Abelian reduction in differential-algebraic and bimeromorphic geometry
[Réduction abélienne en géométrie différentielle algébrique et en géométrie biméromorphe]
Jaoui, Rémi1 ; Moosa, Rahim2
1 Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Blvd. du 11 Novembre 1918, 69622 Villeurbanne (France)
2 University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1 (Canada)
Annales de l'Institut Fourier, Online first, 43 p.

Un nouvel outil pour la théorie des modèles des corps différentiellement clos et des variétés compactes complexes est développé dans cet article. Dans ces théories, il est montré qu’un type interne au corps des constantes (resp. à la droite projective) admet une plus grande image dont le groupe de liaison est une variété abélienne. Les propriétés de ces reductions abéliennes sont étudiées dans le formalisme de la théorie géométrique de la stabilité.

Plusieurs conséquences concernant la géométrie birationnelle des champs de vecteurs algébriques de caractéristique zéro sont alors obtenues. En particulier,

  • (1) il est montré que si une puissance cartésienne d’un champ de vecteurs algébrique admet une intégrale première rationnelle non triviale alors la seconde puissance cartésienne vérifie déjà cette propriété,
  • (2) les champs de vecteurs algébriques isotriviaux de dimension deux sont classifiés à équivalence birationnelle près,
  • (3) les champs de vecteurs algébriques dont tous les recouvrement finis n’admettent aucun facteur non trivial sont étudiés en dimension arbitraire.

Des résultats analogues en géométrie biméromorphe sont aussi obtenus.

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal image whose binding group is an abelian variety. The properties of such abelian reductions are investigated in the Galois-theoretic framework provided by stability theory.

Several geometric consequences for the birational geometry of algebraic vector fields of characteristic zero are then deduced. In particular,

  • (1) it is shown that if some cartesian power of an algebraic vector field admits a nontrivial rational first integral then already the second power does,
  • (2) two-dimensional isotrivial algebraic vector fields are classified up to birational equivalence and
  • (3) algebraic vector fields whose finite covers admit no nontrivial factors are studied in arbitrary dimension.

Analogues of these results in bimeromorphic geometry are also obtained.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3697
Classification : 03C45, 12H05, 32J27
Keywords: algebraic vector fields, differentially closed fields, compact Kähler manifolds, geometric stability theory
Mots-clés : champs de vecteurs algébriques, corps différentielleement clos, variétés kälhériennes compactes, théorie géométrique de la stabilité

Jaoui, Rémi 1 ; Moosa, Rahim 2

1 Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Blvd. du 11 Novembre 1918, 69622 Villeurbanne (France)
2 University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1 (Canada) 
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Jaoui, Rémi; Moosa, Rahim. Abelian reduction in differential-algebraic and bimeromorphic geometry. Annales de l'Institut Fourier, Online first, 43 p.

[1] Brion, Michel Some structure theorems for algebraic groups, Algebraic groups: structure and actions (Proceedings of Symposia in Pure Mathematics), Volume 94, American Mathematical Society, 2017, pp. 53-126 | DOI | MR | Zbl

[2] Buium, Alexandru Differential function fields and moduli of algebraic varieties, Lecture Notes in Mathematics, 1226, Springer, 1986 | DOI | MR | Zbl

[3] Campana, Frédéric Exemples de sous-espaces maximaux isolés de codimension deux d’un espace analytique compact, Institut Élie Cartan, 6 (Inst. Élie Cartan), Volume 6, Université de Nancy 1, Nancy, 1982, pp. 106-127 | MR | Zbl

[4] Campana, Frédéric; Oguiso, Keiji; Peternell, Thomas Non-algebraic hyperkähler manifolds, J. Differ. Geom., Volume 85 (2010) no. 3, pp. 397-424 | DOI | MR | Zbl

[5] Freitag, James; Jaoui, Rémi; Moosa, Rahim When any three solutions are independent, Invent. Math., Volume 230 (2022) no. 3, pp. 1249-1265 | DOI | MR | Zbl

[6] Freitag, James; Jaoui, Rémi; Moosa, Rahim The degree of nonminimality is at most 2, J. Math. Log., Volume 23 (2023) no. 3, 2250031 | DOI | MR | Zbl

[7] Freitag, James; Moosa, Rahim Bounding nonminimality and a conjecture of Borovik-Cherlin, J. Eur. Math. Soc. (2023) (Online first) | DOI

[8] Fujiki, Akira On the structure of compact complex manifolds in C, Algebraic varieties and analytic varieties (Tokyo, 1981) (Advanced Studies in Pure Mathematics), Volume 1, North-Holland, 1983, pp. 231-302 | DOI | MR | Zbl

