an:04019950
Zbl 0627.53052
Donaldson, S. K.
Infinite determinants, stable bundles and curvature
EN
Duke Math. J. 54, 231-247 (1987).
00153597
1987
j
53C55
stable bundle; K??hler manifold; Hermitian Yang-Mills metric; bundles over projective manifolds
For a compact K??hler manifold (X,\(\omega)\) of complex dimension n and a holomorphic r-plane bundle E over X, E is [\(\omega\) ]-stable if every subsheaf \(S\subset \theta (E)\) with torsion-free quotient S/\(\theta\) (E) satisfies the condition \(c_ 1(s)\cdot [\omega]^{n-1}<0\). A conjecture of Hitchin and Kobayashi is that if E is [\(\omega\) ]-stable, then there exists a Hermitian Yang-Mills metric on E. The uniqueness and converse were established by the author in his paper [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)], which also proved the conjecture for bundles over complex algebraic surfaces. The conjecture was proved in full generality by Uhlenbeck and Yau. On the other hand, in this paper an alternative proof is given for bundles over projective manifolds \(X\subset {\mathbb{C}}{\mathbb{P}}^ N\) with a Hodge metric \(\omega\).
A.Stone
Zbl 0547.53019; Zbl 0529.53018