Determinacy of Analytic Foliation Germs

In this paper we consider the determinacy problem for codim 1 complex analytic foliation germs. For function germs (smooth or analytic), the problem has been thoroughly worked out mostly by J. Mather and the results are widely known ([5], [6] see also [8], [14]). Let I be a smooth or analytic function germ at the origin 0 in Rn or en and let J(f) denote the ideal generated by the partial derivatives of I in the ring of function germs at O. We also denote by m the maximal ideal of germs that are 0 at O. Then if (a) mkcmJ(f)+mk+! for some natural number k, (b) lis (right) k-determined, i.e., for any germ g with the same k-jet as f, there is a germ if! of local diffeomorphism or of local biholomorphic map at 0 with if!(O) =0 such that g is equal to the pull-back if!* I of I by ¢ (g is right equivalent to f). Also, (b) implies that (c) mk+!cmJ(f). The condition (c) can be referred to as "infinitesimal (right) k-determinacy", since mJ(f) and mk+! are interpreted as, respectively, the tangent spaces at I to the sets of germs right equivalent to I and of germs with the same k-jet as f In general (c) does not imply (b). Hewever, (c) implies that (d) I is "locally k-determined", i.e., if a germ g has the same k-jet as I and is "close" to f, g is right equivalent to f There are statements corresponding to the above in the right-left case. Note also that the problem is closely related to the unfolding theory. The main result of this paper is (4.l) Theorem, which asserts a statement analogous to the implication (c)::}(d) (local determinacy) for codim 1 foliation germs. As a special case, we also consider multiform functions. In this case we can generalize not only the local determinacy «5.6) Theorem) but also, as already in the work of Cerveau and Mattei [1], the global determinacy (a)::}(b) (Theorems (5.11) and (5.18». The difficulty in the foliation case in general is caused by the fact that the associated algebraic objects have only vector space structures and may not be invariant under multiplication by function germs, which prevents us from using such an algebraic tool as Nakayama's lemma. Thus we obtain the local determinacy by actually solving some differential equations.