2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107625We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle $F$ over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on $F$ coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson. We prove that if there is a $τ$-Hermite-Einstein metric $h_{HE}$ on $F$, then there exists a sequence of such balanced metrics that converges and its limit is $h_{HE}$. As a corollary, we obtain an approximation theorem for coupled Vortex equations that cover in particular the cases of Hermite-Einstein equations, Garcia-Prada and Bradlows's coupled Vortex equations and special Vafa-Witten equations.53 pages. To appear in Math. Annalen. Last section has been rewritten. Comments welcome !Differential GeometryAlgebraic GeometryComplex Variables14L24;53D20;14D21;53C07;32A25text