Canonical metrics on stable vector bundles (Q812180)

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 [\textit{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 [\textit{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.