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 [math]\displaystyle{ p }[/math] from a Kähler manifold [math]\displaystyle{ X }[/math] to the unit disk that has maximal rank everywhere except over 0, each cohomology class on [math]\displaystyle{ p^{-1}(t), t \ne 0 }[/math] is the restriction of some cohomology class on the entire [math]\displaystyle{ X }[/math] if the cohomology class is invariant under a circle action (monodromy action); in short,

[math]\displaystyle{ \operatorname{H}^*(X) \to \operatorname{H}^*(p^{-1}(t))^{S^1} }[/math]

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 [math]\displaystyle{ X \to S }[/math] over the spectrum [math]\displaystyle{ S }[/math] of the henselization of [math]\displaystyle{ k[T] }[/math], [math]\displaystyle{ k }[/math] an algebraically closed field, if [math]\displaystyle{ X }[/math] is essentially smooth over [math]\displaystyle{ k }[/math] and [math]\displaystyle{ X_{\overline{\eta}} }[/math] smooth over [math]\displaystyle{ \overline{\eta} }[/math], then the homomorphism on [math]\displaystyle{ \mathbb{Q} }[/math]-cohomology:

[math]\displaystyle{ \operatorname{H}^*(X_s) \to \operatorname{H}^*(X_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)} }[/math]

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

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]