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 everywhere except over 0, each cohomology class on $p^{-1}(0)$ is the restriction of some cohomology class on the entire $X$ if the cohomology class is invariant under a circle action; in short,

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

is surjective.

In algebraic geometry, Deligne proved the following analog. Given a proper morphism $f:X\to S$ over the spectrum $S$ of the henselization of $k[T]$, $k$ an algebraically closed field, the homomorphism

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

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

References

• Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. 2, 215–290.
• Deligne, Pierre, La conjecture de Weil : II, Publications Mathématiques de l'IHÉS, Tome 52 (1980), pp. 137–252.
• 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]