Group variety

A group variety is an algebraic variety whose multiplication and inversion are morphisms.

An abelian variety is a complete connected group variety. Despite its name, commutativity follows from completeness and the rigidity lemma.

One form of Mumford's rigidity lemma says that if $X$ is complete, $Y$ is connected, and a morphism $f:X\times Y\to Z$ maps one fiber $X\times \{y_0\}$ to a point, then $f$ factors through the projection to $Y$ under the usual pointed hypotheses.

Translation identifies the cotangent sheaf of a group scheme with the constant bundle determined by its cotangent space at the identity:

For a line bundle $L$ on an abelian variety and points $x,y$,

For a line bundle $L$ on an abelian variety $X$,is a homomorphism by the Theorem of the square.

For an abelian variety,is the subgroup of translation-invariant line bundles.

The seesaw theorem determines a line bundle on a product from its restrictions to fibers: if it is trivial on all fibers over one factor and on one transverse section, then it is trivial.