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In mathematics, the '''local invariant cycle theorem''' was originally a conjecture of Griffiths <ref>{{harvnb|Clemens|1977|loc=Introduction}}</ref><ref>{{harvnb|Griffiths|1970|loc=Conjecture 8.1.}}</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. The conjecture was first proved by Clemens.
Deligne also proved the following.<ref>{{harvnb|Deligne|1980|loc=Théorème 3.6.1.}}</ref><ref>{{harvnb|Deligne|1980|loc=(3.6.4.)}}</ref> Given a [[proper morphism]] <math>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<!-- meaning? --> over <math>k</math> and <math>X_{\overline{\eta}}</math> smooth over <math>\overline{\eta}</math>, then the homomorphism on <math>\mathbb{Q}</math>-cohomology:
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