Recent Preprints

• Moduli of Riemann Surfaces, Transcendental Aspects. This is an informal set of lecture notes of a course given at the ICTP (Trieste) in July-August, 1999. Since the file has lots of figures, I have only included a postscript file: PS file

• Regulators and Characteristic Classes of Flat Bundles (with Johan Dupont and Steven Zucker) (35 pages, January, 1998) This is a substantially revised version of an old manuscript we wrote quite a few years ago. The paper concerns the relation between the universal Cheeger-Simons Chern classes for vlat bundles, the universal Beilinson class for flat bundles, and the Borel regulator element. We prove for example, that the first and last are the same (with real coefficients). DVI file * PS file * PDF file (readable with Acrobat)

• Geometric Proofs of Some Results of Morita (with David Reed) (15 pages, October 1998). In this paper we reformulate and give new proofs of three results of S. Morita which relate various 2 dimensional cohomology classes of the standard moduli space of curves and the universal curve over it. The proofs we give are more geometric and should help clarify the geometric significance of Morita's results. We will also use them in our forthcoming paper on the Arakelov geometry of moduli spaces of curves. Download from the mathematics preprint server.

• Locally symmetric famiiies of curves and jacobians (11 pages, March 1998.) In this paper, I consider a question of Frans Oort: Are there any positive dimensional locally symmetric subvarieties of A_g, the moduli space of principally polarized abelian varieties that are contained in the locus of jacobians and contain the jacobian of at least one smooth curve? DVI file * PS file * PDF file (readable with Acrobat)

• Mapping Class Groups and Moduli Spaces of Curves (with Eduard Looijenga) (46 pages: final + version, May, 1997) Appeared in Algebraic Geometry, Santa Cruz, 1995: Proc. Symp. Pure Math 62.2 (1997), 97-142. This is a survey of recent developments in the study of Moduli spaces of curves and mapping class groups. The survey is from the point of view of algebraic geometry. It contains a few new results as well. DVI file * PS file

• Infinitesimal Presentations of the Torelli Groups (55 pages: final version, January 1997. This has appeared in J. Amer. Math. Soc. 10 (1997), pp. 597-651.) In this paper, I give a presentation of the Malcev Lie algebra of each Torelli group in genus 6 or more. The presentation is quadratic as it is for the pure braid group. I also prove that there is a "universal," symplectically invariant, projectively flat connection over Torelli space. It is analogous to the flat connection over the classifying space of the pure braid group that plays a role in the theory of Vassiliev invariants. The methods used are Hodge theory combined with the representation theory of symplectic groups. DVI file * PS file The figures may be obtained by clicking on figure 1 and figure 2. (File in pdf format (Acrobat), with pictures, available from AMS. Click on link to Journal above.)

• The Hode de Rham Theory of Relative Malcev Completion (38 pages: final version, May 1997) Appeared in Ann. Scient. Ecole Norm. Sup. t. 31 (1998), 47-92. In this paper I prove that the coordinate ring of the completion of the fundamental group of a smooth manifold with respect to a Zariski dense reductive representation can be computed using differetial forms. (This result and the corresponding constructions are based on notes of Deligne.) In the case where the representation is trivial, one recovers Chen's de Rham theorem for the unipotent completion of the fundametal group. This result is used to prove that if X is the complement of a normal crossings divisor in a compact Kahler manifold, or is a smooth complex algebraic variety, then the coordinate ring of the fundamental group of (X,x) with respect to the monodromy representation of an admissible variation of Hodge structure over X with Zariski dense monodromy has a canonical MHS. In the case where the variation is trivial, one recovers the result of Morgan that the Malcev completion of the fundamental group of a smooth variety (X,x) has a canonical mixed Hodge structure. This paper is a major technical ingredient in the paper Infinitesimal Presentations of the Torelli Groups. DVI file * PS file

• Betti Number Estimates for Nilpotent Groups (with Michael Freedman and Peter Teichner)(20 pages: final version 10/96) (This has appeared in the book Fields Medalists' Lectures, edited by Sir Michael Atiyah & Daniel Iagolnitzer, World Scientific Series in 20th Century Mathematics - Vol. 5, 1997.) In this paper we prove a generalization of the Golod-Shaferevich Theorem for nilpotent Lie algebras. This theorem gives a lower bound on the second betti number of the Lie algebra in terms of its first betti number and the position of the relations in a minimal prersentation of the Lie algebra in the lower central series of a free Lie algebra. As one application, we prove that the Malcev Lie algebra associated to a compact orientable 3-manifold is either trivial, abelian of dimension 1 or 3, the three dimension Heisenberg Lie algebra, or is infinite dimensional. DVI file * PS file

• Torelli Groups and Geometry of Moduli Spaces of Curves (This has appeared in Current Topics in Complex Algebraic Geometry (C.~H.~Clemens and J.~Kollar, eds.) MSRI publications no.~28, Cambridge University Press, 1995.) In this paper I survey some of Dennis Johnson's work on the Torelli groups and apply it to prove various results about algebraic curves. For example, I prove that when the genus is 3 or more, the Picard group of the moduli space of genus g curves and a level m structure (m arbitrary) is finitely generated. Combining Johnson's results with Saito's theory of Hodge modules, I classify all "natural" normal functions over the moduli space of genus g curves with a level m structure (g at least 3) --- they are all half integer multiples of the normal function of "C - C(-)." This is used to compute the Picard group of the generic curve of genus g with a level m structure. (A generalization of the Francetta conjecture.) DVI file * PS file