In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths ^[1][2] which states that, given a surjective proper map ${\displaystyle p}$ from a Kähler manifold ${\displaystyle X}$ to the unit disk that has maximal rank everywhere except over 0, each cohomology class on ${\displaystyle p^{-1}(t),t\neq 0}$ is the restriction of some cohomology class on the entire ${\displaystyle X}$ if the cohomology class is invariant under a circle action (monodromy action); in short,

${\displaystyle \operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(p^{-1}(t))^{S^{1))}$

is surjective. The conjecture was first proved by Clemens. The theorem is also a consequence of the BBD decomposition.^[3]

Deligne also proved the following.^[4][5] Given a proper morphism ${\displaystyle X\to S}$ over the spectrum ${\displaystyle S}$ of the henselization of ${\displaystyle k[T]}$, ${\displaystyle k}$ an algebraically closed field, if ${\displaystyle X}$ is essentially smooth over ${\displaystyle k}$ and ${\displaystyle X_{\overline {\eta ))}$ smooth over ${\displaystyle {\overline {\eta ))}$, then the homomorphism on ${\displaystyle \mathbb {Q} }$-cohomology:

${\displaystyle \operatorname {H} ^{*}(X_{s})\to \operatorname {H} ^{*}(X_{\overline {\eta )))^{\operatorname {Gal} ({\overline {\eta ))/\eta )))$

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