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Results in Mathematics

, 74:115 | Cite as

On a Resolution of Singularities with Two Strata

  • Vincenzo Di GennaroEmail author
  • Davide Franco
Article
  • 68 Downloads

Abstract

Let X be a complex, irreducible, quasi-projective variety, and \(\pi :{\widetilde{X}}\rightarrow X\) a resolution of singularities of X. Assume that the singular locus \({\text {Sing}}(X)\) of X is smooth, that the induced map \(\pi ^{-1}({\text {Sing}}(X))\rightarrow {\text {Sing}}(X)\) is a smooth fibration admitting a cohomology extension of the fiber, and that \(\pi ^{-1}({\text {Sing}}(X))\) has a negative normal bundle in \({\widetilde{X}}\). We present a very short and explicit proof of the Decomposition Theorem for \(\pi \), providing a way to compute the intersection cohomology of X by means of the cohomology of \({\widetilde{X}}\) and of \(\pi ^{-1}({\text {Sing}}(X))\). Our result applies to special Schubert varieties with two strata, even if \(\pi \) is non-small. And to certain hypersurfaces of \({\mathbb {P}}^5\) with one-dimensional singular locus.

Keywords

Projective variety smooth fibration resolution of singularities derived category intersection cohomology Decomposition Theorem Poincaré polynomial Betti numbers Schubert varieties 

Mathematics Subject Classification

Primary 14B05 Secondary 14E15 14F05 14F43 14F45 14M15 32S20 32S60 58K15 

Notes

References

  1. 1.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux Pervers, Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, 100, Soc. Math., Paris, France, pp. 5–171 (1982)Google Scholar
  2. 2.
    Bertram, A., Ein, L., Lazarsfeld, R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. AMS 4, 587–602 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cheeger, J., Goresky, M., MacPherson, R.: L2-cohomology and intersection homology for singular algebraic varieties. In: Seminar on Differential Geometry, Volume 102 of Annals of Mathematics Studies, pp. 303–340. Princeton University Press, Princeton (1982)CrossRefGoogle Scholar
  4. 4.
    de Cataldo, M.A., Migliorini, L.: The Gysin map is compatible with Mixed Hodge structures, Algebraic structures and moduli spaces. In: CRM Proc. Lecture Notes, vol. 38, , pp. 133–138. Amer. Math. Soc., Providence, RI (2004)Google Scholar
  5. 5.
    de Cataldo, M.A., Migliorini, L.: The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. 4 38(5), 693–750 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    de Cataldo, M.A., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Am. Math. Soc. (N.S.) 46(4), 535–633 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Di Gennaro, V., Franco, D.: Néron-Severi group of a general hypersurface. Commun. Contemp. Math. 19(01), 1650004 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Di Gennaro, V., Franco, D.: On the existence of a Gysin morphism for the blow-up of an ordinary singularity. Ann. Univ. Ferrara Sezione VII Sci. Mat. 63(1), 75–86 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Di Gennaro, V., Franco, D.: On the topology of a resolution of isolated singularities. J. Singul. 16, 195–211 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dimca, A.: Sheaves in Topology. Springer Universitext, Berlin (2004)CrossRefGoogle Scholar
  11. 11.
    Fulton, W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete; 3.Folge, Bd. 2. Springer (1984)Google Scholar
  12. 12.
    Fulton, W., MacPherson, R.: Categorical framework for the study of singular spaces. Mem. Am. Math. Soc. 31(243), vi+165 (1981)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Goresky, M., MacPherson, R.: Intersection homology II. Invent. Math. 71, 77–129 (1983)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. A. Wiley-Interscience, New York (1978)zbMATHGoogle Scholar
  15. 15.
    Jouanolou, J.P.: Cohomologie de quelques schémas classiques et théorie cohomologique des classes de Chern. In: SGA 5, 1965–66. Springer Lecture Notes, vol. 589, pp. 282–350 (1977)Google Scholar
  16. 16.
    Kirwan, F.: An Introduction to Intersection Homology Theory. Longman Scientific & Technical (1988)Google Scholar
  17. 17.
    Lamotke, K.: The topology of complex projective varieties after S. Lefschetz. Topology 20, 15–51 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lazarsfeld, R.: Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete; 3.Folge, vol. 49. Springer (2004)Google Scholar
  19. 19.
    MacPherson, R.: Global questions in the topology of singular spaces. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Warsaw), pp. 213–235 (1983)Google Scholar
  20. 20.
    Massey, D.B.: Intersection cohomology, monodromy and the Milnor fiber. Int. J. Math. 20(4), 491–507 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Perone, M.: Direct sum decomposition and weak Krull-Schmidt Theorems. Ph.D. Thesis, Scuola di Dottorato di Ricerca in Matematica, Ciclo XXIII, Università degli Studi di Padova. http://paduaresearch.cab.unipd.it/3365/1/phdthesis.pdf (2011)
  22. 22.
    Saito, M.: Mixed Hodge modules. Publ. RIMS Kyoto Univ. 26, 221–333 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Spanier, E.H.: Algebraic Topology. McGraw-Hill Series in Higher Mathematics (1966)Google Scholar
  24. 24.
    Yamaguchi, H.: A note on the self-intersection formula. Mem. Nagano Natl. Coll. Technol. 19, 147–149 (1988)Google Scholar
  25. 25.
    Voisin, C.: Hodge Theory and Complex Algebraic Geometry, I, Cambridge Studies in Advanced Mathematics, vol. 76. Cambridge University Press, Cambridge (2002)Google Scholar
  26. 26.
    Williamson, G.: Hodge Theory of the Decomposition Theorem [after M.A. de Cataldo and L. Migliorini], Séminaire BOURBAKI, n. 1115, pp. 31 (2015–2016)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università di Napoli “Federico II”NapoliItaly

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