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Configuration Spaces pp 107–118Cite as

Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds

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Abstract

Let X be a complex algebraic manifold. Let U be the complement of a configuration of submanifolds. We study the Leray spectral sequence of the inclusion U ↪ X computing the cohomology of U. Under some condition posed on the intersections of submanifolds we show that the Leray spectral sequence degenerates on E 3. This result generalizes well known properties of hyperplane arrangements. The main cause which rigidifies the spectral sequence is the weight filtration in cohomology.

Keywords

  • Deligne spectral sequence
  • Hodge structure
  • Leray spectral sequence

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Notes

  1. 1.

    Avoiding the sheaf language and forgetting Hodge structure we have

    $$\displaystyle{\left (R^{k}j'_{{\ast}}\mathbb{Q}_{ U_{1}\cup U_{2}}\right )_{x} \simeq H^{k}(B_{x} \cap (U_{ 1} \cup U_{2})) \simeq H^{k}(\partial B_{x} \cap (U_{ 1} \cup U_{2}))\,,}$$

    where B x is a small ball around x. The stalk of the sheaf of local cohomology \(\mathcal{H}_{\ell}(Y )_{x}\) is isomorphic to

    $$\displaystyle{H_{\ell}(B_{x} \cap Y,\partial B_{x} \cap Y ) \simeq \bar{ H}_{\ell-1}(\partial B_{x} \cap Y )\,.}$$

    By Alexander duality in ∂ B x we have

    $$\displaystyle{\bar{H}^{k}(\partial B_{x} \cap (U_{ 1} \cup U_{2})) \simeq \bar{ H}_{2n-k-2}(\partial B_{x} \cap Y ) \simeq H_{2n-k-1}(B_{x} \cap Y,\partial B_{x} \cap Y )\,.}$$

References

  1. A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100 (Société Mathématique de France, Paris, 1982), pp. 5–171

    Google Scholar 

  2. A. Björner, T. Ekedahl, Subspace arrangements over finite fields: cohomological and enumerative aspects. Adv. Math. 129 (2), 159–187 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M.W. Davis, T. Januszkiewicz, I.J. Leary, The ℓ 2-cohomology of hyperplanes complements. Groups Geom. Dyn. 1, 301–309 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. M.W. Davis, S. Settepanella, Vanishing results for the cohomology of complex toric hyperplane complements. Publ. Mat. 57, 379–392 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. P. Deligne, Théorie de Hodge II. Publications Mathématiques/Institut des Hautes Etudes Scientifiques, vol. 40 (Institut des Hautes études Scientifiques, Bures-sur-Yvette, 1972), pp. 5–57

    Google Scholar 

  6. P. Deligne, Théorie de Hodge III. Publications Mathématiques/Institut des Hautes Etudes Scientifiques, vol. 44 (Institut des Hautes études Scientifiques, Bures-sur-Yvette, 1974), pp. 5–77

    Google Scholar 

  7. P. Deligne, La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. G. Denham, A.I. Suciu, S. Yuzvinsky, Combinatorial covers and vanishing of cohomology. Selecta Math. (N.S.) 22 (2), 561–594 (2016)

    Google Scholar 

  9. H. Esnault, V. Schechtman, E. Viehweg, Cohomology of local systems on the complement of hyperplanes. Invent. math. 109, 557–561 (1992); Erratum. 112 (1993)

    Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  11. K. Jewell, Complements of sphere and subspace arrangements. Topology Appl. 56 (3), 199–214 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. K. Jewell, P. Orlik, B. Shapiro, On the complements of affine subspace arrangements. Topology Appl. 56 (3), 215–233 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Y. Kawahara, Hodge number of cohomology of local systems on the complement of hyperplanes in \(\mathbb{P}^{3}\). J. Math. Sci. Univ. Tokyo 8, 177–199 (2001)

    MathSciNet  MATH  Google Scholar 

  14. M. Saito, Hodge modules. Publ. Res. Inst. Math. Sci. 26 (2), 221–333 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. V. Schechtman, H. Terao, A. Varchenko, Cohomology of local systems and the Kac-Kazhdan condition for singular vectors. J. Pure Appl. Algebra 100, 93–102 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. K. Timmerscheidt, Mixed Hodge theory for unitary local systems. J. Reine Angew. Math. 379, 152–171 (1987)

    MathSciNet  MATH  Google Scholar 

  17. B. Totaro, Configuration spaces of algebraic varieties. Topology 35 (4), 1057–1067 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by NCN grant 2013/08/A/ST1/00804.

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Authors and Affiliations

  1. Department of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    Andrzej Weber

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  1. Andrzej Weber
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Correspondence to Andrzej Weber .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Filippo Callegaro

  2. Department of Mathematics, University of Rochester, Rochester, USA

    Frederick Cohen

  3. Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy

    Corrado De Concini

  4. Department of Mathematics, University of Bremen, Bremen, Germany

    Eva Maria Feichtner

  5. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Giovanni Gaiffi

  6. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Mario Salvetti

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Weber, A. (2016). Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_4

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