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.     

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.     

The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces

In the thick part of the moduli space of Riemann surfaces, we show that the sectional curvature of the Weil-Petersson metric is bounded independently of the genus of the surface.

     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

A compactification of the moduli space of twisted holomorphic maps

Given compact symplectic manifold X with a compatible almost complex structure and a Hamiltonian action of S¹ with moment map , and a real number K?0, we compactify the moduli space of twisted holomorphic maps to X with energy ?K. This moduli space parameterizes equivalence classes of tuples (C,P,A,?), where C is a smooth compact complex curve of fixed genus g, P is a principal S¹ bundle over C, A is a connection on P and ? is a section of PS¹_×X satisfying

     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

A finite group acting on the moduli space of K3 surfaces

We consider the natural action of a finite group on the moduli space of polarized K3 surfaces which induces a duality defined by Mukai for surfaces of this type. We show that the group permutes polarized Fourier-Mukai partners of polarized K3 surfaces and we study the divisors in the fixed loci of the elements of this finite group.

     

Spaces and moduli spaces of Riemannian metrics

These notes present and survey results about spaces and moduli spaces of complete Riemannian metrics with curvature bounds on open and closed manifolds, here focussing mainly on connectedness and disconnectedness properties. They also discuss several open problems and questions in the field.     

The Picard group of the moduli of G-bundles on a curve

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G₂ coarse moduli space and the moduli stack).     

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

We construct a family of polarizations of the moduli space of flat SU(n)-connections on a closed 2-manifold of genus g(≧2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman’s real polarizations in the case of n=2 [17], as well as the Kahler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli Spaces of parabolic bundles are also studied.     

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

We construct a family of polarizations of the moduli space of flatSU(n)-connections on a closed 2-manifold of genusg( 2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman's real polarizations in the case ofn=2 [17], as well as the Kähler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli spaces of parabolic bundles are also studied.Oblatum 9-I-1995 & 19-X-1995     

A remark on the field of moduli of Riemann surfaces

Archiv der Mathematik - Let S be a closed Riemann surface of genus $$gge 2$$ and let $$mathrm{Aut}(S)$$ be its group of conformal automorphisms. It is well known that if either: (i)...     

On moduli spaces for stable bundles on quadrics

We investigate the relation between stable representations of quivers and stable sheaves. A construction of thin smooth compact moduli spaces for stable sheaves on quadrics based on this relation is presented. Translated fromMatematicheskie Zametki, Vol 62, No. 6, pp. 843–864, December, 1997 Translated by S. K. Lando     

A moduli space of non-compact curves on a complex surface

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by finitely many holomorphic functions.     

Composition operators on Hardy spaces of Riemann surfaces

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.     

Automorphism groups of pseudo-real Riemann surfaces of low genus

A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution.We obtain the classifcation of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2,3 and 4.For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is eitherC4orC8or the Fro¨benius group of order 20,and in the case ofC4there are exactly two possible topological actions.Let MK P R,g be the set of surfaces in the moduli space MK g corresponding to pseudo-real Riemann surfaces.We obtain the equisymmetric stratifcation of MK P R,g for generag=2,3,4,and as a consequence we have that MK P R,gis connected forg=2,3 but MK P R,4has three connected components.