1,295 research outputs found

### Monodromy group for a strongly semistable principal bundle over a curve, II

Let $X$ be a geometrically irreducible smooth projective curve defined over a
field $k$. Assume that $X$ has a $k$-rational point; fix a $k$-rational point
$x\in X$. From these data we construct an affine group scheme ${\mathcal G}_X$
defined over the field $k$ as well as a principal ${\mathcal G}_X$-bundle
$E_{{\mathcal G}_X}$ over the curve $X$. The group scheme ${\mathcal G}_X$ is
given by a ${\mathbb Q}$--graded neutral Tannakian category built out of all
strongly semistable vector bundles over $X$. The principal bundle $E_{{\mathcal
G}_X}$ is tautological. Let $G$ be a linear algebraic group, defined over $k$,
that does not admit any nontrivial character which is trivial on the connected
component, containing the identity element, of the reduced center of $G$. Let
$E_G$ be a strongly semistable principal $G$-bundle over $X$. We associate to
$E_G$ a group scheme $M$ defined over $k$, which we call the monodromy group
scheme of $E_G$, and a principal $M$-bundle $E_M$ over $X$, which we call the
monodromy bundle of $E_G$. The group scheme $M$ is canonically a quotient of
${\mathcal G}_X$, and $E_M$ is the extension of structure group of
$E_{{\mathcal G}_X}$. The group scheme $M$ is also canonically embedded in the
fiber ${\rm Ad}(E_G)_{x}$ over $x$ of the adjoint bundle.Comment: This final version includes strengthening of the result by referee's
comments. K-Theory (to appear

### An analogue of the Narasimhan-Seshadri theorem and some applications

We prove an analogue in higher dimensions of the classical
Narasimhan-Seshadri theorem for strongly stable vector bundles of degree 0 on a
smooth projective variety $X$ with a fixed ample line bundle $\Theta$. As
applications, over fields of characteristic zero, we give a new proof of the
main theorem in a recent paper of Balaji and Koll\'ar and derive an effective
version of this theorem; over uncountable fields of positive characteristics,
if $G$ is a simple and simply connected algebraic group and the characteristic
of the field is bigger than the Coxeter index of $G$, we prove the existence of
strongly stable principal $G$ bundles on smooth projective surfaces whose
holonomy group is the whole of $G$.Comment: 42 pages. Theorem 3 of this version is new. Typos have been
corrected. To appear in Journal of Topolog

### Tensor product theorem for Hitchin pairs -An algebraic approach

We give an algebraic approach to the study of Hitchin pairs and prove the
tensor product theorem for Higgs semistable Hitchin pairs over smooth
projective curves defined over algebraically closed fields $k$ of
characteristic $0$ and characteristic $p$, with $p$ satisfying some natural
bounds. We also prove the corresponding theorem for polystable bundles.Comment: To appear in Annales de l'Institut Fourier, Volume 61 (2011

### A splitting theorem for good complexifications

The purpose of this paper is to produce restrictions on fundamental groups of
manifolds admitting good complexifications by proving the following
Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a
good complexification has a finite-sheeted regular covering $M_1$ such that
$M_1$ admits a fiber bundle structure with base $(S^1)^k$ and fiber $N$ that
admits a good complexification and also has zero virtual first Betti number. We
give several applications to manifolds of dimension at most 5.Comment: 13 pgs no fig

### Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulae for the local Euler
obstruction. In particular, a result of this type is proved in [BLS].The
formula proved in that paper turns out to be equivalent to saying that the
local Euler obstruction, as a constructible function, satisfies the local Euler
condition (in bivariant theory) with respect to general linear forms.
The purpose of this work is to understand what prevents the local Euler
obstruction of satisfying the local Euler condition with respect to functions
which are singular at the considered point. This is measured by an invariant
(or ``defect'') of such functions that we define below. We give an
interpretation of this defect in terms of vanishing cycles, which allows us to
calculate it algebraically. When the function has an isolated singularity, our
invariant can be defined geometrically, via obstruction theory. We notice that
this invariant unifies the usual concepts of {\it the Milnor number} of a
function and of the {\it local Euler obstruction} of an analytic set.Comment: 18 page

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