Abelian reduction in differential-algebraic and bimeromorphic geometry

TL;DR

This paper develops a new model-theoretic tool called abelian reduction for differential and bimeromorphic geometry, revealing structural properties of algebraic vector fields and their integrability.

Contribution

It introduces abelian reductions for types internal to constants or the projective line, with applications to birational geometry and classification of algebraic vector fields.

Findings

Nontrivial rational first integrals in powers imply the second power also admits one.

Two-dimensional isotrivial algebraic vector fields are classified up to birational equivalence.

Vector fields with no nontrivial factors in finite covers are characterized in arbitrary dimensions.

Abstract

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.