Nori-semistable vector bundle explained

\pi _{1(X,x)} .

Definition

Let

X be a scheme over a field k and V a vector bundle on X . It is said that V is Nori semistable if for any smooth and proper curve C over \bar k and any morphism j:C\to X the pull back j ^{*(V)} is semistable of degree 0.^[4]

Difference with Nori's original definition

Nori semistable vector bundles were called by Nori semistable causing a lot of confusion with the already existing definition of semistable vector bundles. More importantly Nori simply said that the restriction of

V to any curve in X had to be semistable of degree 0. Then for instance in positive characteristic a morphism j like the Frobenius morphism was not included in Nori's original definition. The importance of including it is that the above definition makes the category of Nori semistable vector bundles tannakian and the group scheme associated to it is the S -fundamental group scheme^[5] \pi ^{S(X,x)} . Instead, Nori's original definition didn't give rise to a Tannakian category but only to an abelian category.

Notes

1. Nori . Madhav V. . On the Representations of the Fundamental Group . Compositio Mathematica . 33 . 1976 . 1 . 29–42 . 417179 . 0337.14016.
2. Book: 10.1017/CBO9780511627064 . Szamuely . Tamás . Galois Groups and Fundamental Groups . Cambridge Studies in Advanced Mathematics . 117 . 2009. 9780521888509 .
3. 1809.06755 . 10.2748/tmj.20200727 . On the fundamental group schemes of certain quotient varieties . 2021 . Biswas . Indranil . Hai . Phùng Hô . Dos Santos . João Pedro . Tohoku Mathematical Journal . 73 . 4 . 54217282. 565-595 .
4. Book: 10.1007/978-3-540-38955-2. Hodge Cycles, Motives, and Shimura Varieties. P. . Deligne. J. M.. Milne . Lecture Notes in Mathematics . 1982 . 900 . 978-3-540-11174-0 . Tannakian Categories. .
5. 10.5802/aif.2667. On the S -fundamental group scheme . 2011 . Langer . Adrian . Annales de l'Institut Fourier . 61 . 5 . 2077–2119 . 0905.4600 . 53506862 .