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Polar Homology

Published online by Cambridge University Press:  20 November 2018

Boris Khesin  and
Alexei Rosly
Boris Khesin
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3 e-mail: [email protected]
Alexei Rosly
Affiliation:
Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow, Russia e-mail: [email protected]
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Abstract

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For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar $k$-chains are subvarieties of complex dimension $k$ with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it. One can also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, which have the properties similar to those of the corresponding topological notions.

Keywords

Primary:14C1014F10secondary:58A14Poincaré residueholomorphic linking
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 5 , 01 October 2003 , pp. 1100 - 1120
Copyright
Copyright © Canadian Mathematical Society 2003

References

[AKSZ] Alexandrov, M., Kontsevich, M., Schwarz, A. and Zaboronsky, O., The Geometry of the Master Equation and Topological Quantum Field Theory. Internat. J. Modern Phys. A 12(1997), 1405–1430, (hep-th/9502010).Google Scholar
[A] Arnold, V. I., Arrangement of ovals of real plane algebraic curves, involutions of smooth four-dimensional manifolds, and on arithmetic of integral-valued quadratic forms. Functional Anal. Appl. 5(1971), 169–176.Google Scholar
[BO] Bloch, S. and Ogus, A., Gersten's conjecture and the homology of schemes. Ann. Sci. École Norm. Sup. 7(1974), 181–201.Google Scholar
[De] Deligne, P., Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40(1971), 5–57.Google Scholar
[DT] Donaldson, S. K. and Thomas, R. P., Gauge theory in higher dimensions. The geometric universe, Oxford, 1996, Oxford Univ. Press, Oxford, 1998, pp. 31–47.Google Scholar
[FK] Frenkel, I. B. and Khesin, B. A., Four Dimensional Realization of Two Dimensional Current Groups. Comm. Math. Phys. 178(1996), 541–561.Google Scholar
[FT] Frenkel, I. B. and Todorov, A. N., in preparation. [Ger] A. Gerasimov, unpublished, 1995.Google Scholar
[Gr] Griffiths, P. A., Variations on a Theorem of Abel. Invent. Math. 35(1976), 321–390.Google Scholar
[GH] Griffiths, P. A. and Harris, J., Principles of Algebraic Geometry. Wiley, New York, 1978.Google Scholar
[KR] Khesin, B. and Rosly, A., Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces. In: The Arnoldfest, Proceedings of a conference in honour of V. I. Arnold for his sixtieth birthday, Toronto, 1997, (eds., Bierstone, E. et al.), Fields Inst. Commun. 24(1999), 311–323, (math.AG/0009013).Google Scholar
[iKR] Khesin, B. and Rosly, A., in preparation.Google Scholar
[ASL] Losev, A. S., private communication, 1999.Google Scholar
[LNS] Losev, A., Nekrasov, N. and Shatashvili, S., Issues in Topological Gauge Theory. Nuclear Phys. B 534(1998), 549–611, (hep-th/9711108).Google Scholar
[T] Thomas, R. P., Gauge theory on Calabi-Yau manifolds. Ph.D. thesis, Oxford, 1997.Google Scholar
[Tju] Tjurina, G. N., On the moduli space of complex surfaces with q = 0 and K = 0 . Chapter IX in Algebraic surfaces, by Shafarevich, I. R., et al., Trudy Steklov Mat. Inst. 75(1965) pp. 163–191.Google Scholar
[W] Witten, E., Chern-Simons gauge theory as a string theory. The Floer memorial volume, Progr. Math. 133, Birkhäuser, Basel, 1995, 637–678 (hep-th/9207094).Google Scholar