[9] Fujiki, Akira Relative algebraic reduction and relative Albanese map for a fiber space in 𝒞, Publ. Res. Inst. Math. Sci., Volume 19 (1983) no. 1, pp. 207-236 | DOI | MR | Zbl

[10] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton University Press, 1981 | DOI | MR | Zbl

[11] Hrushovski, Ehud Almost orthogonal regular types, Ann. Pure Appl. Logic, Volume 45 (1989) no. 2, pp. 139-155 | DOI | MR | Zbl

[12] Hrushovski, Ehud Computing the Galois group of a linear differential equation, Differential Galois theory (Bedlewo, 2001) (Banach Center Publications), Volume 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 97-138 | DOI | MR | Zbl

[13] Hrushovski, Ehud; Itai, Masanori On model complete differential fields, Trans. Am. Math. Soc., Volume 355 (2003) no. 11, pp. 4267-4296 | DOI | MR | Zbl

[14] Hrushovski, Ehud; Sokolović, Željko Strongly minimal sets in differentially closed fields (1993) (unpublished manuscript)

[15] Jaoui, Rémi Corps différentiels et flots géodésiques I—Orthogonalité aux constantes pour les équations différentielles autonomes, Bull. Soc. Math. Fr., Volume 148 (2020) no. 3, pp. 529-595 | DOI | MR | Zbl

[16] Jaoui, Rémi; Jimenez, Léo; Pillay, Anand Relative internality and definable fibrations, Adv. Math., Volume 415 (2023), 108870 | DOI | MR | Zbl

[17] Jimenez, Léo Groupoids and relative internality, J. Symb. Log., Volume 84 (2019) no. 3, pp. 987-1006 | DOI | MR | Zbl

[18] Jin, Ruizhang; Moosa, Rahim Internality of logarithmic-differential pullbacks, Trans. Am. Math. Soc., Volume 373 (2020) no. 7, pp. 4863-4887 | DOI | MR | Zbl

[19] Kolchin, Ellis R. Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, 54, Academic Press Inc., 1973 | MR | Zbl

[20] Kowalski, Piotr; Pillay, Anand Quantifier elimination for algebraic D-groups, Trans. Am. Math. Soc., Volume 358 (2006) no. 1, pp. 167-181 | DOI | MR | Zbl

[21] Lieberman, David I. Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) (Lecture Notes in Mathematics), Volume 670, Springer, 1978, pp. 140-186 | MR | Zbl

[22] Moosa, Rahim A nonstandard Riemann existence theorem, Trans. Am. Math. Soc., Volume 356 (2004) no. 5, pp. 1781-1797 | DOI | MR | Zbl

[23] Moosa, Rahim The model theory of compact complex spaces, Logic Colloquium ’01 (Lecture Notes in Logic), Volume 20, Association for Symbolic Logic, Urbana, IL, 2005, pp. 317-349 | DOI | MR | Zbl

[24] Moosa, Rahim On saturation and the model theory of compact Kähler manifolds, J. Reine Angew. Math., Volume 586 (2005), pp. 1-20 | DOI | MR | Zbl

[25] Moosa, Rahim; Pillay, Anand Model theory and Kähler geometry, Model theory with applications to algebra and analysis. Vol. 1 (London Mathematical Society Lecture Note Series), Volume 349, Cambridge University Press, 2008, pp. 167-195 | DOI | MR | Zbl

[26] Moosa, Rahim; Pillay, Anand Some model theory of fibrations and algebraic reductions, Sel. Math., New Ser., Volume 20 (2014) no. 4, pp. 1067-1082 | DOI | MR | Zbl

[27] Pillay, Anand Geometric Stability Theory, Oxford University Press, 1996 | DOI | MR | Zbl

[28] Pillay, Anand Algebraic D-groups and differential Galois theory, Pac. J. Math., Volume 216 (2004) no. 2, pp. 343-360 | DOI | MR | Zbl

[29] Pillay, Anand; Scanlon, Thomas Compact complex manifolds with the DOP and other properties, J. Symb. Log., Volume 67 (2002) no. 2, pp. 737-743 | DOI | MR | Zbl

[30] Rosenlicht, Maxwell Some basic theorems on algebraic groups, Am. J. Math., Volume 78 (1956), pp. 401-443 | DOI | MR | Zbl

[31] Umemura, Hiroshi Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J., Volume 117 (1990), pp. 125-171 | DOI | MR | Zbl

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