Interaction de strates consécutives III: le cas de la valeur propre 1.

Abstract

In this article we consider an holomorphic germ $f :(\Bbb C^{n+1},0) \longrightarrow (\Bbb C,0)$ such that there exists a germ of curve $(S, 0)$ contained in the singular set Sing $(f)$ of the hypersurface germ $\{f = 0 \}$ with the property that, for any point $x$ of sing $(f) \setminus S$, is not an eigenvalue of the local monodromy of $f$ acting on the reduced cohomology of the Milnor's fiber of $f$ at $x$. This 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 the eigenvalue 1 appears in another stratum (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 is related to the spectral part for the eigenvalue 1 of the vanishing cycles with compact supports. It has the same dimension as $H^n(F) =1$ and $H^n_c(F)=1$ (the case $n = 2$, that is to say the case of an hypersurface in $\Bbb C^3$, is analoguous but special) and it leads to a non trivial monodromic factorization of the canonical map $can : H^n_c(F)=1 \longrightarrow H^n_{c \cap S}(F)=1 \longrightarrow H^n(F)=1$, and to a monodromic isomorphism of variation $var : H^n_{c \cap S}(F)=1 \longrightarrow H^n_c(F)=1$. It gives also a canonical hermitian form $\mathcal H : H^n_{c \cap S}(F)=1 \times H^n(F)=1 \longrightarrow \Bbb C$ which is non-degenerate (here again the n = 2 is analoguous but special). 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 corresponds to the fact that, in the derived category, the vanishing cycle complex is not quasi-isomorphic to the direct sum of its cohomology sheaves. We show that this implies, for instance, 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 point of $S$