The Penrose Transform for Compactly Supported Cohomology

Let the manifold X parametrise a family of compact complex submanifoldsof the complex (or CR) manifold Z. Under mild conditions thePenrose transform typically provides isomorphisms between acohomology group of a holomorphic vector bundle V Z and thekernel of a differential operator between sections of vectorbundles over X. When the spaces in question are homogeneousfor a group G the Penrose transform provides an intertwiningoperator between representations. The paper develops a Penrose transform for compactly supportedcohomology on Z. It provides a number of examples where a compactlysupported cohomology group is shown to be isomorphic to thecokernel of a differential operator between compactly supportedsections of vector bundles over X. It considers also how theSerre duality pairing carries through the transform.     

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization $\tilde{J}^d(Y)$ of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in $\tilde{J}^d(Y)$ . We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa–Intriligator formulae in the nodal case.     

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).     

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

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).     

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III (S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983), 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries.Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically.The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.     

A modification of the Hodge star operator on manifolds with boundary

For smooth compact oriented Riemannian manifolds M of dimension 4k+2, k≥0, with or without boundary, and a vector bundle F on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of M twisted by F. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when k=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Produits dans la cohomologie des variétés de Shimura : quelques calculs

We show how our recent results (arXiv: math.NT/0403407 v1 24 Mar 2004) together with a criterion of Venkataramana can be used to get explicit conditions for the virtual non-vanishing of the product of two cohomology classes of certain Shimura varieties. To cite this article: N. Bergeron, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

The cohomology of the cotangent bundle of Heisenberg groups

Given a parabolic subalgebra g₁×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g₁ invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.     

Stochastic cohomology of Chen–Souriau and line bundle over the Brownian bridge

We define a stochastic cohomology theory related to a stochastic diffeology for the Hoelder loop space. We show that the stochastic de Rham cohomology groups are equal to the deterministic de Rham cohomology groups of the Hoelder loop space. As an application, we show that a stochastic line bundle over the Brownian bridge (with fiber almost surely defined) is isomorphic to a true line bundle over the Hoelder loop space. Received: 9 November 1998 / Revised version: 14 July 2000 / Published online: 26 April 2001     

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the “cone at infinity” but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge *{\star} operator twistors (i.e. vectors of the pseudo-Hermitian space H _2,2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4, 2)/Z ₂. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H _p,q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.     

Class numbers of definite quaternary forms with square discriminant

Let V be a definite quaternary space over Q having square discriminant. We derive an explicit formula for the number of proper classes of maximal integral lattices in V.     

Homotopy classification of graded Picard categories

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.     

Optimal critical mass in the two dimensional Keller–Segel model in R2

The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass there is global existence of classical solutions and for large initial mass blow-up occurs. In this Note we complete this picture and give an explicit value for the critical mass when the system is set in the whole space. To cite this article: J. Dolbeault, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds

LetX be a projective manifold of dimension n ≥ 2 andY →X an infinite covering space. EmbedX into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection ofX by a linear subspace of codimension <n extends toY. The proof uses a Hausdorff duality theorem for L₂ cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein.     

The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits

We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.     

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.