Local invariant cycle theorem: Difference between revisions

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In mathematics, the '''local invariant cycle theorem''' was originally a conjecture of Griffiths <ref>{{harvnb|Clemens|loc=Introduction}}</ref> which states that, given a proper map <math>p</math> from a [[Kähler manifold]] <math>X</math> to the unit disk that has maximal rank except over 0, each cohomology class on <math>p^{-1}(0)</math> is the restriction of some cohomology class on the entierentire <math>X</math> if the cohomology class is invariant under thea circle action of <math>[0, 2\pi] \to e^{i\theta};</math> in short,

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

== Notes ==

{{reflist}}

== References ==

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

*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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