Some remarks on compactifications of commutative algebraic groups.

*(English)*Zbl 0587.14030Let E be a connected commutative algebraic group defined on the algebraically closed field k. The authors wish to embed projectively E by first completing it and then embedding the compactification \(\bar E.\) Let G be the maximal linear subgroup of E and \(A=E/G\) the abelian variety quotient. If P is a projective G-variety and \(G\to P\) an open G- immersion, then the bundle E(P) with fibre P over A associated to the principal G-bundle \(E\to A\) is a compactification. - In fact, if X is an E-variety, \(i: E\to X\) an open E-immersion, and \(\bar G\) the closure of the G-orbit of i(0), there is an E-morphism \(E(\bar G)\to X\). The authors prove that it is birational and proper, and an isomorphism if X is normal and \(\bar G\) has only finitely many G-orbits. This shows that, under the latter conditions, the above compactifications are the only ones.

The G-linear projective embeddings of E(P) are related to the G-linear line bundles on E(P). For P complete, it is shown that the group of G- linear line bundles on E(P) is the product of the groups of G-linear line bundles on P and all line bundles on A. If k is the field of the complex numbers then \(G={\mathbb{G}}_ a^{(p)}\times {\mathbb{G}}_ m^{(q)}\). A completion of G can be constructed by embedding each factor in \(P^ 1\). Different factorizations of G can lead to different compactifications of E, as an example presented shows. Finally, if the compactification P of G comes from such a factorization, and the projective embedding of E uses a normally generated line bundle on A, the homogeneous ideal of the embedded E(P) is shown to be generated by forms of degree \(\leq \dim A+3.\)

The G-linear projective embeddings of E(P) are related to the G-linear line bundles on E(P). For P complete, it is shown that the group of G- linear line bundles on E(P) is the product of the groups of G-linear line bundles on P and all line bundles on A. If k is the field of the complex numbers then \(G={\mathbb{G}}_ a^{(p)}\times {\mathbb{G}}_ m^{(q)}\). A completion of G can be constructed by embedding each factor in \(P^ 1\). Different factorizations of G can lead to different compactifications of E, as an example presented shows. Finally, if the compactification P of G comes from such a factorization, and the projective embedding of E uses a normally generated line bundle on A, the homogeneous ideal of the embedded E(P) is shown to be generated by forms of degree \(\leq \dim A+3.\)

Reviewer: A.R.Magid