There are several natural constructions that give connected algebraic groups that are neither affine nor complete. • If
C is a curve with an effective divisor
m, then it has an associated
generalized Jacobian Jm. This is a commutative algebraic group that maps onto the Jacobian variety
J0 of
C with affine kernel. So
J is an extension of an abelian variety by an affine algebraic group. In general this extension does not split. • The reduced connected component of the relative Picard scheme of a proper scheme over a perfect field is an algebraic group, which is in general neither affine nor proper. • The connected component of the closed fiber of a
Neron model over a discrete valuation ring is an algebraic group, which is in general neither affine nor proper. • For analytic groups some of the obvious analogs of Chevalley's theorem fail. For example, the product of the additive group
C and any elliptic curve has a dense collection of closed (analytic but not algebraic) subgroups isomorphic to
C so there is no unique "maximal affine subgroup", while the product of two copies of the multiplicative group C* is isomorphic (analytically but not algebraically) to a non-split extension of any given elliptic curve by
C. ==Applications==