Related papers: Riemann-Roch for real varieties

Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although…

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result…

We extend our previous results on generalized Dixmier-Douady theory to graded $C^*$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^*$-algebras. For a strongly self-absorbing $C^*$-algebra…

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset \Sigma of simple roots of G, and let E_\phi be a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation \phi of P.…

In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group $\mathrm{G}$, a precise condition under which an…

We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let $B$ be a complex manifold and $\pi_i : M_i \to B$ be a fibration in oriented circles,…

Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and…

Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short…

Given a family $(F,h) \to X \times S$ of Hermite-Einstein bundles on a compact K\"ahler manifold $(X,g)$ we consider the higher direct image sheaves $R^q p_* \mathcal{O}(F)$ on $S$, where $p: X \times S \to S$ is the projection. On the…

A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is…

We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by applying (a virtual version of) the…

We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the…

We show that any compact Kahler manifold with integral Kahler form, parametrizes a natural holomorphic family of Cauchy-Riemann operators on the Riemann sphere such that the Quillen determinant line bundle of this family is isomorphic to a…

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic…

This paper consists in a very brief English summary of the results appearing in French in two previous articles. We omit the proofs and focus on explaining our approach and theorems. This paper is not intended to be published. Questions are…

In the paper I introduce a new characteristic class $c(E)$ for a finite rank vector bundle $E$ on an affine scheme $S:=Spec(A)$ - the fundamental class of $E$. The class $c(E)$ is not a characteristic class in the classical sense in the…

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another…

We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on $H^{p,q}$ under the assumption that $H^{p-2,q-2}$ vanishes. More generally, we study Hodge-Riemann polynomials, which are…

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class…

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and \xi a holomorphic line bundle on it such that r is not a divisor of degree(\xi). Let {\mathcal M}_\xi(r) denote the moduli space of stable…