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

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


14L30 Group actions on varieties or schemes (quotients)
14L10 Group varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
14L35 Classical groups (algebro-geometric aspects)
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[1] F. A. BOGOMOLOV, Holomorphic tensors and vector bundles on projective manifolds (in Russian), Izvestija Akademii Nauk SSR, Ser Mat. 42 (1978), 1227-1287. · Zbl 0439.14002
[2] A. BOREL, Linear Algebraic Groups, W. A. Benjamin, New York, 1969 · Zbl 0186.33201
[3] A. BOREL AND T. A. SPRINGER, Rationality properties of linear algebraic groups II, Tohoku Math. J. 20 (1968), 443-497. · Zbl 0211.53302
[4] A. BOREL AND J. TITS, Groupes reductifs, Pub. Math. I. H. E. S. No. 27 (1965), 55-150
[5] D. GIESEKAR, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), 77-85. JSTOR: · Zbl 0431.14005
[6] A. GROTHENDIECK AND M. DEMAZURE, Schemas en groupes I, II, III (SGA 3), Lectur Notes in Math. 151, 152, 153, Springer-Verlag, Berlin-Hidelberg-New York, 1976.
[7] G. KEMPF, Instability in invariant theory, Ann. of Math. 108 (1978), 299-316 JSTOR: · Zbl 0406.14031
[8] M. MARUYAMA, The theorem of Grauert-Mlich-Spindler, Math. Ann. 255 (1981), 317-333 · Zbl 0438.14015
[9] D. MUMFORD, Geometric Invariant Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1965. · Zbl 0147.39304
[10] M. S. NARASIMHAN AND C. S. SESHADRI, Stable and unitary vector bundles on a compac Riemann surface, Ann. of Math. 82 (1965), 540-567. · Zbl 0171.04803
[11] P. E. NEWSTEAD, Lectures on introduction to moduli problems and orbit spaces, Tat Institute Lecture Notes, Springer-Verlag, 1978. · Zbl 0411.14003
[12] M. V. NORI, On the representation of the fundamental group, Compositio Math. 33 (1976), 29-41. · Zbl 0337.14016
[13] M. V. NORI, Characterisations of the representations of the algebraic fundamental grou of a variety, Thesis, University of Bombay, 1980.
[14] A. RAMANATHAN, Stable principal bundles on a compact Riemann surface, Math. Ann 213 (1975), 129-152. · Zbl 0284.32019
[15] A. RAMANATHAN, Stable principle bundles on a compact Riemann surface–Constructio of moduli space, Thesis, University of Bombay, 1976. · Zbl 0284.32019
[16] G. ROUSSEAU, Immersible spheriques et theorie des invariants, C. R. Acad. Sci. Paris, 286 (1978), 247-250. · Zbl 0375.14013
[17] J. -P. SERRE, Espaces fibres algebriques, in: Anneau de Chow et applications, Seminair Chevalley, 1958.
[18] R. STEINBERG, Regular elements of semisimple algebraic groups, Publ. Math. I. H. E. S. No. 25, (1965). · Zbl 0136.30002
[19] H. TANGO, On the behaviour of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73-89. · Zbl 0239.14007
[20] M. RAYNAUD, Sections des fibres vectoriels sur une courbe, Bull. Soc. Math. France 11 (1982), 103-125. · Zbl 0505.14011
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