Local invariant cycle theorem

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

$\operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(p^{-1})(0))^{S^{1}}$

is surjective.