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

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

is surjective. The conjecture was first proved by Clemens. The theorem is also a consequence of the BBD decomposition.^[3]

Deligne also proved the following.^[4][5] Given a proper morphism${\displaystyle X\to S}$ over the spectrum ${\displaystyle S}$ of the henselization of ${\displaystyle k[T]}$ , ${\displaystyle k}$ an algebraically closed field, if ${\displaystyle X}$ is essentially smooth over ${\displaystyle k}$ and ${\displaystyle X_{\overline {\eta }}}$ smooth over ${\displaystyle {\overline {\eta }}}$ , then the homomorphism on ${\displaystyle \mathbb {Q} }$ -cohomology:

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

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

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

• 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]