Approximation theorems in algebra and Cauchy-Riemann geometry: When Michael Artin meets Henri Poincaré – CRARTIN

In 1968, M. Artin's famous approximation theorem was published: Whenever an analytic system of equations has a formal solution, then this formal solution can be approximated by convergent solutions. Approximation happens in the sense of the Krull topology, that is, if $y(x)$ is a formal solution of $f(x,y)=0$, then for any $kinN$ there exists a convergent power series $y_k (x)$ such that $f(x,y_k(x)) = 0$ and $y - y_k$ vanishes to order $k$ (say at the origin). Whether or not a formal solution to some problem can be approximated by convergent solutions is a fundamental question in many different areas of mathematics. Indeed, the equivalence of given geometric structures or dynamical systems (of analytic type) is often reduced to the question of the formal equivalence of the given objects; then one usually tries to determine conditions (on the given objects) which force the formal equivalence to be an analytic equivalence. In CR geometry, the corresponding problem, that is the main topic of this project, may be stated in its full generality as follows: Assume that we are given two germs of real-analytic submanifolds $(M,0)subset C^N$ and $(M',0)subset (C^{N'},0)$. A formal map $Hcolon (M,0) o (M',0)$ is a formal holomorphic power series map $H$ which has the property that for every real-analytic function $varphi$ vanishing on $M'$, the formal real-analytic function $varphi circ H$ vanishes on $M$. The first natural problem is : Under which conditions on $(M,0)$ and $(M',0)$ does it hold that any formal holomorphic map $Hcolon(M,0) o (M',0)$ can be approximated by convergent holomorphic maps $H_k colon (M,0) o (M',0)$ such that $H$ agrees with $H_k$ up to order $k$? (Problem 1). If, for a pair of manifolds $(M,0)$ and $(M',0)$ the property stated in Problem 1 holds, we shall say that $(M,0)$ and $(M',0)$ have the Artin approximation property. