View source for Hodge conjecture

The statement that for any smooth projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474601.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474602.png" /> of complex numbers and for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474603.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474604.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474606.png" /> is the component of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474607.png" /> in the Hodge decomposition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474608.png" /></td> </tr></table>

is generated by the cohomology classes of algebraic cycles of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h0474609.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746010.png" />. This conjecture was put forth by W.V.D. Hodge in [[#References|[1]]].

In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746011.png" />, the Hodge conjecture is equivalent to the [[Lefschetz theorem|Lefschetz theorem]] on cohomology of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746012.png" />. The Hodge conjecture has also been proved for the following classes of varieties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746013.png" /> is a smooth four-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746015.png" /> 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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746016.png" /> is a smooth Fermat hypersurface of prime order (see [[#References|[4]]], [[#References|[5]]]).

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746017.png" /> is a simple five-dimensional Abelian variety (see [[#References|[6]]]).

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746018.png" /> is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746019.png" />-dimensional Abelian variety, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746021.png" /> is an odd number, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746023.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746025.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746026.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746028.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746029.png" /> be a smooth complex projective variety. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746030.png" /> is a Hodge substructure such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746032.png" />. Then there should exist an algebraic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746034.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047460/h04746036.png" />.

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>