Let $ X $ be a compact complex algebraic manifold of dimension $ n $.
For each integer $ p \in \mathbb{N} $, let $ H^{p,p} (X) $ denotes the subspace of $ H^{2p} (X, \mathbb{C} ) $ of type $ (p,p) $.
The group of rational $ (p,p) $ - cycles : $ H^{p,p} (X , \mathbb{Q} ) = H^{2p} ( X , \mathbb{Q} ) \cap H^{p,p} (X) $ is called the group of rational Hodge classes of type $ (p,p) $.
An $ r $ -cycle of an algebraic variety $X$ is a formal finite linear combination $ \displaystyle \sum_{ i \in [1,h] } m_i Z_i $ of closed irreducible subvarieties $ Z_i $ of dimension $ r $ with integer coefficients $ m_i $.
The group of $ r $ - cycles is denoted by $ \mathcal{Z}_r (X) $.
On a compact complex algebraic manifold, the class of closed irreducible subvarieties of codimension $ p $ extends to a linear morphism :
$$ \mathrm{cl}_{ \mathbb{Q} } \ : \ \mathcal{Z}_p (X) \otimes \mathbb{Q} \to H^{p,p} (X, \mathbb{Q} ) $$ defined by, $$ \mathrm{cl}_{ \mathbb{Q} } \big( \displaystyle \sum_{ i \in [1,h] } m_i Z_i \big) = \displaystyle \sum_{ i \in [1,h] } m_i \eta_{Z_{i}} , \ \ \forall m_i \in \mathbb{Q} $$
The elements of the image of $ \mathrm{cl}_{ \mathbb{Q} } $ are called rational algebraic Hodge classes of type $ (p,p) $.
The Hodge conjecture says :
On a non-singular complex projective variety, any rational Hodge class of type $ (p,p) $ is algebraic, i.e : in the image of $ \mathrm{cl}_{ \mathbb{Q} } $.
- Question,
In the last statement of the Hodge conjecture above, How to prove that, $ \exists Z \in \mathcal{Z}_p (X) \otimes \mathbb{Q} $, such that, $ \mathrm{cl}_{ \mathbb{Q} } (Z) \neq 0 $ ?.
In other words, How to show that, $ \mathrm{Im} \ \mathrm{cl}_{ \mathbb{Q} } \neq \{ 0 \} $ ?
Thanks in advance for your answers.