Degenerations of the moduli spaces of vector bundles on curves I

LetY be a smooth projective curve degenerating to a reducible curveX with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on Ydegenerates to a moduli space onX which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles onY with fixed determinant and arbitrary rank.     

Kähler geometry of Douady spaces

We consider the generalized Petersson–Weil metric on the moduli space of compact submanifolds of a Kähler manifold or a projective variety. It is extended as a positive current to the space of points corresponding to reduced fibers, and estimates are shown. For moduli of projective varieties the Petersson–Weil form is the curvature of a certain determinant line bundle equipped with a Quillen metric. We investigate its extension to the compactified moduli space.     

Moduli spaces of convex projective structures on surfaces

We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for SL₃(R) defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149]. We investigate the cluster structure of this moduli space, and define its quantum version.     

Instanton approximation, periodic ASD connections, and mean dimension

We study a moduli space of ASD connections over S³×R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the mean dimension by using a “Runge-approximation” for ASD connections, and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic ASD connections.     

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the p-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any p-rank stratum. We discuss irreducibility and connectedness of some p-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.     

Espace des modules des variétés hyperboliquement plongéesThe moduli space of hyperbolically imbedded manifolds

In this Note, we construct the moduli space of hyperbolically imbedded manifolds. We recall that the moduli space of compact hyperbolic manifolds has been constructed by Brody and Wright. To construct our moduli space, we use a general criterion to represent analytic functors by coarse moduli spaces due to Schumacher. The objects to deform are couples (X,D) where X is a compact manifold and D is a normal crossing divisor in X such that X?D is hyperbolically imbedded in X. This criterion is based on two ingredients: in our case, the first is the existence of semi-universal logarithmic deformation due to Kawamata. The second is a consequence of a theorem of stability of hyperbolically imbedded spaces through logarithmic deformations. We use the relative-distance of Kobayashi to simplify the proof. To cite this article: A. Khalfallah, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 237–242.     

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.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

Real cubic surfaces and real hyperbolic geometry

The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π²/1080 in the metric of constant curvature ?1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components. To cite this article: D. Allcock et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Energy functions on moduli spaces of flat surfaces with erasing forest

This paper follows on from Nguyen (Geom Funct Anal 20(1):192–228, 2010), in which we study flat surfaces with erasing forest, these surfaces are obtained by deforming the metric structure of translation surfaces, and their moduli space can be viewed as a deformation of the moduli space of translation surfaces. We showed that the moduli spaces of such surfaces are complex orbifolds, and admit a natural volume form μ _Tr. The aim of this paper is to show that the volume of those moduli spaces with respect to μ _Tr, normalized by some energy function involving the area and the total length of the erasing forest, is finite. Note that translation surfaces and flat surfaces of genus zero can be viewed as special cases of flat surfaces with erasing forest, and on their moduli space, the volume form μ _Tr equals the usual ones up to a multiplicative constant. Using this result we obtain new proofs for some classical results due to Masur-Veech, and Thurston concerning the finiteness of the volume of the moduli space of translation sufaces, and of the moduli space of polyhedral flat surfaces.     

Weil-Petersson Areas of The Moduli Spaces Of Tori

We study the Teichmüller spaces of torus with one branch point of order v and of torus with a totally geodesic boundary curve of length m, respectively. Applying the obtained results for the corresponding moduli spaces we find that the Weil-Petersson area of the moduli space of torus with one conical point of order v is (π²/6)(1 - l/v²) and that of the moduli space of torus with a totally geodesic boundary curve of length m is π²/6 + m²/24.     

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g?3 and let M₀ be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M₀ have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan-Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.     

Contractibility of the space of rational maps

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space $\operatorname {Bun}_{G}$ of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.     

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on \mathbb{C}{{\mathbb{P}}^1}

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ??¹ such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by $ \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) $ . We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Navigation in tree spaces

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes, endowing the resolving space with an interesting canonical pseudometric. Indeed, the given map can be reinterpreted as relating the real and the tropical versions of the Deligne–Knudsen–Mumford compactification of the moduli space of Riemann spheres.     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,I

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Transversality for equivariant exact 1-forms and gauge theory on 3-manifolds

An equivariant jet transversality framework is developed for the study of critical sets of invariant functions on G manifolds. Techniques are developed to extend transversality results to the infinite dimensional Fredholm setting. As an application, the generic structure of the SU(4) perturbed flat moduli space of an integral homology three-sphere is described, as well as the generic structure of the parameterized moduli space for a path of perturbations. A similar analysis of the U(3) moduli space for rational homology three-spheres is also carried out.     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,II

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).