Hodge conjecture

The statement that for any smooth projective variety $ X $ over the field $ \mathbf C $ of complex numbers and for any integer $ p \geq 0 $ the $ \mathbf Q $- space $ H ^ {2p} ( X, \mathbf Q ) \cap H ^ {p,p} $, where $ H ^ {p,p} $ is the component of type $ ( p, p) $ in the Hodge decomposition

$$ H ^ {2p} ( X, \mathbf Q ) \otimes _ {\mathbf Q } \mathbf C = \ \oplus _ {r = 0 } ^ { 2p } H ^ {r, 2p - r } , $$

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

In the case $ p = 1 $, the Hodge conjecture is equivalent to the Lefschetz theorem on cohomology of type $ ( 1, 1) $. The Hodge conjecture has also been proved for the following classes of varieties:

1) $ X $ is a smooth four-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree $ P ^ {1} \times Y \rightarrow X $, where $ Y $ 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) $ X $ is a smooth Fermat hypersurface of prime order (see [4], [5]).

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

4) $ X $ is a simple $ d $- dimensional Abelian variety, and $ \mathop{\rm End} ( X) \otimes _ {\mathbf Z } \mathbf R = \mathbf R ^ {l} $, where $ d/l $ is an odd number, or $ \mathop{\rm End} ( X) \otimes _ {\mathbf Z } \mathbf R = [ M _ {2} ( \mathbf R )] ^ {l} $, where $ d/2l $ 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 $ X $ is an element of $ H ^ {2p} ( X , \mathbf Q ) \cap F ^ { p } H ^ {2p} ( X , \mathbf C ) $ for some $ p $, where $ F ^ { j } H ^ {m} ( X , \mathbf C ) = \sum _ {i \geq j } H ^ {i,m-} i $( 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 $ X $ be a smooth complex projective variety. Suppose that $ M \subseteq H ^ {m} ( X , \mathbf C ) $ is a Hodge substructure such that $ M ^ {i,m-} i = 0 $ for $ i \leq p $. Then there should exist an algebraic subset $ Z $ of $ X $ of codimension $ p $ such that $ M \subseteq \mathop{\rm Ker} ( H ^ {m} ( X , \mathbf C ) \rightarrow H ^ {m} ( X \setminus Z , \mathbf C )) $.

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