Strongly minimal groups in o-minimal structures. (English) Zbl 1468.14102

Eleftheriou, Pantelis E.; Hasson, Assaf; Peterzil, Ya’acov

J. Eur. Math. Soc. (JEMS) 23, No. 10, 3351-3418 (2021).

B. Zilber’s trichotomy conjecture introduced in [Sibirsk. Mat. Zh. 25, 71–88 (1984; Zbl 0581.03022)] was disproved by E. Hrushovski [J. Amer. Math. Soc. 62, 147–166 (1993; Zbl 0804.03020)]. However, this conjecture is true in various restricted settings. This paper proves Zilber’s trichotomy conjecture for strongly minimal expansion of \(2\)-dimensional groups, definable in o-minimal structures. The main theorem is as follows:
Let \(\mathcal{M}\) be an o-minimal expansion of a real closed field, \(\langle G;\ +\rangle\) be a \(2\)-dimensional group definable in \(\mathcal{M}\), and \(\mathcal D=\langle G;\ +, \ldots\rangle\) be a strongly minimal structure, all of whose atomic relations definable in \(\mathcal{M}\). If \(\mathcal D\) is not locally modular, then an algebraic closed field \(K\) is interpretable in \(\mathcal D\), and the group \(G\), with all its induced \(\mathcal D\)-structure, is definably isomorphic to an algebraic \(K\)-group with all its induced \(K\)-structure.
It is a generalization of [A. Hassen et. al., Proc. London Math. Soc. (3) 97, 117–154 (2008; Zbl 1153.03011)] which treats the case in which \(G\) is the algebraic closure \(K=R[i]\) and \(\mathcal D\) is a structure generated by an \(\mathcal M\)-definable function, and its proof follows the same strategy as the Hassen’s paper; that is, constructing a field configuration and using Hrushovski’s result that a strongly minimal structure admitting a field construction interprets an algebraically closed field.
A \(\mathcal D\)-definable subset of \(G^2\) whose Morley rank is one is called a plane curve in this paper. The paper establishes the necessary ingredients for the proof of the main theorem in several distinct steps. In each step, resemblances of \(\mathcal D\)-definable sets to complex algebraic sets are demonstrated including finiteness of the frontiers of plane curves, finiteness of their poles and their intersection theory. The algebraically closed field \(K\) is defined as the collection of all Jacobian matrices at zero of local smooth maps from \(G\) to \(G\) whose graph is contained in a plane curve by identifying them with the matrices in \(M_2(R)\).

Reviewer: Fujita Masato (Kure)

MSC:

14P25	Topology of real algebraic varieties
03C64	Model theory of ordered structures; o-minimality
03C45	Classification theory, stability, and related concepts in model theory

Keywords:

o-minimality; strongly minimal groups; Zilber’s conjecture

Citations:

Zbl 0581.03022; Zbl 0804.03020; Zbl 1153.03011

References:

