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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,

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:

is surjective, where are the special and generic points and the homomorphism is the composition

See also


  1. ^ Clemens 1977, Introduction
  2. ^ Griffiths 1970, Conjecture 8.1.
  3. ^ Beilinson, Bernstein & Deligne 1982, Corollaire 6.2.9.
  4. ^ Deligne 1980, Théorème 3.6.1.
  5. ^ Deligne 1980, (3.6.4.)


  • Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers". Astérisque (in French). Paris: Société Mathématique de France. 100. 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]