Complex analytic vanishing cycles for formal schemes

Complex analytic vanishing cycles for formal schemes

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Vladimir Berkovich, IAS
IAS Room S-101

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 {\it 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 $R$.