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 $p^{-1}(t),t\neq 0$ 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,

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

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

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

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

See also

• Hodge theory

References

• Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers". Astérisque (in French). 100. Paris: Société Mathématique de France. MR 0751966.
• Clemens, C. H. (1977). "Degeneration of Kähler manifolds". Duke Mathematical Journal. 44 (2). doi:10.1215/S0012-7094-77-04410-6. S2CID 120378293.
• Deligne, Pierre (1980). "La conjecture de Weil : II" (PDF). Publications Mathématiques de l'IHÉS. 52: 137–252. doi:10.1007/BF02684780. MR 0601520. S2CID 189769469. Zbl 0456.14014.
• Griffiths, Phillip A. (1970). "Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems". Bulletin of the American Mathematical Society. 76 (2): 228–296. doi:10.1090/S0002-9904-1970-12444-2.
• 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]