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Balance point and stability of vector bundles over a projective manifold. (English) Zbl 1011.32016
Here, answering a question of Donaldson, the author gives a differential-geometric interpretation (and also a symplectic quotient one) of the Gieseker stability of a holomorphic rank $$r$$ vector bundle $$E$$ on a polarized projective manifold $$(X,{\mathbf O}_X(1))$$. He shows that $$E$$ is Gieseker stable if and only if for all integers $$k\gg 0$$ the global sections of $$E(k)$$ map $$X$$ to a point of the Grassmannian $$\text{Grass} (r,N)$$, $$N=h^0 (X,E(k))$$, which may be moved to a balanced point.

##### MSC:
 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 32Q20 Kähler-Einstein manifolds 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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