Advertisement

On the blow-up formula of twisted de Rham cohomology

  • Youming ChenEmail author
  • Song Yang
Article
  • 18 Downloads

Abstract

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of \(E_{1}\)-degeneracy of the Hodge–de Rham spectral sequence associated with a local system of complex vector spaces.

Keywords

Local system Twisted de Rham cohomology Blow-up Hodge–de Rham spectral sequence 

Mathematics Subject Classification

32S45 14F40 32C35 

Notes

Acknowledgements

The authors would like to thank the Departments of Mathematics of Pennsylvania State University and Università degli Studi di Milano for the hospitalities during their respective visits, and thank Sheng Rao and Xiangdong Yang for the useful discussions. In particular, the authors would like to thank the referee for introducing Example 1 to them. This work is partially supported by the NSFC (Grant Nos. 11571242, 11701414), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJ1709216) and the China Scholarship Council.

References

  1. 1.
    Angella, D., Suwa, T., Tardini, N., Tomassini, A.: Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms. arXiv:1712.08889v1
  2. 2.
    Barbieri-Viale, L.: \({\mathscr {H}}\)-cohomology versus algebraic cycles. Math. Nachr. 184, 5–57 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics 82. Springer, Berlin (1982)CrossRefGoogle Scholar
  4. 4.
    Deligne, P.: Théorème de Lefschetz et critères de dégenéréscence de suites spectrals. Publ. Math. Inst. Hautes Études Sci. 35, 259–277 (1968)CrossRefzbMATHGoogle Scholar
  5. 5.
    Demailly, J.-P.: Complex Analytic and Differential Geometry. J.-P. Demailly’s CADG e-book (2012)Google Scholar
  6. 6.
    Geiges, H., Pasquotto, F.: A formula for the Chern classes of symplectic blow-ups. J. Lond. Math. Soc. 76, 313–330 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. Inst. Hautes Études Sci. 29, 95–103 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Meng, L.: Morse–Novikov cohomology for blow-ups of complex manfolds. arXiv:1806.06622v3
  9. 9.
    Meng, L.: An explicit formula of blow-ups for Dolbeault cohomology. arXiv:1806.11435v2
  10. 10.
    Rao, S., Yang, S., Yang, X.: Dolbeault cohomology of blowing up complex manifolds. J. Math. Pures Appl. (2019).  https://doi.org/10.1016/j.matpur.2019.01.016 Google Scholar
  11. 11.
    Rao, S., Yang, S., Yang, X.: Dolbeault cohomology of blowing up complex manifolds II: bundle-valued case. J. Math. Pures Appl. (2019).  https://doi.org/10.1016/j.matpur.2019.02.010 Google Scholar
  12. 12.
    Stelzig, J.: The double complex of a blow-up. arXiv:1808.02882v1
  13. 13.
    Voisin, C.: Hodge Theory and Complex Algebraic Geometry I, II. Cambridge Studies in Advanced Mathematics, 76, 77. Cambridge University Press, Cambridge (2002)Google Scholar
  14. 14.
    Wells, R.O.: Comparison of de Rham and Dolbeault cohomology for proper surjective mappings. Pacific J. Math. 53, 281–300 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yang, S., Yang, X.: Bott–Chern blow-up formula and bimeromorphic invariance of the \(\partial {\bar{\partial }}\)-Lemma for threefolds. arXiv:1712.08901v3
  16. 16.
    Yang, X., Zhao, G.: A note on the Morse–Novikov cohomology of blow-ups of locally conformal Kähler manifolds. Bull. Aust. Math. Soc. 91, 155–166 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of ScienceChongqing University of TechnologyChongqingPeople’s Republic of China
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China

Personalised recommendations