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 $p^(t), t \ne 0$ 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,
:$\operatorname^*(X) \to \operatorname^*(p^(t))^$
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 /math>, $k$ an algebraically closed field, if $X$ is essentially smooth over $k$ and $X_$ smooth over $\overline$, then the homomorphism on $\mathbb$-cohomology:
:$\operatorname^*(X_s) \to \operatorname^*(X_)^$
is surjective, where $s, \eta$ are the special and generic points and the homomorphism is the composition $\operatorname^*(X_s) \simeq \operatorname^*(X) \to \operatorname^*(X_) \to \operatorname^*(X_).$

See also

*Hodge theory

Notes

References

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*Morrison, David R. The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), 101-119, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984

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Mathematics