Advertisement

Revista Matemática Complutense

, Volume 31, Issue 2, pp 479–524 | Cite as

Rational homotopy and intersection-formality of complex algebraic varieties

  • David Chataur
  • Joana CiriciEmail author
Article
First Online:
  • 106 Downloads

Abstract

A homotopical treatment of intersection cohomology recently developed by Chataur–Saralegui–Tanré associates a perverse algebraic model to every topological pseudomanifold, extending Sullivan’s presentation of rational homotopy theory to intersection cohomology. In this context, there is a notion of intersection-formality, measuring the vanishing of higher Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We then use mixed Hodge theory to prove some intersection-formality results for large families of complex projective varieties, such as isolated surface singularities and varieties of arbitrary dimension with ordinary isolated singularities.

Keywords

Rational homotopy Formality Intersection cohomology Mixed Hodge theory Weight spectral sequence Isolated singularities 

Mathematics Subject Classification

55P62 55N33 32S35 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Banagl, M.: Intersection Spaces, Spatial Homology Truncation, and String Theory. Lecture Notes in Mathematics, vol. 1997. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chataur, D., Cirici, J.: Rational homotopy of complex projective varieties with normal isolated singularities. Forum Math. 29(1), 41–57 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chataur, D., Saralegi, M., Tanré, D.: Intersection cohomology. Simplicial blow-up and rational homotopy. Mem. Amer. Math. Soc. (to appear)Google Scholar
  4. 4.
    Cirici, J.: Cofibrant models of diagrams: mixed Hodge structures in rational homotopy. Trans. Am. Math. Soc. 367(8), 5935–5970 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cirici, J., Guillén, F.: \(E_1\)-formality of complex algebraic varieties. Algebra Geom. Topol. 14(5), 3049–3079 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cirici, J., Horel, G.: Mixed Hodge structures and formality of symmetric monoidal functors. arXiv:1703.06816 [math.AT] (2017)
  7. 7.
    Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deligne, P.: Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. 44, 5–77 (1974)CrossRefzbMATHGoogle Scholar
  9. 9.
    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dimca, A.: Singularities and Topology of Hypersurfaces. Universitext. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  11. 11.
    Durfee, A.: Mixed Hodge structures on punctured neighborhoods. Duke Math. J. 50(4), 1017–1040 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Durfee, A.: Neighborhoods of algebraic sets. Trans. Am. Math. Soc. 276(2), 517–530 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Durfee, A.H., Hain, R.M.: Mixed Hodge structures on the homotopy of links. Math. Ann. 280(1), 69–83 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Friedman, G.: On the chain-level intersection pairing for PL pseudomanifolds. Homol. Homotopy Appl. 11(1), 261–314 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Friedman, G., McClure, J.E.: Cup and cap products in intersection (co)homology. Adv. Math. 240, 383–426 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goresky, M.: Intersection homology operations. Comment. Math. Helv. 59(3), 485–505 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19(2), 135–162 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goresky, M., MacPherson, R.: Intersection homology. II. Invent. Math. 72(1), 77–129 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hain, R.M.: The de Rham homotopy theory of complex algebraic varieties. II. K-Theory 1(5), 481–497 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hamm, H.: Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191, 235–252 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hovey, M.: Intersection homological algebra. In: New Topological Contexts for Galois Theory and Algebraic Geometry (BIRS 2008), Geometry & Topology Monographs, vol. 16, pp. 133–150. Geometry & Topology Publications, Coventry (2009)Google Scholar
  22. 22.
    Kapovich, M., Kollár, J.: Fundamental groups of links of isolated singularities. J. Am. Math. Soc. 27(4), 929–952 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton (1968)Google Scholar
  24. 24.
    Morgan, J.W.: The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 48, 137–204 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Navarro-Aznar, V.: Sur la théorie de Hodge des variétés algébriques à singularités isolées. Systémes différentiels et singularités, Astérisque, vol. 130, pp. 272–307. Société Mathématique de France, Paris (1985)Google Scholar
  26. 26.
    Navarro-Aznar, V.: Sur la théorie de Hodge-Deligne. Invent. Math. 90(1), 11–76 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Peters, C., Steenbrink, J.: Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 52. Springer, Berlin (2008)Google Scholar
  28. 28.
    Polizzi, F., Rapagnetta, A., Sabatino, P.: On factoriality of threefolds with isolated singularities. Mich. Math. J. 63(4), 781–801 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Saito, M.: Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Simpson, C.: Local systems on proper algebraic \(V\)-manifolds. Pure Appl. Math. Q. 7(4), 1675–1759 (2011). (Special Issue: In memory of Eckart Viehweg)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. In: Singularities, Part 2 (Arcata, California, 1981), Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 513–536. American Mathematical Society, Providence, RI (1983)Google Scholar
  33. 33.
    Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Totaro, B.: Chow groups, Chow cohomology, and linear varieties. Forum Math. Sigma 2, e17, 25 (2014)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.LAMFA Université de Picardie Jules VerneAmiens Cedex 1France
  2. 2.Fachbereich Mathematik und InformatikUniversität MünsterMünsterGermany

Personalised recommendations