2022/07/15 by Rémi Jaoui, Jaoui, Rémi, Rahim Moosa +1 · 5 citations
Mathematics · Physics and Astronomy · #03C45 #12H05 #32J27 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2207.07515
openalex publication_date 2022/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal image whose binding group is an abelian variety. The properties of such "abelian reductions" are investigated in the Galois-theoretic framework provided by stability theory. Several geometric consequences for the birational geometry of algebraic vector fields of characteristic zero are then deduced. In particular, (1) it is shown that if some cartesian power of an algebraic vector field admits a nontrivial rational first integral then already the second power does, (2) two-dimensional isotrivial algebraic vector fields are classified up to birational equivalence, and (3) algebraic vector fields whose finite covers admit no nontrivial factors are studied in arbitrary dimension. Analogues of these results in bimeromorphic geometry are also obtained.