hodge

THE HODGE CONJECTURE
PIERRE DELIGNE

1. Statement
We recall that a pseudo complex structure on a C ∞ -manifold X of dimension
2N is a C-module structure on the tangent bundle TX . Such a module structure
induces an action of the group C∗ on TX , with λ ∈ C∗ acting by multiplication
by λ. By transport of structures, the group C∗ acts also on each exterior power
∧n TX , as well as on thecomplexified dual Ωn := Hom(∧n TX , C). For p + q = n, a
¯ −q .
(p, q)-form is a section of Ωn on which λ ∈ C∗ acts by multiplication by λ−p λ
From now on, we assume X complex analytic. A (p, q)-form is then a form which,
in local holomorphic coordinates, can be written as
ai1 ,...,ip ,j1 ...jq dzi1 ∧ · · · ∧dzip ∧d¯
zj1 ∧ · · · ∧d¯
zjq ,
and the decomposition Ωn = ⊕Ωp,q induces a decomposition d = d + d ofthe
exterior differential, with d (resp. d ) of degree (1, 0) (resp. (0, 1)).
If X is compact and admits a K¨ahler metric, for instance if X is a projective
non-singular algebraic variety, this action of C∗ on forms induces an action on
cohomology. More precisely, H n (X, C) is the space of closed n-forms modulo exact
forms, and if we define H p,q to be the space of closed (p, q)-forms modulo thed d
of (p − 1, q − 1)-forms, the natural map
(1)

⊕
p+q=n

H p,q → H n (X, C)

is an isomorphism. If we choose a K¨ahler structure on X, one can give the following
interpretation to the decomposition (1) of H n (X, C): the action of C∗ on forms
commutes with the Laplace operator, hence induces an action of C∗ on the space
∼
Hn of harmonic n-forms. We have Hn −→ H n (X, C) and H p,q identifieswith the
space of harmonic (p, q)-forms.
When X moves in a holomorphic family, the Hodge filtration F p := ⊕ H a,n−a
a≥p

of H n (X, C) is better behaved than the Hodge decomposition. Locally on the parameter space T , H n (Xt , C) is independent of t ∈ T and the Hodge filtration can
be viewed as a variable filtration F (t) on a fixed vector space. It varies holomorphically with t, and obeysGriffiths transversality: at first order around t0 ∈ T , F p (t)
remains in F p−1 (t0 ).
So far, we have computed cohomology using C ∞ forms. We could as well have
used forms with generalized functions coefficients, that is, currents. The resulting
groups H n (X, C) and H p,q are the same. If Z is a closed analytic subspace of X,
of complex codimension p, Z is an integral cycle and, by Poincar´e duality,defines
a class cl(Z) in H 2p (X, Z). The integration current on Z is a closed (p, p)-form
with generalized function coefficients, representing the image of cl(Z) in H 2p (X, C).
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PIERRE DELIGNE

The class cl(Z) in H 2p (X, Z) is hence of type (p, p), in the sense that its image in
H 2p (X, C) is. Rational (p, p)-classes are called Hodge classes. They form the group
H 2p (X, Q) ∩ H p,p (X) =H 2p (X, Q) ∩ F p ⊂ H 2p (X, C).
In [6], Hodge posed the
Hodge Conjecture. On a projective non-singular algebraic variety over C, any
Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.
2. Remarks
(i) By Chow’s theorem, on a complex projective variety, algebraic cycles are the
same as closed analytic subspaces.
(ii) On a projective non-singular variety X over C, thegroup of integral linear
combinations of classes cl(Z) of algebraic cycles coincides with the group of integral linear combinations of products of Chern classes of algebraic (equivalently by
GAGA: analytic) vector bundles. To express cl(Z) in terms of Chern classes, one
resolves the structural sheaf OZ by a finite complex of vector bundles. That Chern
classes are algebraic cycles holds,basically, because vector bundles have plenty of
meromorphic sections.
(iii) A particular case of (ii) is that the integral linear combinations of classes of
divisors (= codimension 1 cycles) are simply the first Chern classes of line bundles.
If Z + − Z − is the divisor of a meromorphic section of L, c1 (L) = cl(Z + ) − cl(Z − ).
This is the starting point of the proof given by Kodaira and Spencer [7]...