Local invariant cycle theorem: Difference between revisions

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{{Short description|Invariant cycle theorem}}

In mathematics, the '''local invariant cycle theorem''' was originally a conjecture of Griffiths <ref>{{harvnb|Clemens|19971977|loc=Introduction}}</ref><ref>{{harvnb|Griffiths|1970|loc=Conjecture 8.1.}}</ref> which states that, given a surjective [[proper map]] <math>p</math> from a [[Kähler manifold]] <math>X</math> to the unit disk that has maximal rank everywhere except over 0, each cohomology class on <math>p^{-1}(0t), t \ne 0</math> is the restriction of some cohomology class on the entire <math>X</math> if the cohomology class is invariant under a circle action (monodromy action); in short,

:<math>\operatorname{H}^*(X) \to \operatorname{H}^*(p^{-1}(0t))^{S^1}</math>

is surjective.<ref>Editorial note: the first proof of the theorem was given by Clemens, apparently but this needs to be checked.</ref>

is surjective. The conjecture was first proved by Clemens. The theorem is also a consequence of the [[BBD decomposition]].<ref>{{harvnb|Beilinson|Bernstein|Deligne|1982|loc=Corollaire 6.2.9.}}</ref>

InDeligne algebraic geometry, Delignealso proved the following analog.<ref>{{harvnb|Deligne|1980|loc=Théorème 3.6.1.}}</ref><ref>{{harvnb|Deligne|1980|loc=(3.6.4.)}}</ref> Given a [[proper morphism]] <math>f : X \to S</math> over the spectrum <math>S</math> of the henselization of <math>k[T]</math>, <math>k</math> an algebraically closed field, if <math>X</math> is essentially smooth<!-- meaning? --> over <math>k</math> and <math>X_{\overline{\eta}}</math> smooth over <math>\overline{\eta}</math>, then the homomorphism on <math>\mathbb{Q}</math>-cohomology:

:<math>\operatorname{H}^*(XX_s) \to \operatorname{H}^*(X_sX_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)}</math>

is surjective, where <math>s, \eta</math> are the special and generic points and the homomorphism is the composition <math>\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 ==

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== References ==

* {{cite journal

*Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. 2, 215–290.

| last1 = Beilinson

*Deligne, Pierre, La conjecture de Weil : II, Publications Mathématiques de l'IHÉS, Tome 52 (1980), pp. 137–252.

| first1 = Alexander A.

| authorlink1 = Alexander Beilinson

| authorlink2 = Joseph Bernstein

| first2=Joseph |last2=Bernstein

| authorlink3=Pierre Deligne

| first3=Pierre |last3=Deligne

| year = 1982

| title = Faisceaux pervers

| journal = Astérisque

| volume = 100

| publisher = [[Société Mathématique de France]]

| ___location=Paris

| language = French

| mr = 0751966

}}

*{{cite journal|s2cid=120378293 |doi=10.1215/S0012-7094-77-04410-6 |title=Degeneration of Kähler manifolds |year=1977 |last1=Clemens |first1=C. H. |journal=Duke Mathematical Journal |volume=44 |issue=2 }}

*{{cite journal |url=http://www.numdam.org/item/PMIHES_1980__52__137_0.pdf|title=La conjecture de Weil : II |journal=Publications Mathématiques de l'IHÉS |year=1980 |volume=52 |pages=137–252 |last1=Deligne |first1=Pierre |doi=10.1007/BF02684780 |s2cid=189769469|mr=601520 | zbl= 0456.14014 }}

*{{cite journal |doi=10.1090/S0002-9904-1970-12444-2 |title=Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems |year=1970 |last1=Griffiths |first1=Phillip A. |journal=Bulletin of the American Mathematical Society |volume=76 |issue=2 |pages=228–296 |doi-access=free }}

*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. [http://web.math.ucsb.edu/~drm/papers/clemens-schmid.pdf]

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[[Category:Theorems in algebraic geometry]]