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.     

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.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

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.     

Symplectic structures on moduli spaces of framed sheaves on surfaces

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.     

Logarithmic moduli spaces for surfaces of class VII

In this paper we describe logarithmic moduli spaces of pairs (S, D) consisting of a minimal surface S of class VII with second Betti number b ₂ > 0 together with a reduced maximal divisor D of b ₂ rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of S\D and for the associated holomorphic contraction . Part of this work was done while the first author visited the University of Osnabrück under the program “Globale Methoden in der komplexen Geometrie” of the DFG and while the second author visited the Max-Planck-Institut für Mathematik in Bonn and the LATP, Université de Provence. We thank these institutions for their hospitality and for financial support. Furthermore the authors wish to thank Georges Dloussky for numerous discussions on surfaces of class VII.     

Geometric compactification of moduli spaces of half-translation structures on surfaces

In this paper, we give an equivariant compactification of the space \({\mathbb {P}}{\text {Flat}}(\Sigma )\) of homothety classes of half-translation structures on a compact, connected, orientable surface \(\Sigma \). We introduce the space \({\mathbb {P}}{\text {Mix}}(\Sigma )\) of homothety classes of mixed structures on \(\Sigma \), that are \({\text {CAT}}(0)\) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are \({\mathbb {R}}\)-trees and completions of surfaces endowed with half-translation structures. Endowing \({\text {Mix}}(\Sigma )\) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that \({\mathbb {P}}{\text {Mix}}(\Sigma )\) is an equivariant compactification of \({\mathbb {P}}{\text {Flat}}(\Sigma )\), thus allowing us to understand in a geometric way the degenerations of half-translation structures on \(\Sigma \). We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on \(\Sigma \), by the mean of the translation distances of the elements of the covering group of \(\Sigma \).     

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     

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 irreducible components of moduli spaces of vector bundles on surfaces

Oblatum 14-III-1992 & 16-XI-1992     

Bubble tree compactification of moduli spaces of vector bundles on surfaces

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c ₁ = 0, c ₁ = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.     

Smoothness and Poisson structures of Bridgeland moduli spaces on Poisson surfaces

Let X be a projective smooth holomorphic Poisson surface, in other words, whose anti-canonical bundle is effective. We show that moduli spaces of certain Bridgeland stable objects on X are smooth. Moreover, we construct Poisson structures on these moduli spaces.

     

On moduli spaces of sheaves on K3 or abelian surfaces

We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.     

A note on singular moduli spaces of sheaves on K3 surfaces

This paper studies deformations and birational maps between singular moduli spaces of torsion free semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that when the greatest common divisor of the rank and the first Chern class is 2, two such moduli spaces of the same dimension can be connected by deformations and birational maps.     

Witten's formulas for intersection pairings on moduli spaces of flat G-bundles

In a 1992 paper (J. Geom. Phys. 9 (1992) 303), Witten gave a formula for the intersection pairings of the moduli space of flat G-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.     

On rationality of moduli spaces of vector bundles on real Hirzebruch surfaces

Let X be a real form of a Hirzebruch surface. Let M _H (r,c ₁, c ₂) be the moduli space of vector bundles on X. Under some numerical conditions on r, c ₁ and c ₂, we identify those M _H (r,c ₁,c ₂) that are rational.