# Complex analytic vanishing cycles for formal schemes

Let $$R={\cal O}_{{\bf C},0}$$ be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme $$\cal X$$ of finite type over $$R$$ defines a complex analytic space $${\cal X}^h$$ over an open disc $$D$$ of small radius with center at zero. The preimage of the punctured disc $$D^\ast=D\backslash\{0\}$$ is denoted by $${\cal X}^h_\eta$$, and the preimage of zero coincides with the analytification $${\cal X}_s^h$$ of the closed fiber $${\cal X}_s$$ of $$\cal X$$. The complex analytic vanishing cycles functor associates to every abelian sheaf $$F$$ on $${\cal X}^h_\eta$$ a complex $$R\Psi_\eta(F)$$ in the derived category of abelian sheaves on $${\cal X}_s^h$$ provided with an action of the fundamental group $$\Pi=\pi_1(D^\ast)$$. In this talk I'll explain a result from my work in progress which implies that, if $$F$$ is the locally constant sheaf $$\Lambda_{{\cal X}^h_\eta}$$ associated to an arbitrary finitely generated abelian group $$\Lambda$$ provided with an action of $$\Pi$$, the restriction of the complex $$R\Psi_\eta(\Lambda_{{\cal X}^h_\eta})$$ to the analytification $${\cal Y}^h$$ of a subscheme $${\cal Y}\subset{\cal X}_s$$ depends only on the formal completion $$\widehat{\cal X}_{/\cal Y}$$ of $$\cal X$$ along $$\cal Y$$. The result itself tells that the construction of the vanishing cycles complexes can be extended to the category of special formal schemes over the completion $$\widehat R$$ of $$R$$.

### Affiliation

Weizmann Institute of Sciences; Member, School of Mathematics