Kinds of differentials and algebraic groups

Question

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess differentials of the first kind and abelian integrals & Jacobian are related, but it seems still like a loose connection to me and I do not see how this can be extended to other kinds of differentials. Is there a clean picture for this?

Answer 1

Two notes: First, Wikipedia says affine spaces, not linear algebraic groups, and affine spaces (together with the vector space group structure) are really meant here, not more general linear algebraic groups. Second, Wikipedia mentions the parenthetically the generalized Jacobian, and that really is key to understanding what they are talking about.

By definition, the generalized Jacobian of a curve $X$ with respect to a divisor $D$ is the moduli space of pairs of a line bundle together with a trivialization over $D$.

This is a commutative algebraic group, the extension of the Jacobian of $X$ (an abelian variety), first by an algebraic torus of rank the size of the support of $D$ minus $1$, and then by a commutative unipotent algebraic group (in characteristic zero, an affine space) of dimension the degree of $D$ minus the size of the support of $D$.

So the generalized Jacobian really combines abelian varieties, algebraic tori, and affine spaces. What is the relation to differentials?

The cotangent space of the generalized Jacobian (at any point) is naturally isomorphic to the space of differentials on $X$ with divisor of poles at most $D$, i.e. at a point of $D$ with multiplicity $m$, we allow poles of order at most $m$.

So how do we match up the three kinds of differentials with the three kinds of algebraic groups?

Well, I don't see how it can be made exactly one-to-one, but:

• The space of differentials of the first kind, i.e. differentials with no poles, is naturally isomorphic to the cotangent space of the Jacobian, i.e. the cotangent space of the abelian variety part of the generalized Jacobian, since it's the same as the generalized Jacobian with divisor zero.

• The space of differentials of the second kind with poles $\leq D$, i.e. differentials with only simple poles, only contained in the support of $D$, is naturally isomorphic to the cotangent space of the maximal quotient of the generalized Jacobian that's an extension of an abelian variety by a torus. So the space of differentials of the second kind, modulo differentials of the first kind, is isomorphic to the contangent space of the torus part of the generalized Jacobian.

• Finally, the space of differentials of the third kind with poles $\leq D$, modulo differentials of the first kind, is isomorphic to the contangent space of the unipotent/affine space part of the generalized Jacobian. To see this, one can check using the residue theorem that the natural map from differentials of the third kind with poles $\leq D$ modulo differentials of the first kind to all differentials with poles $\leq D$ modulo differentials of the second kind is an isomorphism. Since the differentials of the second kind form the cotangent space of the quotient of the generalized Jacobian by the maximal unipotent subgroup, the quotient of all differentials by differentials of the second kind gives the cotangent space of the maximal unipotent subgroup.

Answer 2

Actually there is. Differentials of the first kind are slightly different from abelian and Jacobian integrals. Abelian differentials are quite simply holomorphic/meromorphic on a closed Riemann space. They are related to the Jacobian in that inverting Abelian integrals of the first kind results in the compact Riemann surface $S$ of genus $p\geq 1$ that corresponds to a given algebraic equation $F(z,w)=0$.

Here are the sources:

https://encyclopediaofmath.org/wiki/Abelian_differential.

https://encyclopediaofmath.org/wiki/Jacobi_inversion_problem.