Limits of balanced metrics on vector bundles and polarised manifolds

Details Limits of balanced metrics on vector bundles and polarised manifolds Limits of balanced metrics on vector bundles and polarised manifolds

2011-11-11

arxiv.org/abs/1111.2819v1

Mathematics

Differential Geometry

31 pages

Scientific paper

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter \alpha, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of \alpha, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einstein metric on E and a constant scalar curvature K\"ahler metric in c_1(L). For special values of \alpha, limits of balanced metrics are solutions of a system of coupled equations relating a Hermitian-Einstein metric on E and a K\"ahler metric in c_1(L). For this, we compute the top two terms of the density of states expansion of the Bergman kernel of E \otimes L^k.

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