# 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.