Hodge conjecture

The statement that for any smooth projective variety over the field of complex numbers and for any integer the -space , where is the component of type in the Hodge decomposition

is generated by the cohomology classes of algebraic cycles of codimension over . This conjecture was put forth by W.V.D. Hodge in [1].

In the case , the Hodge conjecture is equivalent to the Lefschetz theorem on cohomology of type . The Hodge conjecture has also been proved for the following classes of varieties:

1) is a smooth four-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree , where is a smooth variety (see [2]). Uniruled varieties are, for example, the unirational varieties and the four-dimensional complete intersections with an ample anti-canonical class (see [3]).

2) is a smooth Fermat hypersurface of prime order (see [4], [5]).

3) is a simple five-dimensional Abelian variety (see [6]).

4) is a simple -dimensional Abelian variety, and , where is an odd number, or , where is an odd number.

References

[1]	W.V.D. Hodge, "The topological invariants of algebraic varieties" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 1 , Amer. Math. Soc. (1952) pp. 182–192
[2]	A. Conte, J.P. Murre, "The Hodge conjecture for fourfolds admitting a covering by rational curves" Math. Ann. , 238 (1978) pp. 79–88
[3]	A. Conte, J.P. Murre, "The Hodge conjecture for Fano complete intersections of dimension four" , J. de Géométrie Algébrique d'Angers, juillet 1979 , Sijthoff & Noordhoff (1980) pp. 129–141
[4]	Z. Ran, "Cycles on Fermat hypersurfaces" Compositio Math. , 42 : 1 (1980–1981) pp. 121–142
[5]	T. Shioda, "The Hodge conjecture and the Tate conjecture for Fermat varieties" Proc. Japan. Acad. Ser. A , 55 : 3 (1979) pp. 111–114
[6]	S.G. Tankeev, "On algebraic cycles on simple -dimensional abelian varieties" Math. USSR Izv. , 19 (1982) pp. 95–123 Izv. Akad. Nauk SSSR Ser. Mat. , 45 : 4 (1981) pp. 793–823

Comments

A Hodge class on a smooth complex projective variety is an element of for some , where (the Hodge filtration, cf. Hodge structure). The Hodge conjecture regards the algebraicity of the Hodge classes.

A weaker form is the variational Hodge conjecture. Suppose one has a smooth family of complex projective varieties and a locally constant cohomology class in the fibres which is everywhere a Hodge class and is algebraic at one fibre. Then it should be algebraic in nearby fibres. This has been verified in certain cases [a1], [a2].

An absolute Hodge class on a projective variety over a number field is a certain compatible system of cohomology classes in Betti, de Rham and étale cohomology. On an Abelian variety, every Hodge class is a Betti component of an absolute Hodge class [a3]. Absolute Hodge classes are used to define a weak notion of motif for algebraic varieties.

Hodge has formulated a more general conjecture, corrected by A. Grothendieck [a4]. Let be a smooth complex projective variety. Suppose that is a Hodge substructure such that for . Then there should exist an algebraic subset of of codimension such that .

More general conjectures of this type are due to A. Beilinson [a5].

References

[a1]	S. Bloch, "Semi-regularity and de Rham cohomology" Invent. Mat. , 17 (1972) pp. 51–66
[a2]	J.H.M. Steenbrink, "Some remarks about the Hodge conjecture" E. Cattani (ed.) F. Guillán (ed.) A. Kaplan (ed.) et al. (ed.) , Hodge theory , Lect. notes in math. , 1246 , Springer pp. 165–175
[a3]	P. Deligne (ed.) J.S. Milne (ed.) A. Ogus (ed.) K. Shih (ed.) , Hodge cycles, motives and Shimura varieties , Lect. notes in math. , 900 , Springer (1982)
[a4]	A. Grothendieck, "Hodge's general conjecture is false for trivial reasons" Topology , 8 (1969) pp. 299–303
[a5]	A.A. Beilinson, "Notes on absolute Hodge cohomology" Contemp. Math. , 55 : 1 (1986) pp. 35–68