[1]	Bakker, B., Klingler, B., Tsimerman, J.: Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Amer. Math. Soc.33, 917-939 (2020) Zbl07268734 MR4155216 · Zbl 1460.14027
[2]	Berarducci, A., Otero, M.: Intersection theory for o-minimal manifolds. Ann. Pure Appl. Logic107, 87-119 (2001) Zbl0968.03044MR1807841 · Zbl 0968.03044
[3]	Bouscaren, E.: Model theoretic versions of Weil’s theorem on pregroups. In: The Model Theory of Groups (Notre Dame, IN, 1985-1987), Notre Dame Math. Lectures 11, Univ. Notre Dame Press, Notre Dame, IN, 177-185 (1989) Zbl0794.03050MR985345 · Zbl 0794.03050
[4]	Bouscaren, E.: The group configuration—after E. Hrushovski. In: The Model Theory of Groups (Notre Dame, IN, 1985-1987), Notre Dame Math. Lectures 11, Univ. Notre Dame Press, Notre Dame, IN, 199-209 (1989) Zbl0792.03017MR985348 · Zbl 0792.03017
[5]	Hasson, A., Kowalski, P.: Strongly minimal expansions of.C;C/definable in o-minimal fields. Proc. London Math. Soc. (3)97, 117-154 (2008) Zbl1153.03011MR2434093 · Zbl 1153.03011
[6]	Hasson, A., Onshuus, A., Peterzil, Y.: Definable one dimensional structures in o-minimal theories. Israel J. Math.179, 297-361 (2010) Zbl1213.03049MR2735046 · Zbl 1213.03049
[7]	Hasson, A., Sustretov, D.: Incidence systems on Cartesian powers of algebraic curves. arXiv:1702.05554(2017)
[8]	Hrushovski, E.: Locally modular regular types. In: Classification Theory (Chicago, IL, 1985), Lecture Notes in Math. 1292, Springer, Berlin, 132-164 (1987) Zbl0643.03024 MR1033027 · Zbl 0643.03024
[9]	Hrushovski, E.: Strongly minimal expansions of algebraically closed fields. Israel J. Math. 79, 129-151 (1992) Zbl0773.12005MR1248909 · Zbl 0773.12005
[10]	Hrushovski, E.: A new strongly minimal set. J. Amer. Math. Soc.62, 147-166 (1993) Zbl0804.03020MR1226304 · Zbl 0804.03020
[11]	Hrushovski, E.: The Mordell-Lang conjecture for function fields. J. Amer. Math. Soc.9, 667-690 (1996) Zbl0864.03026MR1333294 · Zbl 0864.03026
[12]	Hrushovski, U., Pillay, A.: Weakly normal groups. In: Logic Colloquium ’85 (Orsay, 1985), Stud. Logic Found. Math. 122, North-Holland, Amsterdam, 233-244 (1987) Zbl0636.03028MR895647 · Zbl 0636.03028
[13]	Hrushovski, E., Zilber, B.: Zariski geometries. J. Amer. Math. Soc.9, 1-56 (1996) Zbl0843.03020MR1311822 · Zbl 0843.03020
[14]	Johns, J.: An open mapping theorem for o-minimal structures. J. Symbolic Logic66, 1817-1820 (2001) Zbl0993.03052MR1877024 · Zbl 0993.03052
[15]	Knight, J. F., Pillay, A., Steinhorn, C.: Definable sets in ordered structures. II. Trans. Amer. Math. Soc.295, 593-605 (1986) Zbl0662.03024MR833698 · Zbl 0662.03024
[16]	Kowalski, P., Randriambololona, S.: Strongly minimal reducts of valued fields. J. Symbolic Logic81, 510-523 (2016) Zbl1436.03186MR3519443 · Zbl 1436.03186
[17]	Marker, D.: Zariski geometries. In: Model Theory and Algebraic Geometry, Lecture Notes in Math. 1696, Springer, Berlin, 107-128 (1998) Zbl0925.03171MR1678535 · Zbl 0925.03171
[18]	Marker, D.: Model Theory. Grad. Texts in Math. 217, Springer, New York (2002) Zbl1003.03034MR1924282 · Zbl 1003.03034
[19]	Marker, D., Pillay, A.: Reducts of.C;C;/which containC. J. Symbolic Logic55, 1243-1251 (1990) Zbl0721.03023MR1071326 · Zbl 0721.03023
[20]	Moosa, R.: A nonstandard Riemann existence theorem. Trans. Amer. Math. Soc.356, 1781-1797 (2004) Zbl1038.03042MR2031041 · Zbl 1038.03042
[21]	Otero, M., Peterzil, Y.:G-linear sets and torsion points in definably compact groups. Arch. Math. Logic48, 387-402 (2009) Zbl1177.03043MR2505431 · Zbl 1177.03043
