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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
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<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>
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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]]].
+
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
  
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:
+
$$
 +
H  ^ {2p} ( X, \mathbf Q ) \otimes _ {\mathbf Q }  \mathbf C  = \
 +
\oplus _ {r = 0 } ^ { 2p }  H ^ {r, 2p - r } ,
 +
$$
  
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]]]).
+
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]]].
  
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]]]).
+
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:
  
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]]]).
+
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]]]).
  
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.
+
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====
 
====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>
 
<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====
 
====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 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]]].
 
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]]].
Line 27: Line 66:
 
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.
 
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" />.
+
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]]].
 
More general conjectures of this type are due to A. Beilinson [[#References|[a5]]].

Revision as of 22:10, 5 June 2020


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
How to Cite This Entry:
Hodge conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodge_conjecture&oldid=16196
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article