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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
Full Text: DOI
[1] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces , Phil. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 524-615. JSTOR: · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 · links.jstor.org
[2] J.-M. Bismut and D. S. Freed, The analysis of elliptic families, I, II , to appear in Commun. Math. Phys.
[3] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles , Proc. London Math. Soc. (3) 50 (1985), no. 1, 1-26. · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1
[4] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[5] H. Gillet and C. Soulé, Classes caractéristiques en théorie d’Arakelov , C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 9, 439-442. · Zbl 0613.14007
[6] Yu. I. Manin, New dimensions in geometry , Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., vol. IIII, Springer, Berlin, 1985, pp. 59-101. · Zbl 0579.14002
[7] V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group , Invent. Math. 77 (1984), no. 1, 163-172. · Zbl 0525.55012 · doi:10.1007/BF01389140 · eudml:143143
[8] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface , Ann. of Math. (2) 82 (1965), 540-567. JSTOR: · Zbl 0171.04803 · doi:10.2307/1970710 · links.jstor.org
[9] D. B. Ray and I. M. Singer, Analytic Torsion for complex manifolds , Ann. of Math. (2) 98 (1973), 154-177. JSTOR: · Zbl 0267.32014 · doi:10.2307/1970909 · links.jstor.org
[10] K. K. Uhlenbeck and S. T. Yau, The existence of Hermitian Yang-Mills connections on stable bundles over Kahler manifolds , · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
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