2005/09/09 by D. Barlet, Barlet, D.
Mathematics · #32S25 #32S40 #32S50 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:32S25 #msc:32S40 #msc:32S50
paper · pdf · doi:10.48550/arxiv.math/0509203
With an appendix by C. Sabbah
arxiv created 2005/09/09 · arxiv updated 2009/12/01
This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 case. Of course this is a more involved situation because the existence of the smooth stratum for the hypersurface f = 0 forces to consider three strata for the nearby cycles. And we already know that the smooth stratum is always "tangled" if it is not alone (see [B.84b] and the introduction of [B.03]). The new phenomenon is the role played here by a "new" cohomology group, denote by Hnc∩ S(F)=1, of the Milnor's fiber of f at the origin. It has the same dimension as Hn(F)=1 and Hnc(F)=1, and it leads to a non trivial factorization of the canonical map can : Hnc∩ S(F)=1 → Hnc(F)=1, and to a monodromic isomorphism of variation var :Hnc∩ S(F)=1→ Hnc(F)=1. It gives a canonical hermitian form H : Hnc∩ S(F)=1 × Hn(F )=1 → ℂ which is non degenerate. This generalizes the case of an isolated singularity for the eigenvalue 1 (see [B.90] and [B.97]). The "overtangling" phenomenon for strata associated to the eigenvalue 1 implies the existence of triple poles at negative integers (with big enough absolute value) for the meromorphic continuation of the distribution ∫X |f |2λ\square for functions f having semi-simple local monodromies at each singular point of f =0.