Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?

Edit: In fact, the following is true: $(\mathrm{Pic}^0(X)_{\mathrm{red}})^\vee = \mathrm{Alb}(X)$ and the Picard scheme is reduced (and then smooth and an Abelian scheme) iff $\dim H^1(X,\mathcal{O}_X) = \dim \mathrm{Pic}^0(X)$.

Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?

Edit: In fact, the following is true: $(\mathrm{Pic}^0(X)_{\mathrm{red}})^\vee = \mathrm{Alb}(X)$, and the Picard scheme is reduced (and then smooth and an Abelian scheme) iff equality holds in $\dim H^1(X,\mathcal{O}_X) \geq \dim \mathrm{Pic}^0(X)$ ("defect of smoothness").

Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?

Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?

Edit: In fact, the following is true: $(\mathrm{Pic}^0(X)_{\mathrm{red}})^\vee = \mathrm{Alb}(X)$ and the Picard scheme is reduced (and then smooth and an Abelian scheme) iff $\dim H^1(X,\mathcal{O}_X) = \dim \mathrm{Pic}^0(X)$.