# Equations de type Vortex et métriques canoniques

Abstract : Let $M$ be a smooth projective manifold. Let $\mathcal{F}$ be a filtered holomorphic vector bundle over $M$. We introduce a notion of Gieseker stability for such objects and relate it to an analytic condition in terms of hermitian metrics on $\mathcal{F}$, called balanced metrics by S.K Donaldson, that come from the world of Geometric Invariant Theory (G.I.T). If there is a metric $h$ on $\mathcal{F}$ that satisfies the $\boldsymbol{\tau}$-Hermite-Einstein equation studied by \'{A}lvarez-C\'{o}nsul and Garc\'{i}a-Prada:
$\sqrt{-1}\Lambda F_h = \sum_i \widetilde{\tau}_i\pi^{\mathscr{F}}_{h,i}$
then we prove that the sequence of balanced metrics exists, converges and its limit, up to a conformal change, is a smooth hermitian metric on $\mathcal{F}$ that satisfies the previous equation. As a corollary, we give by dimensional reduction a theorem of approximation for Vortex equations introduced by Bradlow and their generalizations to coupled Vortex equations.
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Theses
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https://tel.archives-ouvertes.fr/tel-00012107
Contributor : Julien Keller <>
Submitted on : Monday, April 10, 2006 - 5:10:00 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Saturday, April 3, 2010 - 9:01:43 PM

### Identifiers

• HAL Id : tel-00012107, version 1

### Citation

Julien Keller. Equations de type Vortex et métriques canoniques. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2005. Français. ⟨tel-00012107⟩

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