Tangent bundle of hypersurfaces in  G/ P

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology / Volume 4 / Issue 01 / August 2009, pp 91-100
Copyright © ISOPP 2008
DOI: http://dx.doi.org/10.1017/is008002001jkt052 (About DOI), Published online: 12 May 2008

Table of Contents - 2009 - Volume 4, Issue 01

Research Article

Tangent bundle of hypersurfaces in G/P

Article author query
biswas i [Google Scholar]
schumacher g [Google Scholar]

Indranil Biswas^a1 and Georg Schumacher^a2

^a1School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, indranil@math.tifr.res.in.

^a2Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Lahnberge, Hans-Meerwein-Strasse, D-35032 Marburg, Germany, schumac@mathematik.uni-marburg.de.

Abstract

Let G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H xs21AA G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism S1755069608000522_inline1 is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and S1755069608000522_inline2 hold; here n = dimH, and τ(G/P) xs2208 S1755069608000522_inline3 is the index of G/P which is defined by the identity S1755069608000522_inline4 = S1755069608000522_inline5 where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and S1755069608000522_inline6 then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.