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Keywords:
F-fundamental group scheme; Frobenius-finite Vector bundles
F-fundamental group scheme; Frobenius-finite Vector bundles
Summary:
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.
References:
[1] Amrutiya, S., Biswas, I.: On the $F$-fundamental group scheme. Bull. Sci. Math. 34 (2010), 461–474. DOI 10.1016/j.bulsci.2009.12.002 | MR 2665455
[2] Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. London Math. Soc. 3 (1957), 414–452. MR 0131423
[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 10.1016/j.crma.2007.10.010 | MR 2375109
[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 10.1090/S1056-3911-07-00449-3 | MR 2306280
[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 2219253
[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 10.1007/978-3-540-38955-2 | MR 0654325
[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 10.4310/PAMQ.2011.v7.n4.a12 | MR 2918164
[8] Ishimura, S.: A descent problem of vector bundles and its applications. J. Math. Kyoto Univ. 23 (1983), 73–83. DOI 10.1215/kjm/1250521611 | MR 0692730
[9] Lange, H., Stuhler, U.: Vektorbündel auf Kurven und Darstellungen der algebraischen Fudamentalgruppe. Math. Z. 156 (1977), 73–83. DOI 10.1007/BF01215129 | MR 0472827
[10] Langer, A.: Semistable sheaves in positive characteristic. Ann. of Math. 159 (2004), 251–276. DOI 10.4007/annals.2004.159.251 | MR 2051393
[11] Langer, A.: On S-fundamental group scheme. Ann. Inst. Fourier (Grenoble) 61 (5) (2011), 2077–2119. DOI 10.5802/aif.2667 | MR 2961849
[12] Lekaus, S.: Vector bundles of degree zero over an elliptic curve. C. R. Math. Acad. Sci. Paris 335 (2002), 351–354. DOI 10.1016/S1631-073X(02)02478-0 | MR 1931515
[13] Nori, M.V.: The fundamental group-scheme. Proc. Indian Acad. Sci. Math. Sci. 91 (2) (1982), 73–122. DOI 10.1007/BF02967978 | MR 0682517
[14] Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J. 43 (1971), 41–72. DOI 10.1017/S0027763000014367 | MR 0318151
[15] Otabe, S.: An extension of Nori fundamental group. Comm. Algebra 45 (2017), 3422–3448. DOI 10.1080/00927872.2016.1236936 | MR 3609350
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