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").



Albanese dual to the Picard scheme
The nicest modern reference for the theory of the Albanese that I know of is the appendix to [this](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20I.pdf) article of S. Mochizuki.


