Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective proper map $$p$$ from a Kähler manifold $$X$$ to the unit disk that has maximal rank everywhere except over 0, each cohomology class on $$p^{-1}(t), t \ne 0$$ is the restriction of some cohomology class on the entire $$X$$ if the cohomology class is invariant under a circle action (monodromy action); in short,
 * $$\operatorname{H}^*(X) \to \operatorname{H}^*(p^{-1}(t))^{S^1}$$

is surjective. The conjecture was first proved by Clemens. The theorem is also a consequence of the BBD decomposition.

Deligne also proved the following. Given a proper morphism $$X \to S$$ over the spectrum $$S$$ of the henselization of $$k[T]$$, $$k$$ an algebraically closed field, if $$X$$ is essentially smooth over $$k$$ and $$X_{\overline{\eta}}$$ smooth over $$\overline{\eta}$$, then the homomorphism on $$\mathbb{Q}$$-cohomology:
 * $$\operatorname{H}^*(X_s) \to \operatorname{H}^*(X_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)}$$

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