Vortex type equations and canonical metrics

Details Vortex type equations and canonical metrics Vortex type equations and canonical metrics

2006-01-19

arxiv.org/abs/math/0601485v2

Mathematics

Differential Geometry

53 pages. To appear in Math. Annalen. Last section has been rewritten. Comments welcome !

Scientific paper

We 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 $\tau$-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.

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