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A natural graded Lie algebra sheaf over Riemann surfaces. (English) Zbl 0738.30029
A simultaneous uniformization of a family of Riemann surfaces introduced by Bers is realized here by means of a sheaf of \(\mathbb{Z}_ 2\)-graded Lie algebras defined via the bundles of projective connections of the surfaces. This allows to define a structure equivalent to that of the so called super Riemann surfaces defined by Manin. The present approach is deeply connected with the classical theory, the construction is explicit and thereby the moduli space of super Riemann surfaces can be globally given.
Reviewer: A.Kaneko (Komaba)

MSC:
30F10 Compact Riemann surfaces and uniformization
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
17B70 Graded Lie (super)algebras
58A50 Supermanifolds and graded manifolds
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References:
[1] R. Gunning , Special Coordinate Covering of Riemann Surfaces , Math. Ann. , Vol. 170 , 1967 , pp. 67 - 86 . MR 207978 | Zbl 0144.33501 · Zbl 0144.33501
[2] K. Kodaira and D. Spencer , On Deformations of Complex Analytic Structures , Ann. Math. , Vol. 67 , 1958 , pp. 328 - 466 . MR 112154 | Zbl 0128.16901 · Zbl 0128.16901
[3] L. Bers , Simultaneous Uniformizations , Bull. Am. Math. Soc. , Vol. 66 , 1960 , pp. 94 - 97 . Article | MR 111834 | Zbl 0090.05101 · Zbl 0090.05101
[4] Yu.I. Manin , Critical Dimensions of the String Theories and the Dualizing Sheaf on the Moduli Space of (Super) Curves , Funct. Anal. Appl. , Vol. 20 , 1986 , pp. 244 - 245 . MR 868568 | Zbl 0639.14015 · Zbl 0639.14015
[5] C. Le Brun and M. Rothstein , Moduli of Super Riemann Surfaces , Comm. Math. Phys. , Vol. 117 , 1988 , p. 159 . Article | MR 946998 | Zbl 0662.58008 · Zbl 0662.58008
[6] L. Crane and J. Rabin , Super Riemann Surfaces: Uniformization and Teichmüller Theory , Comm. Math. Phys. , Vol. 113 , 1988 , p. 601 . Article | MR 923633 | Zbl 0659.30039 · Zbl 0659.30039
[7] L. Hodgkin , A Direct Calculation of Super Teichmüller Space , Lett. Math. Phys. , Vol. 14 , 1987 , p. 74 . MR 901699 | Zbl 0627.58004 · Zbl 0627.58004
[8] N. Hawley and M. Shiffer , Half Order Differential on Riemann Surfaces , Acta Math. , Vol. 115 , 1966 , pp. 119 - 236 . MR 190326 | Zbl 0136.06701 · Zbl 0136.06701
[9] D. Leites , Seminars on Supermanifolds , NO 30 1988-n13, Matematiska institutionen , Stockholms , ISSN 0348-7652.
[10] L. Ahlforz , Lecture on Quasi Conformal Mappings , Van Nostrand Math. Studies, N 10, Princeton , 1966 .
[11] P. Sipe , Roots of the Canonica Bundle Over the Universal Teichmüller Curve , Math. Ann. , Vol. 260 , 1982 , pp. 67 - 92 . MR 664367 | Zbl 0502.32017 · Zbl 0502.32017
[12] G. Trautmann , Deformations of Sheaves and Bundles , Lect. Notes Math. , Vol. 683 , 1978 , pp. 29 - 41 . MR 517519 | Zbl 0388.32011 · Zbl 0388.32011
[13] I. Kra , Automorphic Forms and Kleinian Groups , Benjamin , Reading, Mass ., 1972 . MR 357775 | Zbl 0253.30015 · Zbl 0253.30015
[14] F. Hirzebruch , Topological Methods in Algebraic Geometry , Springer-Verlag , Berlin - Heidelberg - New York , 1966 . MR 202713 | Zbl 0138.42001 · Zbl 0138.42001
[15] R. Gunning , Riemann Surfaces and Generalized Theta Functions , Berlin , Heidelberg , New York , Springer , 1976 . MR 457787 | Zbl 0341.14013 · Zbl 0341.14013
[16] K. Gawedzki , Supersymmetries-Mathematics of Supergeometry , Ann. Inst. H. Poincaré, Sect. A. , vol. XXVII , 1977 , p. 355 - 366 . Numdam | MR 489701 | Zbl 0369.53061 · Zbl 0369.53061
[17] G. Segal , Comm. Math. Phys. , 80 , 1981 , p. 301 . Article | MR 626704 | Zbl 0495.22017 · Zbl 0495.22017
[18] D. Friedan , in Supersymmetry, Supergravity and Superstrings 86 , B. DE WITT ed., World Scientific , Singapore , 1986 . MR 851575 · Zbl 0648.53057
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