Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain “closure” of the aforementioned Lie algebra.
Revised:
Accepted:
Published online:
Indranil Biswas 1; Phùng Hô Hai 2; Joao Pedro dos Santos 3
CC-BY 4.0
@article{CRMATH_2024__362_G3_309_0,
author = {Indranil Biswas and Ph\`ung H\^o Hai and Joao Pedro dos Santos},
title = {Connections on trivial vector bundles over projective schemes},
journal = {Comptes Rendus. Math\'ematique},
pages = {309--325},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.532},
language = {en},
}
TY - JOUR AU - Indranil Biswas AU - Phùng Hô Hai AU - Joao Pedro dos Santos TI - Connections on trivial vector bundles over projective schemes JO - Comptes Rendus. Mathématique PY - 2024 SP - 309 EP - 325 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.532 LA - en ID - CRMATH_2024__362_G3_309_0 ER -
Indranil Biswas; Phùng Hô Hai; Joao Pedro dos Santos. Connections on trivial vector bundles over projective schemes. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 309-325. doi : 10.5802/crmath.532. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.532/
[1] On certain Tannakian categories of integrable connections over Kähler manifolds, Can. J. Math., Volume 74 (2022) no. 4, pp. 1034-1061 | DOI | Zbl
[2] Vector bundles on curves admitting a connection, Q. J. Math., Volume 57 (2006) no. 2, pp. 143-150 | DOI | MR | Zbl
[3] Éléments de Mathématiques. Groupes et algèbres de Lie. Chapitres 2 et 3, Springer, 2006 | Zbl
[4] Éléments de Mathématiques. Groupes et algèbres de Lie. Chapitres 7 et 8, Springer, 2006 | Zbl
[5] Élément de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6, Springer, 2007 | Zbl
[6] Éléments de mathématique. Groupes et algèbres de Lie. Chapitre 1, Springer, 2007 | Zbl
[7] Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math., Inst. Hautes Étud. Sci., Volume 35 (1968), pp. 107-126 | DOI | Numdam | Zbl
[8] Le groupe fondamental de la droite projective moins trois points, Galois Groups over , Berkeley, CA (Mathematical Sciences Research Institute Publications), Volume 16, Springer, 1987, pp. 79-297 | DOI | Zbl
[9] Tannakian categories, Hodge cycles, motives, and Shimura varieties (Lecture Notes in Mathematics), Volume 900, Springer, 1982, pp. 101-228 | DOI | Zbl
[10] Groupes algébriques. Tome I: Géométrie algébrique. Généralités. Groupes commutatifs, Masson; North-Holland, 1970 (avec un appendice ‘Corps de classes local’ par Michiel Hazewinkel) | Zbl
[11] The nullcone of the Lie algebra of , Indag. Math., New Ser., Volume 30 (2019) no. 4, pp. 623-648 | DOI | MR | Zbl
[12] Algebraic Lie algebras and representative functions, Ill. J. Math., Volume 3 (1959), pp. 499-529 | MR | Zbl
[13] Lie algebra cohomology and affine algebraic groups, Ill. J. Math., Volume 18 (1974), pp. 170-176 | MR | Zbl
[14] Théorèmes de Bertini et applications, Progress in Mathematics, 42, Birkhäuser, 1983 | Zbl
[15] Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math., Inst. Hautes Étud. Sci., Volume 39 (1970), pp. 175-232 | DOI | Numdam | Zbl
[16] Two elements generations on semi-simple Lie groups, Kōdai Math. Semin. Rep., Volume 1949 (1949) no. 5-6, pp. 89-90 | MR | Zbl
[17] Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford Science Publications, 2002 | DOI | Zbl
[18] Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer, 1971 | Zbl
[19] Algebraic Groups. The Theory of Group Schemes of Finite Type Over a Field, Cambridge Studies in Advanced Mathematics, 170, Cambridge University Press, 2017 | DOI | Zbl
[20] Hopf algebras and their actions on rings, Regional Conference Series in Mathematics, 82, American Mathematical Society, 1993 (Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992) | DOI | Zbl
[21] Abelian varieties, Tata Institute of Fundamental Research. Studies in Mathematics, 5, Tata Institute of Fundamental Research; Oxford University Press, 1970 | Zbl
[22] Lie algebras of pro-affine algebraic groups, Can. J. Math., Volume 54 (2002) no. 3, pp. 595-607 | DOI | MR | Zbl
[23] Construction of Hilbert and Quot schemes, Fundamental algebraic geometry (Mathematical Surveys and Monographs), Volume 123, American Mathematical Society, 2005, pp. 105-137 | MR
[24] Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | DOI | Zbl
[25] Hopf algebras, Benjamin, 1969 | Zbl
[26] Lie algebras and algebraic groups, Springer Monographs in Mathematics, Springer, 2005 | DOI | Zbl
[27] Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Springer, 1979 | DOI | Zbl
Cited by Sources:
Comments - Policy