Local invariant cycle theorem: Difference between revisions

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{{Short description|Invariant cycle theorem}}

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. The theorem is also a consequence of the [[BBD decomposition]].<ref>{{harvnb|Beilinson|Bernstein|Deligne|1982|loc=Corollaire 6.2.9.}}</ref>