Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths ^[1] which states that, given a proper map $p$ from a Kähler manifold $X$ to the unit disk that has maximal rank except over 0, each cohomology class on $p^{-1}(0)$ is the restriction of some cohomology class on the entire $X$ if the cohomology class is invariant under a circle action; in short,

\operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(p^{-1}(0))^{S^{1}}

is surjective.^[2]

Notes

^ Clemens, Introduction
^ Editorial note: the first proof of the theorem was given by Clemens, apparently but this needs to be checked.

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. [1]

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[1] Clemens, Introduction

[2] Editorial note: the first proof of the theorem was given by Clemens, apparently but this needs to be checked.

[1]

[2]