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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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##### References:
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