Mathematics

A complex torus T is a complex variety that is the quotient of a complex vector space of dimension n by a discrete subgroup of rank 2n (a "lattice"); if T can be realized as a closed subvariety of some complex projective space then it is called an abelian variety.A complex torus/abelian variety  is  in some sense a linear object, since it has a natural group structure and its main  geometric invariants  can be explicitly described in terms of the lattice. A  smooth complex projective variety  is called *irregular* if admits a non constant map  to a complex torus.  I will sketch  the construction of the Albanese map  of an irregular variety X, namely of  the  "maximal" map from X to a complex torus T, and  discuss some instances of how it  can be used to analyze geometrical properties of X.