Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths{{harvnb|Clemens|1977|loc=Introduction}}{{harvnb|Griffiths|1970|loc=Conjecture 8.1.}} which states that, given a surjective proper map $p$ from a Kähler manifold $X$ to the unit disk that has maximal rank everywhere except over 0, each cohomology class on $p^{-1}(t), t \ne 0$ is the restriction of some cohomology class on the entire $X$ if the cohomology class is invariant under a circle action (monodromy action); in short,

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

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

Deligne also proved the following.{{harvnb|Deligne|1980|loc=Théorème 3.6.1.}}{{harvnb|Deligne|1980|loc=(3.6.4.)}} Given a proper morphism $X \to S$ over the spectrum $S$ of the henselization of $k[T]$ , $k$ an algebraically closed field, if $X$ is essentially smooth over $k$ and $X_{\overline{\eta}}$ smooth over $\overline{\eta}$ , then the homomorphism on $\mathbb{Q}$ -cohomology:

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

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

Notes

References

{{cite journal

| last1 = Beilinson

| 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]

Category:Theorems in algebraic geometry

Local invariant cycle theorem

See also

Notes

References