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The geometry of the moduli space of Riemann surfaces. (English) Zbl 0565.32011
Let \(\hat R\) be a marked Riemann surface of genus \(g\geq 2\) endowed with the hyperbolic metric. \(T_ g\) is the genus g Teichmüller space. \(M_ g\) is the classical moduli space of Riemann surfaces of genus g and \(\bar M_ g\) the moduli space of genus g stable curves (hyperbolic Riemann surfaces with nodes). This research announcement briefly sketches past and current work of the author in describing how the hyperbolic geometry of \(\hat R\) leads to a symplectic geometry on both \(T_ g\) and \(\bar M_ g\). For example, Weil introduced a Kähler metric for \(T_ g\) based on the Petersson product for automorphic forms. The Weil-Petersson Kähler form \(\omega\) extends to a symplectic form on \(\bar M_ g\). The author has shown that an integral multiple of \(\omega /\pi^ 2\) is the Chern form of a positive line bundle over \(\bar M_ g\). This associated positive line bundle gives rise to a projective embedding of \(\bar M_ g\) into complex projective space via Baily’s version of the Kodaira imbedding theorem. Consult this paper for a listing of the seven papers of the author in which this program has been advanced.
Reviewer: D.Minda

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
30F10 Compact Riemann surfaces and uniformization
30F30 Differentials on Riemann surfaces
Full Text: DOI
[1] Lars V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171 – 191. · Zbl 0146.30602 · doi:10.2307/1970309 · doi.org
[2] W. L. Baily, On the imbedding of \?-manifolds in projective space, Amer. J. Math. 79 (1957), 403 – 430. · Zbl 0173.22706 · doi:10.2307/2372689 · doi.org
[3] Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 43 – 55. Ann. of Math. Studies, No. 79. · Zbl 0045.42501
[4] John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221 – 239. · Zbl 0533.57003 · doi:10.1007/BF01389321 · doi.org
[5] Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623 – 635. · Zbl 0358.32017
[6] W. P. Thurston, The geometry and topology of 3-manifolds, notes.
[7] Scott Wolpert, The Fenchel-Nielsen deformation, Ann. of Math. (2) 115 (1982), no. 3, 501 – 528. · Zbl 0496.30039 · doi:10.2307/2007011 · doi.org
[8] Scott Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), no. 2, 207 – 234. · Zbl 0518.30040 · doi:10.2307/2007075 · doi.org
[9] Scott Wolpert, On the Kähler form of the moduli space of once punctured tori, Comment. Math. Helv. 58 (1983), no. 2, 246 – 256. · Zbl 0527.30031 · doi:10.1007/BF02564634 · doi.org
[10] Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969 – 997. · Zbl 0578.32039 · doi:10.2307/2374363 · doi.org
[11] Scott Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), no. 3, 491 – 523. · Zbl 0575.14024 · doi:10.2307/2006980 · doi.org
[12] Scott A. Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. Math. 107 (1985), no. 6, 1485 – 1507 (1986). · Zbl 0581.14022 · doi:10.2307/2374413 · doi.org
[13] Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119 – 145. · Zbl 0595.32031 · doi:10.1007/BF01388794 · doi.org
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