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5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

208 + + + Associated to

208 + + + Associated to each ~N there is a tangent space, TN~) -- ~N.F, where I~ N is the + subspace of ~'~G defined by :- + + G G G : jl(~,~)(0)+i e N }. + + The notions of M-~N-determinacy and finite 5~ N determinacy are defined in exactly + + the same way as for 5~ G equivalence. Clearly if F is M-5~N-determined then it is also + M-determined with respect to the ~G action. Moreover ,we have the following result. Proposition 3.2 G + F e tpq is finitely ~G determined Proof + + G ~ ¢=~ TG(F) has finite codimension in Epq ¢=~ TN(F) has finite codimension in TN(F) + ¢~ F is finitely ~Ndetermined. The equivalence of finite ~G (resp. ~N ) - determinacy and the finite codimension of TG(F) (resp. TN(F) ) follows from the Determinacy Theorem in [D]. Some modification G G is needed to take account of the additive action of Eq on Epq, but this is straightforward. The equivalence of the finite codimension of TG(F) and that of TN(F ) follows from the + + fact that L~N has finite codimension in L2G" We shall say that F is finitely determined if these conditions hold. Note, in particular, + that if F is infinitesimaUy 5~ G stable then it is finitely determined. Before stating the main theorem of this section we need one further definition.

209 Definition 3.3 G Let ~ be any group of equivalences acting linearly on ~pq. G (i) A vector subspace M of ~pqiS ~-intrinsic if it is invariant under the action of ~ on G G (ii)If M is any subset of l~lNthen its ~-intrinsic part, denoted Itr~M, is defined by :- Itrl~M ; (~.0e~.t.M. Itr~M is the unique, maximal (with respect to inclusion), intrinsic vector space contained in M. The result we shall use to obtain determinacy estimates is the following. Theorem 3.4 + G Let ~ = ~N for some N. Then, ifF ~ ~ixliS finitely determined, it is Itr~ TN(F )- -determined. Outline of Proof k Because F has finite codimension we may work entirely mod (mpqEpq) G for some k. G The action of G on ~pqiS then essentially algebraic. If p E Itr~ TN(F) then tp ~ Itr~ TN(F ) for all t in ~. It follows from the fact that Itr~ TN(F) is invariant under ~ that TN(t p) c TN(F ). So:- TN(F+tp) c TN(F ) + TN(tP) c TN(F ). Since the codimension of the orbits of an algebraic group action is semicontinous, this implies that TN(F+tp) = TN(F ) for all t in some interval (-~,e). From the Mather lemma [M, Lemma 3.1] we deduce that F+tp is ~-equivalent to F for all t in (-e,e). Finally, because ~ is unipotent, and hence has closed orbits [BPW, Proposition 1.3], we can conclude that F+tp is ~-equivalent to F for all t.

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