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Canonical metrics on stable vector bundles. (English) Zbl 1093.32008

Let \(X\) be a projective manifold polarized by an ample line bundle \(O_X(1)\) and \(E\) a rank \(r\) irreducible vector bundle on \(X\). In [X. Wang, Math. Res. Letter No. 2–3, 393–411 (2002; Zbl 1011.32016)] the author proved that \(E\) is Gieseker stable if and only if for all \(k \gg 0\) the associated global sections embedding \(X\) in a Grassmannian \(G(r,N)\) can be moved to a balance plane, i.e. up to an element of SL\((N)\) it satisfies the balance equation.
Here the author shows that when \(k\) goes to \(+\infty\) the metric obtained on \(E(k)\) goes to a metric solving a weakly Hermitian-Einstein equation. This result solves the second question raised in [S. K. Donaldson, Asian J. Math. 3, No. 1, xliii–xlvii (1999; Zbl 0957.01030)]. The result shows how Atiyah-Bott’s infinite dimensional symplectic quotient is approximated by Mumford’s GIT quotient.

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)
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