Some remarks on the instability flag. (English) Zbl 0567.14027

Ramanan, S.; Ramanathan, A.

Tôhoku Math. J., II. Ser. 36, 269-291 (1984).

Let G be a reductive group acting on a projective variety \(M\subseteq {\mathbb{P}}(V)\). G. R. Kempf [Ann. Math., II. Sér. 108, 299-316 (1978; Zbl 0406.14031)] and G. Rousseau [C. R. Acad. Sci., Paris, Ser. A 286, 247-250 (1978; Zbl 0375.14013)] proved that to any non- semistable point \(m\in M\) one can associate a canonical parabolic subgroup P(m). One can also associate to m a conjugacy class of 1- parameter subgroups in P(m). Let \(\lambda\) be such a subgroup and decompose V as \(\oplus_{i\in {\mathbb{Z}}}V_ i\) where \(V_ i\) is the space of weight vectors of weight i for \(\lambda\). Write \(m=m_ 0+m_ 1\) with \(0\neq m_ 0\in V_ j\) and \(m_ 1\in \oplus_{i>j}V_ i\). The authors give a proof of Kempf’s and Rousseau’s results and refining Kempf’s techniques they prove that \(P(m_ 0)=P(m)\). Moreover they prove that if U is the unipotent radical of P(m) then, for the natural action of P(m)/U on \(V_ j\), the point \(m_ 0\) becomes semi-stable after the polarization is replaced by a multiple and the action is twisted by a dominant character. The authors also investigate the existence of instability flag, that is of groups P(m), over non-perfect fields. They introduce the concept of separable action of G on M over the ground field k and show that if G act separably and m is k-rational non-semistable, then P(m) is defined over k. Finally they apply the results to the study of vector bundles and give among other results an algebraic proof of the result that any bundle associated to a semistable vector bundle on a projective curve in characteristic 0, by extension of the structure group, is itself semistable.

Reviewer: D.Laksov

MSC:

14L30	Group actions on varieties or schemes (quotients)
14L10	Group varieties
14M15	Grassmannians, Schubert varieties, flag manifolds
14L35	Classical groups (algebro-geometric aspects)

Keywords:

1-parameter groups; reductive group acting on a projective variety; non- semistable point; parabolic subgroup; instability flag

Citations:

Zbl 0406.14031; Zbl 0375.14013

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