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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.

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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