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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

MSC:
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
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