Related papers: Riemann-Roch for real varieties

Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose…

Let $\pi: X \longrightarrow C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback $\pi^*E$ is a $H-$stable…

Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the…

We extend the concept of Segre's Invariant to vector bundles on a surface $X$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern's classes. The irreducibility of…

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete…

Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $\xi$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $\chi(E\otimes F)= 0$ for $E \in \mathcal M$.…

We give a classification theorem for a class of C*-algebras which are direct limits of extensions of circle algebras by purely infinite C*-algebras. The invariant consists of the following: (1) the set of Murray-von Neumann equivalence…

A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M, whose fibers are the spaces of symmetric, positive-definite bilinear forms on the tangent spaces of M, which represent the Rieman?nian metrics. In this work, we aim…

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $\alpha : E \rightarrow F$ which, for smooth…

In this paper, we prove the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level.

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian…

In this paper we consider the category $C (\tilde k, \tilde H)$ of the $(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$ are the…

We introduce an elliptic version of the Grothendieck-Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the…

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras $\mathcal{B}(p;q, 0)$ (over a ring of polynomials in one variable) are…

Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag variety over a compact complex manifold $B$. We shall obtain a description of the cohomology of $E$ when $B=X_\Gamma:=\Gamma\backslash X, E=Y_\Gamma:=\Gamma\backslash Y$…

This paper proves an integral version of the Riemann-Roch theorem for surface bundles, comparing the standard cohomology classes with the cohomology classes coming from the symplectic group.

We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $\mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or…

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal…

This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian…

Let $X$ be a non-singular quasi-projective variety over a field, and let $\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$ subbundles of…