Abelian reduction in differential-algebraic and bimeromorphic geometry
Abstract : Several results on the birational geometry of algebraic vector fields in characteristic zero are obtained. 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 admitting no nontrivial rational first integrals are classified up to birational equivalence, (3) a structural dichotomy is established for algebraic vector fields (of arbitrary dimension) whose finite covers admit no nontrivial factors, and (4) a necessary condition is given on the Albanese map of a smooth projective algebraic variety in order for it to admit an algebraic vector field having no nontrivial rational first integrals. Analogues in bimeromorphic geometry, for families of compact Kaehler manifolds parametrised by Moishezon varieties, are also obtained. These theorems are applications of a new tool here introduced into the model theory of differentially closed fields of characteristic zero (and of compact complex manifolds). In such settings, it is shown that a finite rank type that is internal to the field of constants, C, admits a maximal image whose binding group is an abelian variety in C. The properties of such abelian reductions are investigated. Several consequences for types over constant parameters are deduced by combining the use of abelian reductions with the failure of the inverse differential Galois problem for linear algebraic groups over C. One such consequence is that if p is over constant parameters and not C-orthogonal then the second Morley power of p is not weakly C-orthogonal. Statements (1) through (4) are geometric articulations of these consequences.