On the canonical base property
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On the canonical base property Ehud Hrushovski · Daniel Palacín · Anand Pillay
Published online: 14 June 2013 © Springer Basel 2013
Abstract We give an example of a finite rank, in fact ℵ1 -categorical, theory where the canonical base property (CBP) fails. In fact, we give a “group-like” example in a sense that we will describe below. We also prove, in a finite Morley rank context, that if all definable Galois groups are “rigid,” then T has the CBP. Keywords
Stable theory · ℵ1 -categoricity · Definable Galois group · CBP
Mathematics Subject Classification (2000)
03C45
1 Introduction and preliminaries The canonical base property (CBP) is a property appropriate for finite rank theories, the formulation of which was motivated by results of Campana in bimeromorphic geometry and analogous results by Pillay and Ziegler in differential and difference algebraic geometry in characteristic 0. The notion has been studied by Chatzidakis [1], Moosa and Pillay [3] (where the expression CBP was introduced) and in a somewhat
E. Hrushovski Hebrew University of Jerusalem, Jerusalem, Israel e-mail: [email protected] D. Palacín Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Spain e-mail: [email protected] A. Pillay (B) University of Leeds, Leeds, England e-mail: [email protected]
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more general framework by Palacín and Wagner [4]. The notion makes sense for arbitrary supersimple theories of finite SU -rank. But to avoid unnecessary abstraction, we will restrict ourselves here to stable theories T of finite Morley rank (and even more). We now state the CBP giving definitions of some of the ingredients in an appropriate context later in this section. Definition 1.1 T has the CBP if (working in M eq for a saturated M | T , and over any small set of parameters), for any a, b such that tp(a/b) is stationary and b is the canonical base of tp(a/b), tp(b/a) is semiminimal, namely almost internal to the family of U -rank 1 types. Here, the canonical base of a stationary type p(x) = tp(a/B), written Cb( p), is the smallest definably closed subset B0 of dcl(B) such that p does not fork over B0 and p|B0 is stationary. Given our totally transcendental hypothesis on T, B0 will be the definably closure of a finite subtuple b0 and we write b0 for Cb( p). Remark 1.2 Chatzidakis shows in [1] that with notation as above tp(b/a) is always analyzable in the family of nonmodular U -rank 1 types (which we know to be of Morley rank 1). Hence, the CBP is equivalent to saying that tp(b/a) is almost internal to the family of nonmodular strongly minimal definable sets. When T is the theory of the many-sorted structure CCM of compact complex manifolds (with predicates for analytic subvarieties of products of sorts), Pillay [6] noted that results of Campana yield that the CBP holds in the strong sense that tp(b/a) is internal to the sort of the projective line over C. This gives another proof that this sort is the only nonmodula
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