Local invariant cycle theorem

Short description: 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 $p^{-1}(t),t\ne 0$ 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,

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

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

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

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

See also

• Hodge theory

References

• Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers" (in French). Astérisque (Paris: Société Mathématique de France) 100. 
• Clemens, C. H. (1977). "Degeneration of Kähler manifolds". Duke Mathematical Journal 44 (2). doi:10.1215/S0012-7094-77-04410-6. 
• Deligne, Pierre (1980). "La conjecture de Weil : II". Publications Mathématiques de l'IHÉS 52: 137–252. doi:10.1007/BF02684780. http://www.numdam.org/item/PMIHES_1980__52__137_0.pdf. 
• 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. https://doi.org/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]