Abstract
The cohomology groups of the fibres of a polynomial mapping are locally constant except over a finite number of points. The change of the cohomology is caused by singularities if one includes those at infinity. In this paper conditions are given such that the rank of certain cohomology groups is still constant or behaves in a semicontinuous way. For the description of the change of cohomology it is important to look at direct image sheaves. The nature of the change can be understood using notions from the theory of covering projections.
Partially supported by Deutsche Forschungsgemeinschaft.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982), 1–172.
H.A. Hamm, Lê D.T., Rectified homotopical depth and Grothendieck conjectures, in: The Grothendieck Festschrift, Vol. II, 311–351, Progr. Math. 87, Birkhäuser, Boston, MA, 1990.
H.A. Hamm, Lê D.T., Vanishing Theorems for constructible sheaves I, J. reine angew. Math. 471 (1996), 115–138.
H.A. Hamm, Lê D.T., Vanishing Theorems for constructible sheaves II, Kodai Math. J. 21 (1998), 208–247.
H.A. Hamm, Lê D.T., Théorèmes d’annulation pour les faisceaux algébriquement constructibles, C.R.Acad.Sci Paris 327 (1998), sér. I, 759–762.
M. Kashiwara, P. Schapira, Sheaves on Manifolds, Springer-Verlag, Berlin 1990.
J. Schürmann, Morse theory for constructible sheaves, preprint.
D. Siersma, J. Smeltink, Classification of singularities at infinity of polynomials of degree.4 in two variables, Georgian Math. J. 7 (2000), 179–190.
D. Siersma, M. Tibár, Singularities at infinity and their vanishing cycles, Duke Math. J. 80 (1995), 771–783.
M. Tibár, Topology at infinity of polynomial mappings and Thom regularity condition, Compositio Math. 111 (1998), 89–109.
M. Tibár, Singularities and topology of meromorphic functions, this volume.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this chapter
Cite this chapter
Hamm, H.A. (2002). On the Cohomology of Fibres of Polynomial Maps. In: Libgober, A., Tibăr, M. (eds) Trends in Singularities. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8161-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8161-6_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9461-6
Online ISBN: 978-3-0348-8161-6
eBook Packages: Springer Book Archive