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 is complete, is connected, and a morphism maps one fiber to a point, then factors through the projection to 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:
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.
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An **Abelian variety** is a fundamental concept in algebraic geometry and is defined as a projective algebraic variety that has the structure of a group variety. More formally, an Abelian variety can be described as follows: 1. **Projective Variety**: It is a complex manifold that can be embedded in projective space \(\mathbb{P}^n\) for some integer \(n\). This means it can be described in terms of polynomial equations.