \(X=P_ 2({\mathbb{C}})\) and \(L={\mathcal O}(m+3)\), we fix the way to compute the orbifold holomorphic conformal structure on the double plane branched over a curve of degree 2m with at worst simple singularities and Hilbert modular cusps. This enable us to compute explicitly the GHCS’s on \(P_ 2({\mathbb{C}})\) corresponding to certain Hilbert modular surfaces and to construct a compact surface with simple singularities which fulfills \=c\({}_ 1^ 2=2\bar c_ 2\) (in the modified sense) but is not covered by the bidisc. The relation between singular Einstein-Kähler metrics and GHCS’s is also discussed.

MSC:

32J15 Compact complex surfaces
32G99 Deformations of analytic structures
32L05 Holomorphic bundles and generalizations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14H15 Families, moduli of curves (analytic)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [A] Aubin, T.: Equation du type Aonge-Ampère sur les variétés Káhlériennes compactes. C.R. Acad. Paris283, 119-121 (1976)
[2] [B-G-S] Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature Progress in Mathematics Vol. 61. Basel, Boston, Stuttgart: Birkhäuser 1985 · Zbl 0591.53001
[3] [Br] Brieskorn, E.: Die Auflösung der rationalen Singularitäten holomorpher Abbildungen. Math. Ann.178, 255-270 (1968) · Zbl 0159.37703 · doi:10.1007/BF01352140
[4] [Bu] Buyalo, S.V.: Manifolds of nonpositive curvature with small volume. Math. Zam.29, 243-525 (1981) · Zbl 0464.53037
[5] [H1] Hirzebruch, F.: The ring of Hilbert modular forms for real quadratic fields of small discriminant. Lect. Notes Math.627, 287-323. Berlin, Heidelberg, New York: Springer 1977
[6] [H2] Hirzebruch, F.: Hilbert modular group of the field \(Q (\sqrt 5 )\) and the cubic diagonal surface of Clebsch and Klein. Russ. Math. Surv.3, 96-110 (1976) · Zbl 0356.14010 · doi:10.1070/RM1976v031n05ABEH004190
[7] [I] Ivinskis, K.: Normale Flächen und die Miyaoka-Kobayashi-Ungleichung. Diplomarbeit, Bonn (1985)
[8] [K] Klein, F.: Weitere Untersuchungen über das Ikosaeder. Gesammelte Math. Abhandlungen Bd. 2, 321-384. Berlin, Heidelberg, New York: Springer 1973
[9] [Ko] Kobayashi, R.: Einstein-KälerV-metrics on open SatakeV-surfaces with isolated quotient singularities. Math. Ann.272, 385-398 (1985) · Zbl 0556.14019 · doi:10.1007/BF01455566
[10] [Ko-T] Kobayashi, R., Todorov, A. N.: Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics. Tohoku Math. J. (to appear) · Zbl 0646.14029
[11] [Kob] Kobayashi, S.: The first Chern class and holomorphic symmetric tensor fiels. J. Math. Soc. Japan32, 325-329 (1980) · Zbl 0447.53055 · doi:10.2969/jmsj/03220325
[12] [Kob-O1] Kobayashi, S., Ochiai, T.: Holomorphic structures modeled after hyperquadratics. Tohoku Math. J.34, 587-629 (1982) · Zbl 0508.32007 · doi:10.2748/tmj/1178229159
[13] [Kob-O2] Kobayashi, S., Ochiai, T.: Holomorphic projective structures on complex surfaces. Math. Ann.249, 75-94 (1980), ibid. II. Math. Ann.255, 519-522 (1981) · Zbl 0422.32023 · doi:10.1007/BF01387081
[14] [M] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numberical invariants. Math. Ann.268, 159-171 (1984) · Zbl 0534.14019 · doi:10.1007/BF01456083
[15] [N] Naruki, I.: Some invariants for conics and their applications. Publ. RIMS, Kyoto Univ.19, 1139-1151 (1983) · Zbl 0539.14028 · doi:10.2977/prims/1195182023
[16] [Y1] Yau, S. T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equations. I. Comm. Pure Appl. Math.31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[17] [Y2] Yau, S. T.: A general Schwarz lemma for Kähler manifolds. Amer. J. Math.100, 197-203 (1978) · Zbl 0424.53040 · doi:10.2307/2373880
[18] [S-Y] Sasaki, T., Yoshida, M.: Linear differential equations in two variables of Rank four. Preprint, Max-Planck-Institut für Mathematik · Zbl 0627.35014
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.