Complex analytic vanishing cycles for formal schemes

Complex analytic vanishing cycles for formal schemes - Vladimir Berkovich

Vladimir Berkovich
Weizmann Institute of Sciences; Member, School of Mathematics
December 12, 2013
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 \(R\).