Semi-simple Carrousels and the Monodromy

David B. Massey[1]

  • [1] Northeastern University Dept. of Mathematics Boston MA, 02115 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 1, page 85-100
  • ISSN: 0373-0956

Abstract

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Let 𝒰 be an open neighborhood of the origin in ℂ n + 1 and let f : ( 𝒰 , 0 ) → ( ℂ , 0 ) be complex analytic. Let z 0 be a generic linear form on ℂ n + 1 . If the relative polar curve Γ f , z 0 1 at the origin is irreducible and the intersection number ( Γ f , z 0 1 · V ( f ) ) 0 is prime, then there are severe restrictions on the possible degree n cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when ( Γ f , z 0 1 · V ( f ) ) 0 is not prime.

How to cite

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Massey, David B.. "Semi-simple Carrousels and the Monodromy." Annales de l’institut Fourier 56.1 (2006): 85-100. <http://eudml.org/doc/10144>.

@article{Massey2006,
abstract = {Let $\mathcal\{U\}$ be an open neighborhood of the origin in $\mathbb\{C\}^\{n+1\}$ and let $f:(\mathcal\{U\}, \mathbf\{0\})\rightarrow (\mathbb\{C\}, 0)$ be complex analytic. Let $z_0$ be a generic linear form on $\mathbb\{C\}^\{n+1\}$. If the relative polar curve $\Gamma ^1_\{f, z_0\}$ at the origin is irreducible and the intersection number $\big (\Gamma ^1_\{f, z_0\}\cdot V(f))_\mathbf\{0\}$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $\big (\Gamma ^1_\{f, z_0\}\cdot V(f))_\mathbf\{0\}$ is not prime.},
affiliation = {Northeastern University Dept. of Mathematics Boston MA, 02115 (USA)},
author = {Massey, David B.},
journal = {Annales de l’institut Fourier},
keywords = {Carrousel; polar curve; monodromy; Milnor fiber; carrousel},
language = {eng},
number = {1},
pages = {85-100},
publisher = {Association des Annales de l’institut Fourier},
title = {Semi-simple Carrousels and the Monodromy},
url = {http://eudml.org/doc/10144},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Massey, David B.
TI - Semi-simple Carrousels and the Monodromy
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 1
SP - 85
EP - 100
AB - Let $\mathcal{U}$ be an open neighborhood of the origin in $\mathbb{C}^{n+1}$ and let $f:(\mathcal{U}, \mathbf{0})\rightarrow (\mathbb{C}, 0)$ be complex analytic. Let $z_0$ be a generic linear form on $\mathbb{C}^{n+1}$. If the relative polar curve $\Gamma ^1_{f, z_0}$ at the origin is irreducible and the intersection number $\big (\Gamma ^1_{f, z_0}\cdot V(f))_\mathbf{0}$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $\big (\Gamma ^1_{f, z_0}\cdot V(f))_\mathbf{0}$ is not prime.
LA - eng
KW - Carrousel; polar curve; monodromy; Milnor fiber; carrousel
UR - http://eudml.org/doc/10144
ER -

References

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  9. D. Massey, Lê Cycles and Hypersurface Singularities, Lecture Notes in Mathematics 1615 (1995) Zbl0835.32002MR1441075
  10. D. Massey, The Sebastiani-Thom Isomorphism in the Derived Category, Compos. Math. 125 (2001), 353-362 Zbl0986.32004MR1818986
  11. D. Massey, The Nexus Diagram and Integral Restrictions on the Monodromy, (2004) 
  12. J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies 61 (1968) Zbl0184.48405MR239612
  13. D. Siersma, Isolated Line Singularities, Proc. Symp. Pure Math. 40 (1983), 485-496 Zbl0514.32007MR713274
  14. B. Teissier, Cycles évanescents, sections planes et conditions de Whitney, Astérisque 7-8 (1973), 285-362 Zbl0295.14003MR374482
  15. M. Tibăr, The Lefschetz Number of a Monodromy Transformation, (1992) Zbl1009.32501
  16. M. Tibăr, Carrousel monodromy and Lefschetz number of Singularities, Enseign. Math. (2) 39 (1993), 233-247 Zbl0809.32010MR1252066

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