Related papers: Secondary characteristic classes and the Euler cla…

The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the…

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction…

We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen-Ruan orbifold cohomology…

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \chi of P. We prove that a holomorphic principal G-bundle E over a connected complex…

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…

Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically…

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's…

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of…

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral…

We study the positivity of the first Chern class of a rank r Ulrich vector bundle E on a smooth n-dimensional variety $X \subseteq \mathbb P^N$. We prove that $c_1(E)$ is very positive on every subvariety not contained in the union of lines…

Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms c_{i}(E;\nabla) in \Gamma(X, \cal{A}^{2i}_{X}). Here \cal{A}^{.}_{X} is the sheaf Beilinson adeles and \nabla is an adelic…

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum…

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic…

We study characteristic classes on hyperk\"ahler manifolds with a view towards the Verbitsky component. The case of the second Chern class leads to a conditional upper bound on the second Betti number in terms of the Riemann--Roch…

We study the rank stratification for the differential of a Lagrangian fibration over a smooth basis. We also introduce and study the notion of Lagrangian morphism of vector bundles. As a consequence, we prove some of the vanishing, in the…

Let $M$ be a smooth algebraic variety of dimension $2(p+q)$ with an algebraic symplectic form and a compatible deformation quantization $\mathcal{O}_h$ of the structure sheaf. Consider a smooth coisotropic subvariety $j: Y \to M$ of…

This paper examines a number of related questions about Euler characteristics and characteristic classes with values in Witt cohomology. We establish a motivic version of the Becker-Gottllieb transfer, generalizing a construction of Hoyois.…

We construct Euler and Stiefel-Whitney classes of vector bundles with quadratic form by analyzing the intersection theory of the associated quadric bundles. We also compute the Chow rings of quadric and isotropic flag bundles. Along the…

We classify nef vector bundles on a smooth hyperquadric of dimension $\geq 4$ with first Chern class two over an algebraically closed field of characteristic zero.

Let $\sE$ be an ample rank $r$ bundle on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. In this article, we prove a number of surprisingly strong lower bounds for $c_1(\sE)^2$ and $c_2(\sE)$. We…