 
Summary: NOTES ON THE HODGE CONJECTURE
DONU ARAPURA
Chennai, Jan. 2008
We work over C throughout. Unless stated otherwise, X, Y can be assumed to
be smooth projective varieties. Given a variety X, let Hi
(X) stand for singular
cohomology with rational coefficients for the classical topology. Recall [GH, W]
that we have a Hodge decomposition
Hi
(X) C =
p+q=i
Hpq
(X)
where ¯Hqp
(X) = Hpq
(X) = Hq
(X, p
X) is the space of de Rham cohomology
classes represented by forms of type (p, q). It will be convenient to abstract things
at this point. A (pure rational) Hodge structure H is a rational vector space with a
