Local invariant cycle theorem

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Short description: Invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths [1][2] 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.[3]

Deligne also proved the following.[4][5] 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

Notes

  1. Clemens 1977, Introduction
  2. Griffiths 1970, Conjecture 8.1.
  3. Beilinson, Bernstein & Deligne 1982, Corollaire 6.2.9.
  4. Deligne 1980, Théorème 3.6.1.
  5. Deligne 1980, (3.6.4.)

References