Interaction de strates consecutives pour les cycles evanescents III : Le cas de la valeur propre 1

Abstract

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 $H^n_{c\cap S}(F)_{=1}$, of the Milnor's fiber of f at the origin. It has the same dimension as $H^n(F)_{=1}$ and $H^n_c(F)_{=1}$, and it leads to a non trivial factorization of the canonical map $$ can : H^n_{c\cap S}(F)_{=1} \to H^n_c(F)_{=1},$$ and to a monodromic isomorphism of variation $$ var :H^n_{c\cap S}(F)_{=1}\to H^n_c(F)_{=1}.$$ It gives a canonical hermitian form $$ \mathcal{H} : H^n_{c\cap S}(F)_{=1} \times H^n(F )_{=1} \to \mathbb{C}$$ 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 $\int_X |f |^{2\lambda}\square $ for functions f having semi-simple local monodromies at each singular point of {f =0}.