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
from a Kähler manifold
to the unit disk that has maximal rank everywhere except over 0, each cohomology class on
is the restriction of some cohomology class on the entire
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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
over the spectrum
of the henselization of
,
an algebraically closed field, if
is essentially smooth over
and
smooth over
, then the homomorphism on
-cohomology:
![{\displaystyle \operatorname {H} ^{*}(X_{s})\to \operatorname {H} ^{*}(X_{\overline {\eta }})^{\operatorname {Gal} ({\overline {\eta }}/\eta )}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is surjective, where
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 }}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
Notes
- ^ Clemens 1977, Introduction
- ^ Griffiths 1970, Conjecture 8.1.
- ^ Beilinson, Bernstein & Deligne 1982, Corollaire 6.2.9.
- ^ Deligne 1980, Théorème 3.6.1.
- ^ 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]