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Extendable cohomologies for complex analytic varieties

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Abstract

We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern–Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As an application of our theory, we prove a Camacho–Sad type index theorem for holomorphic foliations of singular complex varieties.

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References

  1. Abate M., Bracci F., Tovena F.: Index theorems for holomorphic self-maps. Ann. Math. 159, 819–864 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abate, M., Bracci, F., Tovena, F.: Index theorems for holomorphic maps and foliations. Indiana Uni. Math. J. (to appear)

  3. Baum P., Bott R.: Singularities of holomorphic foliations. J. Differ. Geom. 7, 279–342 (1972)

    MATH  MathSciNet  Google Scholar 

  4. Bloom T., Herrera M.: De Rham cohomology of an analytic space. Inv. Math. 7, 275–296 (1969)

    Article  MathSciNet  Google Scholar 

  5. Bott R., Tu L.W.: Differential Forms in Algebraic Topology. Springer, Berlin (1982)

    MATH  Google Scholar 

  6. Bracci F.: First order extension of holomorphic foliations. Hokkaido Math. J. 33(2), 473–490 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Bracci F., Suwa T.: Residues for singular pairs and dynamic of Biholomorphic maps of singular surfaces. Int. J. Math. 15(5), 443–455 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Camacho C., Sad P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. 115(2), 579–595 (1982)

    Article  MathSciNet  Google Scholar 

  9. Checcucci V., Tognoli A., Vesentini E.: Lezioni di Topologia Generale. Feltrinelli, Milano (1968)

    MATH  Google Scholar 

  10. Ferrari A.: Cohomology and holomorphic differential forms on complex analytic spaces. Ann. Scuola Norm. Sup. di Pisa XXIV(1), 65–77 (1970)

    Google Scholar 

  11. Ferrari A.: Coomologia e forme differenziali sugli spazi analitici complessi. Ann. Scuola Norm. Sup. di Pisa XXV(3), 469–480 (1971)

    Google Scholar 

  12. Goresky M., MacPherson R.: Stratified Morse Theory. Springer, Berlin (1988)

    MATH  Google Scholar 

  13. Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables, vol. 1, 2, 3. Wadsworth and Brooks, Belmont (1990)

  14. Hardt R.: Triangulation of subanalytic sets and proper light subanalytic maps. Inv. Math. 38, 207–217 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hartshorne R.: Algebraic geometry. Springer, Berlin (1977)

    MATH  Google Scholar 

  16. Herrera M.: Integration on a semianalytic set. Bull. Soc. Math. France 94, 141–180 (1966)

    MATH  MathSciNet  Google Scholar 

  17. Herrera M.: De Rham theorems on semianalytic sets. Bull. Am. Math. Soc. 73(3), 414–418 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kobayashi S.: Differential Geometry of Complex Vector Bundles. Princeton University Press, Princeton (1987)

    MATH  Google Scholar 

  19. Lehmann D., Suwa T.: Residues of holomorphic vector fields relative to singular invariant subvarieties. J. Differ. Geom. 42, 165–192 (1995)

    MATH  MathSciNet  Google Scholar 

  20. Lojasiewicz S.: Triangulation of semianalytic sets. Ann. Scuola Norm. Sup. di Pisa 18, 449–474 (1964)

    MATH  MathSciNet  Google Scholar 

  21. Munkres J.R.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984)

    MATH  Google Scholar 

  22. Perrone, C.: Extendable cohomologies for complex analytic varieties. Ph.D. Thesis, Università di Roma “Tor Vergata” (2008)

  23. Steenrod N.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  24. Suwa T.: Indices of Vector Fields and Residues of Singular Holomorphic Foliations. Hermann, Paris (1998)

    MATH  Google Scholar 

  25. Suwa T.: Residues of Chern classes on singular varieties. In: Brasselet, J.P., Suwa, T. (eds) Singularités Franco-Japonaise (Séminaires et Congrès, 10), pp. 265–285. Société Mathematique de France, Paris (2005)

    Google Scholar 

  26. Suwa, T.: Residue theoretical approach to intersection theory. Notes for a course given at “The 9th Workshop on Real and Complex Singularities” held in São Carlos in July 2006, Niigata (2007)

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  1. Dipartimento di Matematica della II Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Carlo Perrone

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  1. Carlo Perrone
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Correspondence to Carlo Perrone.

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This article is dedicated to Citro Cucurbitæque maximis.

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Perrone, C. Extendable cohomologies for complex analytic varieties. Math. Ann. 345, 83–132 (2009). https://doi.org/10.1007/s00208-009-0343-7

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  • DOI: https://doi.org/10.1007/s00208-009-0343-7

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