Let $ M $ be a compact kahler manifold. Then, we have : $$ H^k ( M , \mathbb{Z} ) \otimes_{ \mathbb{Z} } \mathbb{C} = \bigoplus_{ p + q = k } H^{p,q} ( M ) $$ and : $$ H^{p,q} ( M ) = \overline{ H^{q,p} ( M ) } $$ We know also that : $$ \mathrm{Hom} ( H_k ( M , \mathbb{Z} ) , \mathbb{Z} ) \otimes_{ \mathbb{Z} } \mathbb{C} \simeq H^k ( M , \mathbb{Z} ) \otimes_{ \mathbb{Z} } \mathbb{C} $$
My question is to know if there exists a Hodge decomposition dual to the objet $ \mathrm{Hom} ( H_k ( M , \mathbb{Z} ) , \mathbb{Z} ) $ which we apply the tensorisation functor, $ \bullet \otimes_{ \mathbb{Z} } \mathbb{C} $ in order to have : $$ \mathrm{Hom} ( H_k ( M , \mathbb{Z} ) , \mathbb{Z} ) \otimes_{ \mathbb{Z} } \mathbb{C} = \bigoplus_{ p + q = k } \mathrm{Hom} ( H_{p,q} ( M ) , \mathbb{C} ) $$ and : $$ H_{p,q} ( M ) = \overline{ H_{q,p} ( M )} $$ and : $$ \mathrm{Hom} ( H_{p,q} ( M ) , \mathbb{C} ) \simeq H^{p,q} ( M ) $$
Thanks in advance for your help.