Infinite determinants, stable bundles and curvature. (English) Zbl 0627.53052

For a compact Kähler 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 \(\omega\).
Reviewer: A.Stone


53C55 Global differential geometry of Hermitian and Kählerian manifolds
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