Non-commutative geometry and new stable structures

نویسنده

  • Boris Zilber
چکیده

1 This paper grew out of an observation that some new stable structures discovered in the 1990' as counterexamples to well-known conjectures in pure model theory might be related to non-commutative geometry. The general meaning of the conjectures was that " very good " , or more technically, very stable structures must be in a certain way reducible to algebraic geometry over algebraically closed fields or to linear structures (Tri-chotomy conjecture and Algebraicity conjecture for groups, see [Z0]). This proved to be true to some extent (see [HZ]) but still two types of counterexamples signal the necessity to reconsider the connection between model theoretic classification principles and classical mathematics. The first class of counterexamples shows that nonlinear one-dimensional Zariski geometries are not necessarily algebraic curves. Given a smooth algebraic curve C with big enough group of regular automorphisms one can produce a " smooth " Zariski curve˜C along with a finite cover p : ˜ C → C. ˜ C can not be identified with any algebraic curve because the construction produces an unusual subgroup of the group of regular automorphisms of˜C ([HZ, section 10). The main theorem of [HZ] states that it is the biggest deviation from an algebraic curve that can happen to a Zariski curve. Typical example of an unusual subgroup of a such˜C is the nilpotent group of two generators U and V with the central commutator = [U, V] of finite order N. So, the defining relations are UV = VU, N = 1. This, of course, hints towards the known structure of non-commutative geometry , the non-commutative (quantum) torus at the N th root of unity. We call this example T N. The other example is of a different nature. B.Poizat constructed in [P] a multiplicative subgroup G of an algebraically closed field (we may assume this to be the field C of complex numbers) such that (C, +, ·, G) has ω-stable theory of rank half of that of C (so called " bad field " , related to the Algebraicity conjecture). The present author has shown in [Z2] that, assuming Schanuel's conjecture, one can construct G by means of real analytic geometry. More specifically one can consider G of the form G = exp(αZ) · exp(βR), α and β linearly independent over R, β / ∈ R ∪ iR, and see that (C, +, ·, G) is superstable of dimension half of …

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تاریخ انتشار 2005