In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective proper map p {\displaystyle p} from a Kähler manifold X {\displaystyle X} to the unit disk that has maximal rank everywhere except over 0, each cohomology class on p − 1 ( t ) , t ≠ 0 {\displaystyle p^{-1}(t),t\neq 0} is the restriction of some cohomology class on the entire X {\displaystyle X} if the cohomology class is invariant under a circle action (monodromy action); in short,

H ∗ ⁡ ( X ) → H ∗ ⁡ ( p − 1 ( t ) ) S 1 {\displaystyle \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 → S {\displaystyle X\to S} over the spectrum S {\displaystyle S} of the henselization of k [ T ] {\displaystyle k[T]}, k {\displaystyle k} an algebraically closed field, if X {\displaystyle X} is essentially smooth over k {\displaystyle k} and X η ¯ {\displaystyle X_{\overline {\eta }}} smooth over η ¯ {\displaystyle {\overline {\eta }}}, then the homomorphism on Q {\displaystyle \mathbb {Q} }-cohomology:

H ∗ ⁡ ( X s ) → H ∗ ⁡ ( X η ¯ ) Gal ⁡ ( η ¯ / η ) {\displaystyle \operatorname {H} ^{*}(X_{s})\to \operatorname {H} ^{*}(X_{\overline {\eta }})^{\operatorname {Gal} ({\overline {\eta }}/\eta )}}

is surjective, where s , η {\displaystyle s,\eta } are the special and generic points and the homomorphism is the composition H ∗ ⁡ ( X s ) ≃ H ∗ ⁡ ( X ) → H ∗ ⁡ ( X η ) → H ∗ ⁡ ( X η ¯ ) . {\displaystyle \operatorname {H} ^{*}(X_{s})\simeq \operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(X_{\eta })\to \operatorname {H} ^{*}(X_{\overline {\eta }}).}

See also

Notes

  • Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers". Astérisque (in French). 100. Paris: Société Mathématique de France. MR.
  • Clemens, C. H. (1977). "Degeneration of Kähler manifolds". Duke Mathematical Journal. 44 (2). doi:. S2CID.
  • Deligne, Pierre (1980). (PDF). Publications Mathématiques de l'IHÉS. 52: 137–252. doi:. MR. S2CID. Zbl.
  • Griffiths, Phillip A. (1970). . Bulletin of the American Mathematical Society. 76 (2): 228–296. doi:.
  • 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.