Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.The theory was first put into a good form when K was the field of complex numbers. In that case there is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil had given a general definition of Albanese variety. For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic0(A), the connected component of what in contemporary terminology is the Picard scheme.