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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 [[#References|[1]]].

In the case  $  p = 1 $, 
the Hodge conjecture is equivalent to the [[Lefschetz theorem|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 [[#References|[2]]]). Uniruled varieties are, for example, the unirational varieties and the four-dimensional complete intersections with an ample anti-canonical class (see [[#References|[3]]]).

2)  $  X $
is a smooth Fermat hypersurface of prime order (see [[#References|[4]]], [[#References|[5]]]).

3)  $  X $
is a simple five-dimensional Abelian variety (see [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.V.D. Hodge,   "The topological invariants of algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''1''' , Amer. Math. Soc.  (1952)  pp. 182–192</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Conte,   J.P. Murre,   "The Hodge conjecture for fourfolds admitting a covering by rational curves"  ''Math. Ann.'' , '''238'''  (1978)  pp. 79–88</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Conte,   J.P. Murre,   "The Hodge conjecture for Fano complete intersections of dimension four" , ''J. de Géométrie Algébrique d'Angers, juillet 1979'' , Sijthoff &amp; Noordhoff  (1980)  pp. 129–141</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Z. Ran,   "Cycles on Fermat hypersurfaces"  ''Compositio Math.'' , '''42''' :  1  (1980–1981)  pp. 121–142</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T. Shioda,   "The Hodge conjecture and the Tate conjecture for Fermat varieties"  ''Proc. Japan. Acad. Ser. A'' , '''55''' :  3  (1979)  pp. 111–114</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.G. Tankeev,   "On algebraic cycles on simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746024.png" />-dimensional abelian varieties"  ''Math. USSR Izv.'' , '''19'''  (1982)  pp. 95–123  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''45''' :  4  (1981)  pp. 793–823</TD></TR></table>

====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|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 [[#References|[a1]]], [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bloch,   "Semi-regularity and de Rham cohomology"  ''Invent. Mat.'' , '''17'''  (1972)  pp. 51–66</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Grothendieck,   "Hodge's general conjecture is false for trivial reasons"  ''Topology'' , '''8'''  (1969)  pp. 299–303</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.A. Beilinson,   "Notes on absolute Hodge cohomology"  ''Contemp. Math.'' , '''55''' :  1  (1986)  pp. 35–68</TD></TR></table>