TY - JOUR
T1 - On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0
AU - Dill, Gabriel Andreas
N1 - Funding Information:
Open Access funding enabled and organized by Projekt DEAL. This work was supported by the Swiss National Science Foundation through the Early Postdoc.Mobility grant no. P2BSP2_195703. I thank the Mathematical Institute of the University of Oxford and my host there, Jonathan Pila, for hosting me as a visitor for the duration of this grant. During the final revisions of this manuscript, I was employed by the Leibniz Universität Hannover on the ERC grant of Ziyang Gao.
Gabriel Dill has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n 945714).
PY - 2022/7/26
Y1 - 2022/7/26
N2 - Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.
AB - Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.
UR - http://www.scopus.com/inward/record.url?scp=85134811721&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2107.14667
DO - 10.48550/arXiv.2107.14667
M3 - Article
AN - SCOPUS:85134811721
JO - Transformation Groups
JF - Transformation Groups
SN - 1083-4362
ER -