Grothendieck and vanishing cycles
The paper is a survey of classical results of Grothendieck on vanishing cycles. Consider a regular, proper and flat curve \(X\) over a strictly local trait \(S=(S,s,\eta )\), whose generic fiber is smooth and whose reduced special fiber is a divisor with normal crossings. One can analyse the difference between the (étale) cohomology of the special fiber \(H^*(X_s)\) and that of the generic geometric fiber \(H^*(X_{\eta})\), the coefficients ring being \(\mathbb{Z}_{\ell}\), \(\ell\) a prime number invertible on \(S\). Assuming that the action of the inertia group \(I\) on \(H^*(X_{\bar{\eta}})\) is tame, one shows that the defect of the specialization map \(H^*(X_s)\rightarrow H^*(X_{\bar{\eta}})\) is an isomorphism controlled by the vanishing cycles groups. This construction leads to Grothendieck's local monodromy theorem and his monodromy pairing for abelian varieties over local fields. The author discusses related current developments and questions and includes the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients.
scientific article; zbMATH DE number 7382715
scientific article