On definable Galois groups and the strong canonical base property

Abstract

In [3], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that T has the canonical base property in a strong form; " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than [3], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that 1-based groups are rigid.

Relative internality and definable fibrations

“…We prove that DCF 0 does not have the strong canonical base property, correcting a proof in [14]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles.…”

“…In Section 4, we relate uniform (almost) relative internality to the canonical base property, or more accurately the strong canonical base property from [14]. We give a correct proof that DCF 0 does not have the strong canonical base property.…”

“…

We first elaborate on the theory of relative internality in stable theories from [7], focusing on the notion of uniform relative internality (called collapse of the groupoid in [7]), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA.We prove that DCF 0 does not have the strong canonical base property, correcting a proof in [14]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles.

…”

Additive covers and the Canonical Base Property

“…The question whether a Galois-theoretic interpretation of the CBP exists arose in [13]. We conclude this section by showing that no pure Galois-theoretic account of the CBP can be provided.…”

“…As we will see below this does not hold for all imaginary types. Palacín and Pillay [13] considered a strengthening of the CBP, called the strong canonical base property, which we reformulate in the setting of additive covers: Given a (possibly imaginary) type p = stp(a/B), its canonical base Cb(p) is algebraic over a, d, where stp( d) is P -internal. If we denote by Q the family types over acl eq (∅) which are P -internal, then the strong CBP holds if and only if every Galois group G relative to Q is rigid [13, Theorem 3.4], that is, the connected component of every definable subgroup of G is definable over acl( G ).…”