A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles

Weinkove, Ben

doi:10.1090/S0002-9947-06-03985-7

A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles
HTML articles powered by AMS MathViewer

by Ben Weinkove PDF

Trans. Amer. Math. Soc. 359 (2007), 1577-1592 Request permission

Abstract:

A complex Frobenius theorem is proved for subsheaves of a holomorphic vector bundle satisfying a finite generation condition and a differential inclusion relation. A notion of ‘multiplier ideal sheaf’ for a sequence of Hermitian metrics is defined. The complex Frobenius theorem is applied to the multiplier ideal sheaf of a sequence of metrics along Donaldson’s heat flow to give a construction of the destabilizing subsheaf appearing in the Donaldson-Uhlenbeck-Yau theorem, in the case of algebraic surfaces.

References

Enrico Bombieri, Algebraic values of meromorphic maps, Invent. Math. 10 (1970), 267–287. MR 306201, DOI 10.1007/BF01418775
Paolo De Bartolomeis and Gang Tian, Stability of complex vector bundles, J. Differential Geom. 43 (1996), no. 2, 231–275. MR 1424426
Mark Andrea A. de Cataldo, Singular Hermitian metrics on vector bundles, J. Reine Angew. Math. 502 (1998), 93–122. MR 1647555, DOI 10.1515/crll.1998.091
Jean-Pierre Demailly, Effective bounds for very ample line bundles, Invent. Math. 124 (1996), no. 1-3, 243–261. MR 1369417, DOI 10.1007/s002220050052
Jean-Pierre Demailly and János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556 (English, with English and French summaries). MR 1852009, DOI 10.1016/S0012-9593(01)01069-2
S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), no. 2, 269–277. MR 710055, DOI 10.4310/jdg/1214437664
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. MR 765366, DOI 10.1112/plms/s3-50.1.1
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
Min-Chun Hong and Gang Tian, Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections, Math. Ann. 330 (2004), no. 3, 441–472. MR 2099188, DOI 10.1007/s00208-004-0539-9
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
J. J. Kohn, Subellipticity of the $\bar \partial$-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), no. 1-2, 79–122. MR 512213, DOI 10.1007/BF02395058
J.-L. Koszul and B. Malgrange, Sur certaines structures fibrées complexes, Arch. Math. (Basel) 9 (1958), 102–109 (French). MR 131882, DOI 10.1007/BF02287068
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
Alan Michael Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549–596. MR 1078269, DOI 10.2307/1971429
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. MR 184252, DOI 10.2307/1970710
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 88770, DOI 10.2307/1970051
Pali, N. Faisceaux $\overline {\partial }$-cohérents sure les variétés complexes, preprint, arXiv: math.AG/0301146 (2003).
Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577, DOI 10.1090/S0894-0347-1988-0944577-9
Yum Tong Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53–156. MR 352516, DOI 10.1007/BF01389965
Yum Tong Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, vol. 8, Birkhäuser Verlag, Basel, 1987. MR 904673, DOI 10.1007/978-3-0348-7486-1
Yum Tong Siu, An effective Matsusaka big theorem, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1387–1405 (English, with English and French summaries). MR 1275204, DOI 10.5802/aif.1378
Yum-Tong Siu, Effective very ampleness, Invent. Math. 124 (1996), no. 1-3, 563–571. MR 1369428, DOI 10.1007/s002220050063
Siu, Y.-T. Multiplier ideal sheaves in algebraic and complex geometry, Samuel Eilenberg Lectures at Columbia University (2002), unpublished.
Henri Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans $\textbf {C}^{n}$, Bull. Soc. Math. France 100 (1972), 353–408 (French). MR 352517, DOI 10.24033/bsmf.1743
Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356, DOI 10.1007/BF01947069
K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257–S293. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861491, DOI 10.1002/cpa.3160390714
Weinkove, B. The J-flow, the Mabuchi energy, the Yang-Mills flow and multiplier ideal sheaves, Ph.D. thesis, Columbia University, 2004.