Geodesic
A note on certain Tannakian group schemes
Archivum mathematicum, Tome 56 (2020) no. 1, pp. 21-29 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this note, we prove that the $F$-fundamental group scheme is a birational invariant for smooth projective varieties. We prove that the $F$-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the $F$-fundamental group scheme and the Nori fundamental group scheme coincide. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles, and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.
In this note, we prove that the $F$-fundamental group scheme is a birational invariant for smooth projective varieties. We prove that the $F$-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the $F$-fundamental group scheme and the Nori fundamental group scheme coincide. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles, and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.
DOI : 10.5817/AM2020-1-21
Classification : 14F05, 14L15
Keywords: F-fundamental group scheme; Frobenius-finite Vector bundles 
@article{10_5817_AM2020_1_21,
     author = {Amrutiya, Sanjay},
     title = {A note on certain {Tannakian} group schemes},
     journal = {Archivum mathematicum},
     pages = {21--29},
     year = {2020},
     volume = {56},
     number = {1},
     doi = {10.5817/AM2020-1-21},
     mrnumber = {4075885},
     zbl = {07177877},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2020-1-21/}
}
TY  - JOUR
AU  - Amrutiya, Sanjay
TI  - A note on certain Tannakian group schemes
JO  - Archivum mathematicum
PY  - 2020
SP  - 21
EP  - 29
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2020-1-21/
DO  - 10.5817/AM2020-1-21
LA  - en
ID  - 10_5817_AM2020_1_21
ER  - 
%0 Journal Article
%A Amrutiya, Sanjay
%T A note on certain Tannakian group schemes
%J Archivum mathematicum
%D 2020
%P 21-29
%V 56
%N 1
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2020-1-21/
%R 10.5817/AM2020-1-21
%G en
%F 10_5817_AM2020_1_21
Amrutiya, Sanjay. A note on certain Tannakian group schemes. Archivum mathematicum, Tome 56 (2020) no. 1, pp. 21-29. doi: 10.5817/AM2020-1-21

[1] Amrutiya, S., Biswas, I.: On the $F$-fundamental group scheme. Bull. Sci. Math. 34 (2010), 461–474. | DOI | MR

[2] Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. London Math. Soc. 3 (1957), 414–452. | MR

[3] Biswas, I., Ducrohet, L.: An analog of a theorem of Lange and Stuhler for principal bundles. C. R. Acad. Sci. Paris, Ser. I 345 (2007), 495–497. | DOI | MR

[4] Biswas, I., Holla, Y.: Comparison of fundamental group schemes of a projective variety and an ample hypersurface. J. Algebraic Geom. 16 (2007), 547–597. | DOI | MR

[5] Biswas, I., Parameswaran, A.J., Subramanian, S.: Monodromy group for a strongly semistable principal bundle over a curve. Duke Math. J. 132 (2006), 1–48. | MR

[6] Deligne, P., Milne, J.S.: Tannakian Categories. Hodge cycles, motives and Shimura varieties, Lecture Notes in Math, vol. 900, Springer–Verlag, Berlin–Heidelberg–New York, 1982. | DOI | MR

[7] Hogadi, A., Mehta, V.B.: Birational invariance of the S-fundamental group scheme. Pure Appl. Math. Q. 7 (4) (2011), 1361–1369, Special Issue: In memory of Eckart Viehweg. | DOI | MR

[8] Ishimura, S.: A descent problem of vector bundles and its applications. J. Math. Kyoto Univ. 23 (1983), 73–83. | DOI | MR

[9] Lange, H., Stuhler, U.: Vektorbündel auf Kurven und Darstellungen der algebraischen Fudamentalgruppe. Math. Z. 156 (1977), 73–83. | DOI | MR

[10] Langer, A.: Semistable sheaves in positive characteristic. Ann. of Math. 159 (2004), 251–276. | DOI | MR

[11] Langer, A.: On S-fundamental group scheme. Ann. Inst. Fourier (Grenoble) 61 (5) (2011), 2077–2119. | DOI | MR

[12] Lekaus, S.: Vector bundles of degree zero over an elliptic curve. C. R. Math. Acad. Sci. Paris 335 (2002), 351–354. | DOI | MR

[13] Nori, M.V.: The fundamental group-scheme. Proc. Indian Acad. Sci. Math. Sci. 91 (2) (1982), 73–122. | DOI | MR

[14] Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J. 43 (1971), 41–72. | DOI | MR

[15] Otabe, S.: An extension of Nori fundamental group. Comm. Algebra 45 (2017), 3422–3448. | DOI | MR

Cité par Sources :