Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties — Examples

The group laws on $A$ and ${\mathbf{G}}_m$ give the group laws on their product. The latter group's affine Hopf formulas are $t\mapsto t\otimes t$, $t\mapsto t^{-1}$, and $t\mapsto 1$, so the group laws are regular. The projection $q(a,t)=a$ is a homomorphism, split by $a\mapsto (a,1)$, and its scheme-theoretic kernel is $\{e_A\}\times {\mathbf{G}}_m$. It is normal because conjugation in the product preserves this factor. It is affine by its displayed spectrum, smooth because it is an open subscheme of the affine line, and geometrically connected because $K[t,t^{-1}]$ is a domain for every field extension $K\mathrm{/}k$.

For every $k$-scheme $T$, $q:G(T)=A(T)\times \mathrm{\Gamma }(T,{\mathcal{O}}_T{)}^{\times }\to A(T)$ is onto, and two elements have the same image exactly when they differ by the action of a unique ${\mathbf{G}}_m(T)$ element. Thus the presheaf quotient is already the represented functor $A(-)$, naturally in $T$; by [F4], its fppf sheafification is the same represented functor. This proves the asserted scheme quotient.

The section $A\times \{1\}\subset G$ is closed since $\{1\}\subset \mathrm{Spec}k[t,t^{-1}]$ is defined by $t-1$. If $G$ were affine, [F3] would make that copy of $A$ affine. For $\dim A>0$ this contradicts [F2]. Hence $G$ is a nonaffine algebraic group in that case. AC is carried through [F2].

By [F1] and [F3], $E$ is a smooth projective one-dimensional scheme over $\mathbf{C}$, hence proper: a closed subscheme of proper projective space is finite type and separated, and its projection remains closed after arbitrary base change. Its complex manifold is connected by the biholomorphism with the connected torus. The irreducible components of a smooth algebraic scheme cannot meet, since its regular local rings are domains; the finitely many components are therefore both algebraically and analytically open and closed. Connectedness leaves exactly one, so $E$ is integral and, over the algebraically closed field $\mathbf{C}$, geometrically integral.

Transported addition on $E(\mathbf{C})$ is jointly holomorphic: around any two points lift to torus coordinates $z,w$, use the holomorphic map $(z,w)\mapsto z+w$, and compose with the biholomorphism $\mathrm{\Phi }$ and its inverse from [F1]. On the dense algebraic open where $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ are affine and $x_1\mathrm{\ne }{x}_2$, set $m=(y_2-y_1)\mathrm{/}(x_2-x_1)$. The chord meets the cubic at a third point with $x$-coordinate $m^2\mathrm{/}4-x_1-x_2$, by comparison of the cubic's quadratic coefficient. Negating its $y$-coordinate gives the rational formulas $x(P+Q)=m^2\mathrm{/}4-x_1-x_2$ and $y(P+Q)=-y_1+m(x_1-x(P+Q))$. They agree with the holomorphic group law by [F1]. Apply [F2] to get a regular algebraic multiplication $E\times E\to E$. Inverse is the regular projective map $[X:Y:Z]\mapsto [X:-Y:Z]$, and the identity is the rational point $O$.

Associativity, inverse, and identity hold on every complex point by transport from the torus. They hold as scheme morphism identities: the relevant product schemes are reduced, their complex closed points are Zariski dense, and equality is a closed condition because the target is separated. Consequently $E$ is a group variety, and properness and geometric connectedness from step 1.1 make it an abelian variety by [F4]. Since $\dim E=1$, the nonaffineness criterion in [F4] proves that $E$ is not affine. This verification explicitly establishes regularity of the algebraic law; analytic uniformization alone was not treated as that supplier. AC is carried through [F2]–[F4].