Divisor spaces on punctured Riemann surfaces

In this paper, we study the topology of spaces of -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

     

An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces

An explicit upper bound for the Weil-Petersson volumes of punctured Riemann surfaces is obtained using Penner's combinatorial integration scheme from [4]. It is shown that for a fixed number of punctures n and for genus g increasing, while this limit is exactly equal to two for n=1. Received: 17 May 2000 / Revised version: 9 August 2000 / Published online: 23 July 2001     

Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

This paper studies intersection theory on the compactified moduli space of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.     

Triangulations and moduli spaces of Riemann surfaces with group actions

Summary We study that subset of the moduli space of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ₂ actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ₂ actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically. This work has been supported by the European Communities Science Plan Project No SCI^*-CT91 (TSTS) “Computational Methods in the Theory of Riemann Surfaces and Algebraic Curves,” by Academy of Finland and by the Swiss National Science Foundation Grant 20-34099.92. We thank M. C. Petrus for providing excellent motivation for this work. This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Canonical measures on the moduli spaces of compact Riemann surfaces

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.     

Poisson structures on moduli spaces of sheaves over Poisson surfaces

Summary We introduce and study the notion of Poisson surface. We prove that the choice of a Poisson structure on a surfaceS canonically determines a Poisson structure on the moduli space of stable sheaves onS. This result generalizes previous results obtained by Mukai [14], for abelian orK3 surfaces, and by Tyurin [16].Oblatum 13-VI-1994 & 22-III-1995This article was processed by the author using thepjourlm style file from Springer-Verlag     

Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

This work was supported by grants from the Sloan Foundation, the National Science Foundation and the CNR     

Rationality of moduli spaces of torsion free sheaves over rational surfaces

In this paper we show that for rational ruled surfaces many moduli spaces of torsion free sheaves with given Chern classes are rational. We deal with the case that the first Chern classc ₁ satisfiesc ₁.F=0 for a fibreF of the ruling. The main tool are priority sheaves introduced by Hirschowitz-Laszlo and Walter, which enable us to reduce the problem to the construction of a family of sheaves over a big enough rational base.     

Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals

The known examples of explicit equations for Riemann surfaces whose field of moduli is different from their field of definition, are all hyperelliptic. In this paper we construct a family of equations for non-hyperelliptic Riemann surfaces, each of them is isomorphic to its conjugate Riemann surface, but none of them admit an anticonformal automorphism of order 2; that is, each of them has its field of moduli, but not a field of definition, contained in \mathbb R{{\mathbb R}} . These appear to be the first explicit such examples in the non-hyperelliptic case.     

Duistermaat—Heckman measures and moduli spaces of flat bundles over surfaces

We introduce Liouville measures and Duistermaat—Heckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces.     

The moduli spaces of rank 2 stable vector bundles over Veronesean surfaces

Partially supported by DGICYT PB88-0224     

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.