Why is : $ H^p ( X ( \mathbb{C} ) , \mathbb{Q} ) = \mathrm{Hom}_{ D^b ( \mathrm{pt} ) } ( \mathbb{Q} , ( a_X )_* \mathbb{Q}_X [p] ) $?

Question

There is a paragraph on page $ 133 $ appearing in the following link : https://www-fourier.ujf-grenoble.fr/~peters/Books/Motieven/PureMotives-final.pdf that i don't understand.

The paragraph says :

Recall that if $ X $ is defined over $ \mathbb{C} $ and

$$ a_X \ : \ X \to \mathrm{Spec} \mathbb{C} $$

is the structure morphism, the Betti cohomology $ H^p ( X ( \mathbb{C} ) , \mathbb{Q} ) $ is given by : $$ H^p ( X ( \mathbb{C} ) , \mathbb{Q} ) = \mathrm{Hom}_{ D^b ( \mathrm{pt} ) } ( \mathbb{Q} , ( a_X )_* \mathbb{Q}_X [p] ) $$

• Question :

How to show that : $$ H^p ( X ( \mathbb{C} ) , \mathbb{Q} ) = \mathrm{Hom}_{ D^b ( \mathrm{pt} ) } ( \mathbb{Q} , ( a_X )_* \mathbb{Q}_X [p] ) $$ ?

Thanks in advance for your help.