Abelian variety is commutative. The classical proof for is based on considering the adjoint representation of in the tangent space at . In the general case we consider . Consider . This induce an automorphism of the vector space . Thus we obtain a set-theoretic map . If we put on the target the natural structure of an algebraic variety over , then we can check this is a morphism of varieties. Since the latter is affine and is complete and connected, this map must be constant! Since this means that is identity so reduce to the identity in a neighborhood of in . Since is irreducible, we are done.
The following is true for general group variety: We have an isomorphism of sheaves where . Since is complete, , so everywhere regular forms on are precisely the invariant forms.
We can show given by is surjective for by inspecting it on and using the dimension formula.
Rigidity lemma: If complete variety and is a morphism of varieties such that is mapped to a single point , then there exists such that .
Idea: Define by choosing any and . By the completeness of , for any affine open neighborhood of of , we have is closed in and is nonempty oopen (since ) and satisfies the requirement that gets mapped by into the affine , hence is mapped to a single point.
Corollary: Any morphism between abelian variety is a group homomorphism up to translation.
The functor is linear on the category of complete varieties with base point.
There is even a weaker characterization of abelian varieties: A morphism on a complete variety with a point satisfying makes an abelian variety. See page 44-45.
Any proper morphism of Noetherian schemes with affine, coherent sheaf on , flat over . There is a finite complex of f.g. projective -modules and an isomorphism of functors: on the category of -algebra .
Corollary 1: The dimension of cohomology groups is upper semicontinuous on and the euler characteristic is locally constant on .
Corollary 2: We can derive the proper base change for coherent sheaves, and it also follows that if the dimension of is constant in (so is locally free sheaf and base change holds) then base change holds for (by right exactness of ).
Seesaw theorem (family of line bundles on complete varieties). We use here the homological criterion for triviality of a line bundle and the upper semicontinuity.
Corollary: Theorem of the cube (it more or less says that is a quadratic functor, which is obvious from the exponential exact sequence if , since Kunneth implies that is of order and is of order 1 when is complete so ). Morally the quadratic nature of follows from that of the component group, which is naturally subgroup of .
Corollary: If is a line bundle on , then there is for such that is tensor product of pullback of under projection .
Corollary: computation of (use Theorem of the cube)
Corollary 1 of (14): computation of ((14) implies the second difference is independent of and isomorphic to .
Corollary 2 of (14): Theoerem of square (For any line bundle we have the Abel Jacobi map given by , the fudge factor is because is second order)
A generic effective divisor on is ample (in the sense that the closed subgroup is finite). This implies that abelian varieties are projective.