Zilber’s restricted trichotomy in characteristic zero

Castle, Benjamin

doi:10.1090/jams/1037

Zilber’s restricted trichotomy in characteristic zero
HTML articles powered by AMS MathViewer

by Benjamin Castle;

J. Amer. Math. Soc. 37 (2024), 1041-1120

DOI: https://doi.org/10.1090/jams/1037

Published electronically: October 3, 2023

HTML | PDF | Request permission

Abstract:

We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if $\mathcal M$ is any non-locally modular strongly minimal structure interpreted in an algebraically closed field $K$ of characteristic zero, then $\mathcal M$ itself interprets $K$; in particular, any non-1-based structure interpreted in $K$ is mutually interpretable with $K$. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.

References

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 82463
Elisabeth Bouscaren, The group configuration—after E. Hrushovski, The model theory of groups (Notre Dame, IN, 1985–1987) Notre Dame Math. Lectures, vol. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 199–209. MR 985348
Steven Buechler, The geometry of weakly minimal types, J. Symbolic Logic 50 (1985), no. 4, 1044–1053 (1986). MR 820131, DOI 10.2307/2273989
Steven Buechler, One theorem of Zil′ber’s on strongly minimal sets, J. Symbolic Logic 50 (1985), no. 4, 1054–1061 (1986). MR 820132, DOI 10.2307/2273990
Steven Buechler, Locally modular theories of finite rank, Ann. Pure Appl. Logic 30 (1986), no. 1, 83–94. Stability in model theory (Trento, 1984). MR 831438, DOI 10.1016/0168-0072(86)90038-2
Benjamin Castle and Assaf Hasson, Very ampleness in strongly minimal sets, to appear in Model Theory, 2022. arXiv:2212.03774 [math.LO].
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
Pantelis E. Eleftheriou, Assaf Hasson, and Ya’acov Peterzil, Strongly minimal groups in o-minimal structures, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 10, 3351–3418. MR 4275476, DOI 10.4171/jems/1095
Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331, DOI 10.1007/978-3-642-69582-7
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
Assaf Hasson and Piotr Kowalski, Strongly minimal expansions of $(\Bbb C,+)$ definable in o-minimal fields, Proc. Lond. Math. Soc. (3) 97 (2008), no. 1, 117–154. MR 2434093, DOI 10.1112/plms/pdm052
Assaf Hasson and Dmitry Sustretov, Incidence systems on cartesian powers of algebraic curves, 2021. arXiv:1702.05554 [math.LO].
Ehud Hrushovski, Locally modular regular types, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 132–164. MR 1033027, DOI 10.1007/BFb0082236
Ehud Hrushovski, Strongly minimal expansions of algebraically closed fields, Israel J. Math. 79 (1992), no. 2-3, 129–151. MR 1248909, DOI 10.1007/BF02808211
Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147–166. Stability in model theory, III (Trento, 1991). MR 1226304, DOI 10.1016/0168-0072(93)90171-9
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR 1854232, DOI 10.1016/S0168-0072(01)00096-3
Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), no. 1, 1–56. MR 1311822, DOI 10.1090/S0894-0347-96-00180-4
Ehud Hrushovski and Željko Sokolovic, Minimal subsets of differentially closed fields, Preprint, 1994.
U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244. MR 895647, DOI 10.1016/S0049-237X(09)70556-7
János Kollár, Max Lieblich, Martin Olsson, and William Sawin, The Zariski topology, linear systems, and algebraic varieties, manuscript, 2021.
Angus Macintyre, On $\omega _{1}$-categorical theories of fields, Fund. Math. 71 (1971), no. 1, 1–25. (errata insert). MR 290954, DOI 10.4064/fm-71-1-1-25
D. Marker and A. Pillay, Reducts of $(\textbf {C},+,\cdot )$ which contain $+$, J. Symbolic Logic 55 (1990), no. 3, 1243–1251. MR 1071326, DOI 10.2307/2274485
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Gary A. Martin, Definability in reducts of algebraically closed fields, J. Symbolic Logic 53 (1988), no. 1, 188–199. MR 929384, DOI 10.2307/2274437
David Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR 971985, DOI 10.1007/978-3-662-21581-4
Ya’acov Peterzil and Sergei Starchenko, Expansions of algebraically closed fields in o-minimal structures, Selecta Math. (N.S.) 7 (2001), no. 3, 409–445. MR 1868299, DOI 10.1007/PL00001405
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
Anand Pillay and Martin Ziegler, Jet spaces of varieties over differential and difference fields, Selecta Math. (N.S.) 9 (2003), no. 4, 579–599. MR 2031753, DOI 10.1007/s00029-003-0339-1
Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI 10.2307/2273680
Bruno Poizat, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein. MR 1827833, DOI 10.1090/surv/087
E. D. Rabinovich, Definability of a field in sufficiently rich incidence systems, QMW Maths Notes, vol. 14, Queen Mary and Westfield College, School of Mathematical Sciences, London, 1993. With an introduction by Wilfrid Hodges. MR 1213456
Eugenia Rabinovich and Boris Zilber, Additive reducts of algebraically closed fields, Manuscript, 1988.
Thomas Scanlon, Local André-Oort conjecture for the universal abelian variety, Invent. Math. 163 (2006), no. 1, 191–211. MR 2208421, DOI 10.1007/s00222-005-0460-1
The Stacks Project Authors, Stacks Project, 2018, https://stacks.math.columbia.edu.
Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley-Los Angeles, Calif., 1951. 2nd ed. MR 44472, DOI 10.1525/9780520348097
Ravi Vakil, The rising sea: foundations of algebraic geometry, 2017, Online notes.
Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791–796. MR 95846, DOI 10.1073/pnas.44.8.791
B. I. Zil′ber, The structure of models of uncountably categorical theories, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 359–368. MR 804692
Boris Zilber, A curve and its abstract Jacobian, Int. Math. Res. Not. IMRN 5 (2014), 1425–1439. MR 3178604, DOI 10.1093/imrn/rns262