Local invariant cycle theorem: Difference between revisions

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In mathematics, the '''local invariant cycle theorem''' was originally a conjecture of Griffiths <ref>{{harvnb|Clemens|1997|loc=Introduction}}</ref> which states that, given a surjective [[proper map]] <math>p</math> from a [[Kähler manifold]] <math>X</math> to the unit disk that has maximal rank everywhere except over 0, each cohomology class on <math>p^{-1}(t), t \ne 0</math> is the restriction of some cohomology class on the entire <math>X</math> if the cohomology class is invariant under a circle action (monodromy action); in short,

:<math>\operatorname{H}^*(X) \to \operatorname{H}^*(p^{-1}(t))^{S^1}</math>

is surjective.<ref>Editorial note: the first proof of the theorem was given by Clemens, apparently but this needs to be checked.</ref>

In algebraic geometry, Deligne proved the following analog.<ref>{{harvnb|Deligne|1980|loc=Théorème 3.6.1.}}</ref><ref>{{harvnb|Deligne|1980|loc=(3.6.4.)}}</ref> Given a [[proper morphism]] <math>f : X \to S</math> over the spectrum <math>S</math> of the henselization of <math>k[T]</math>, <math>k</math> an algebraically closed field, if <math>X</math> is essentially smooth over <math>k</math> and <math>X_{\overline{\eta}}</math> is smooth over <math>\overline{\eta}</math>, then the homomorphism on <math>\mathbb{Q}</math>-cohomology:

:<math>\operatorname{H}^*(X_s) \to \operatorname{H}^*(X_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)}</math>

is surjective, where <math>s, \eta</math> are the special and generic points and the homomorphism is the composition <math>\operatorname{H}^*(X_s) \simeq \operatorname{H}^*(X) \to \operatorname{H}^*(X_{\eta}) \to \operatorname{H}^*(X_{\overline{\eta}}).</math>