Related papers: Hodge theory of holomorphic vector bundle on compa…

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral…

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=\P_C (\mathcal{E})$ with projection map $\pi:X\to C$. In this paper we investigate the minimal free resolution of homogeneous coordinate…

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group…

Let $p:\sXS$ be a proper K\"ahler fibration and $\sE\sX$ a Hermitian holomorphic vector bundle. As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basic Hodge theory, we derive several general curvature formulas for the…

Call a normal complex projective variety $X$ Koll\'ar-hyperbolic if any nonconstant map from a smooth projective curve to $X$ induces a nontrivial homomorphism of \'etale fundamental groups. Examples include (a) smooth varieties with finite…

One describes, using a detailed analysis of Atiyah--Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This…

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on…

In previous papers we introduced the notion of special Bohr - Sommerfeld lagrangian cycles on a compact simply connected symplectic manifold with integer symplectic form, and presented the main interesting case: compact simply connected…

The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact K\"ahler manifold $(X, \omega )$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space…

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…

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We show that if X is a smooth affine scheme of dimension d over a field k of finite 2-cohomological dimension (with char(k)…

For a very ample line bundle L on a compact connected complex manifold X, with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L.

For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$…

In this paper, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, i.e., lie…

Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle $\mathcal{P}$ over…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress towards…

In this paper, we investigate the existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles over rational homogeneous varieties. Using Cartan's highest weight theory, we establish an explicit algebraic…

Let $X$ be a compact K\"ahler manifold. The set $\cha(X)$ of one-dimensional complex valued characters of the fundamental group of $X$ forms an algebraic group. Consider the subset of $\cha(X)$ consisting of those characters for which the…