Local invariant cycle theorem

Template:Short description 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,

${\mathrm{H}}^{\ast }(X)\to {\mathrm{H}}^{\ast }(p^{-1}(t){)}^{S^1}$

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 $X_{\bar{\eta }}$ smooth over $\bar{\eta }$, then the homomorphism on $\mathbb{Q}$-cohomology:

${\mathrm{H}}^{\ast }(X_s)\to {\mathrm{H}}^{\ast }(X_{\bar{\eta }}{)}^{\mathrm{Gal}(\bar{\eta }/\eta )}$

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

See also

• Hodge theory

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. [1]

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