[22]	Otero, M., Peterzil, Y., Pillay, A.: On groups and rings definable in o-minimal expansions of real closed fields. Bull. London Math. Soc.28, 7-14 (1996) Zbl0834.12006MR1356820 · Zbl 0834.12006
[23]	Peterzil, Y., Pillay, A., Starchenko, S.: Definably simple groups in o-minimal structures. Trans. Amer. Math. Soc.352, 4397-4419 (2000) Zbl0952.03046MR1707202 · Zbl 0952.03046
[24]	Peterzil, Y., Pillay, A., Starchenko, S.: Simple algebraic and semialgebraic groups over real closed fields. Trans. Amer. Math. Soc.352, 4421-4450 (2000) Zbl0952.03047 MR1779482 · Zbl 0952.03047
[25]	Peterzil, Y., Starchenko, S.: Definable homomorphisms of abelian groups in o-minimal structures. Ann. Pure Appl. Logic101, 1-27 (2000) Zbl0949.03033MR1729742 · Zbl 0949.03033
[26]	Peterzil, Y., Starchenko, S.: Expansions of algebraically closed fields in o-minimal structures. Selecta Math. (N.S.)7, 409-445 (2001) Zbl1010.03027MR1868299 · Zbl 1010.03027
[27]	Peterzil, Y., Starchenko, S.: Computing o-minimal topological invariants using differential topology. Trans. Amer. Math. Soc.359, 1375-1401 (2007) Zbl1108.03046MR2262855 · Zbl 1108.03046
[28]	Peterzil, Y., Starchenko, S.: Tame complex analysis and o-minimality. In: Proc. International Congress of Mathematicians (Hyderabad, 2010), Vol. II, Hindustan Book Agency, New Delhi, 58-81 (2010) Zbl1246.03061MR2827785 · Zbl 1246.03061
[29]	Peterzil, Y., Starchenko, S.: Topological groups,-types and their stabilizers. J. Eur. Math. Soc.19, 2965-2995 (2017) Zbl1423.03134MR3712999 · Zbl 1423.03134
[30]	Peterzil, Y., Steinhorn, C.: Definable compactness and definable subgroups of o-minimal groups. J. London Math. Soc. (2)59, 769-786 (1999) Zbl0935.03047MR1709079 · Zbl 0935.03047
[31]	Pillay, A.: On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra53, 239-255 (1988) Zbl0662.03025MR961362 · Zbl 0662.03025
[32]	Pillay, A.: Geometric Stability Theory. Oxford Logic Guides 32, Clarendon Press, Oxford Univ. Press, New York (1996) Zbl0871.03023MR1429864 · Zbl 0871.03023
[33]	Pillay, A., Steinhorn, C.: Definable sets in ordered structures. Bull. Amer. Math. Soc. (N.S.) 11, 159-162 (1984) Zbl0542.03016MR741730 · Zbl 0542.03016
[34]	Poizat, B.: Stable Groups. Math. Surveys Monogr. 87, Amer. Math. Soc., Providence, RI (2001) Zbl0969.03047MR1827833 · Zbl 0969.03047
[35]	Rabinovich, E. D.: Definability of a field in sufficiently rich incidence systems. QMW Maths Notes 14, Queen Mary and Westfield College, School of Math. Sci., London (1993) Zbl0752.03017MR1213456
[36]	Strzebonski, A. W.: Euler characteristic in semialgebraic and other o-minimal groups. J. Pure Appl. Algebra96, 173-201 (1994) Zbl0815.03024MR1303545 · Zbl 0815.03024
[37]	van den Dries, L.: Remarks on Tarski’s problem concerning.R;C;;exp/. In: Logic Colloquium ’82 (Florence, 1982), Stud. Logic Found. Math. 112, North-Holland, Amsterdam, 97-121 (1984) Zbl0585.03006MR762106 · Zbl 0585.03006
[38]	van den Dries, L.: Tame Topology and o-Minimal Structures. London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, Cambridge (1998) Zbl0953.03045MR1633348 · Zbl 0953.03045
[39]	Wilkie, A. J.: A theorem of the complement and some new o-minimal structures. Selecta Math. (N.S.)5, 397-421 (1999) Zbl0948.03037MR1740677 · Zbl 0948.03037
[40]	Woerheide, A.: Topology of definable sets in o-minimal structures. PhD thesis, Univ. of Illinois, Urbana-Champaign (1996).
[41]	Zil’ber, B. I.: Strongly minimal countably categorical theories. II. Sibirsk. Mat. Zh.25, 71-88 (1984) (in Russian) Zbl0581.03022MR746943
[42]	Zilber, B.: Zariski Geometries. London Math. Soc. Lecture Note Ser. 360, Cambridge Univ. Press, Cambridge (2010) Zbl1190 · Zbl 1208.03042