Lê modules and traces

Massey, David

doi:10.1090/S0002-9939-06-08208-6

Lê modules and traces
HTML articles powered by AMS MathViewer

by David B. Massey PDF

Proc. Amer. Math. Soc. 134 (2006), 2049-2060 Request permission

Abstract:

We show how some of our recent results clarify the relationship between the Lê numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The Lê numbers are actually the ranks of the free Abelian groups—the Lê modules—appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the Lê module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.

References

Norbert A’Campo, Le nombre de Lefschetz d’une monodromie, Nederl. Akad. Wetensch. Proc. Ser. A 76 = Indag. Math. 35 (1973), 113–118 (French). MR 0320364
C. H. Clemens Jr., Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities, Trans. Amer. Math. Soc. 136 (1969), 93–108. MR 233814, DOI 10.1090/S0002-9947-1969-0233814-9
Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072, DOI 10.1007/978-3-642-18868-8
Grothendieck, A., Séminaire de Géométrie Algébrique (SGA VII-1), volume 288 of Lect. Notes. Math. Springer-Verlag, 1972. Résumé des premiers exposés de A. Grothendieck, rédigé par P. Deligne.
Helmut A. Hamm and Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup. (4) 6 (1973), 317–355. MR 401755, DOI 10.24033/asens.1250
Alan Landman, On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities, Trans. Amer. Math. Soc. 181 (1973), 89–126. MR 344248, DOI 10.1090/S0002-9947-1973-0344248-1
Lê Dũng Tráng, Calcul du nombre de cycles évanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 261–270 (French, with English summary). MR 330501, DOI 10.5802/aif.491
Lê Dũng Tráng, The geometry of the monodromy theorem, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 157–173. MR 541020
Lê, D. T. and Massey, D., Hypersurface Singularities and the Swing. preprint, 2005.
David B. Massey, Lê cycles and hypersurface singularities, Lecture Notes in Mathematics, vol. 1615, Springer-Verlag, Berlin, 1995. MR 1441075, DOI 10.1007/BFb0094409
David B. Massey, Singularities and enriched cycles, Pacific J. Math. 215 (2004), no. 1, 35–84. MR 2060494, DOI 10.2140/pjm.2004.215.35
C. Sabbah, Proximité évanescente. I. La structure polaire d’un ${\scr D}$-module, Compositio Math. 62 (1987), no. 3, 283–328 (French, with English summary). MR 901394
Dirk Siersma, Isolated line singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 485–496. MR 713274, DOI 10.1090/pspum/040.2/713274
Dirk Siersma, Variation mappings on singularities with a $1$-dimensional critical locus, Topology 30 (1991), no. 3, 445–469. MR 1113689, DOI 10.1016/0040-9383(91)90025-Y
Bernard Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972) Astérisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362 (French). MR 0374482