Projective structures, flat bundles, and Kähler metrics on moduli spaces

N. V. Ivanov

Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 211-224

Résumé

This paper deals mainly with the geometrical and topological aspects of the connection recently discovered by Zograf and Takhtadzhyan between the accessory parameters of Schwarzian differential equations and the Weil–Petersson metric on Teichmüller space. In addition it is shown that there is an analogous effect in the theory fo flat bundles. Bibliography: 23 titles.

Bibliographie

[1] Zograf P. G., Takhtadzhyan L. A., “Deistvie uravneniya Liuvillya – proizvodyaschaya funktsiya dlya aktsessornykh parametrov i potentsial metriki Veilya–Petersona na prostranstve Teikhmyullera”, Funktsion. analiz i ego pril., 19:3 (1985), 67–68 | MR | Zbl

[2] Zograf P. G., Takhtadzhyan L. A., Ob uravnenii Liuvillya, aktsessornykh parametrakh i geometrii prostranstva Teikhmyullera I. Rimanovy poverkhnosti roda 0., Preprint LOMI R-11-85, Leningrad, 1985 | MR | Zbl

[3] Zograf P. G., Takhtadzhyan L. A., Ob uniformizatsii rimanovykh poverkhnostei i metrike Veilya–Petersona na prostranstvakh Teikhmyullera i Shottki, Preprint LOMI R-12-85, Leningrad, 1985 | MR | Zbl

[4] Zograf P. G., Takhtadzhyan L. A., “O svyaznosti Narasimkhana–Seshadri i kelerovoi strukture prostranstva modulei golomorfnykh vektornykh rassloenii nad rimanovymi poverkhnostyami”, Funktsion. analiz i ego pril., 20:3 (1986), 84–85 | MR | Zbl

[5] Kvillen D., “Determinanty operatorov Koshi–Rimana na rimanovykh poverkhnostyakh”, Funktsion. analiz i ego pril., 19:1 (1985), 37–41 | MR

[6] Tyurin A. N., “O periodakh kvadratichnykh differentsialov”, UMN, 33:6 (1978), 149–195 | MR | Zbl

[7] Ahlfors L. V., Lectures on quasiconformal mappings, Van Nostrand, N.-Y., 1966 | MR | Zbl

[8] Atiyah M. F., “The signature of fibre-bundles Global Analysis”, Papers in Honour of K. Kodaira, eds. P. C. Spencer, S. Iyanaga, Princeton Univ. Press, Princeton, 1969, 73–84 | MR

[9] Atiyah M. F., Bott R., “The Yang-Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London A, 308 (1983), 523–615 | DOI | MR | Zbl

[10] Bers L., “On boundaries of Teichmüller spaces and on Kleinian groups: I”, Ann. Math., 91:3 (1970), 570–600 | DOI | MR | Zbl

[11] Bers L., Ehrenpreis L., “Holomorphic convexity of Teichmüller spaces”, Bull. Amer. Math. Soc., 70 (1964), 761–764 | DOI | MR | Zbl

[12] Goldman W. M., “The symplectic nature of fundamental groups of surfaces”, Adv. Math., 54:2 (1984), 200–225 | DOI | MR | Zbl

[13] Goto M., Grosshans F. D., Semisimple Lie algebras, Marcel Dekker, Inc., New York and Basel, 1978 | MR | Zbl

[14] Harer J., “The second homology group of the mapping class group of an orientable surface”, Invent. Math., 72:2 (1983), 221–239 | DOI | MR | Zbl

[15] Harris J., “Recent work on Mg”, Proc. of the International Congress of Mathematicians (August 16–24 1983. Warszawa), 1, North Holland, Amsterdam–New York–Oxford, 1984, 719–726 | MR

[16] Hejhal D., “Monodromy groups and linearly polymorhic functions”, Acta Math., 135:1–2 (1975), 1–55 | DOI | MR | Zbl

[17] Hirzebruch F., “The signature of ramified coverings”, Global Analysis. Papers in Honour of K. Kodaira, eds. D. C. Spencer, S. Iyanaga, Princeton Univ. Press, Princeton, 1969, 253–265 | MR

[18] Hubbard J. H., “The monodromy of projective structures”, Ann. of Math. Studies, 1981, no. 97, 257–275 | MR | Zbl

[19] Kodaira K., “A certain type of irregular algebraic surfaces”, J. Anal. Math., 19 (1967), 207–215 | DOI | MR | Zbl

[20] Narasimhan M. S., Seshadri C. S., “Holomorphic vector bundles on compact Riemann surface”, Math. Ann., 155:1 (1964), 69–80 | DOI | MR | Zbl

[21] Wolpert S., “On the homology of the moduli space of stable curves”, Ann. Math., 118:3 (1983), 491–523 | DOI | MR | Zbl

[22] Wolpert S., “The topology and geometry of the moduli space of riemann surfaces”, Lect. Notes in Math., no. 1111, 1985, 431–451 | MR | Zbl

[23] Wolpert S., Chern forms and the riemann tensor for the moduli space of curves, Preprint. Univ. of Maryland, 1984 | MR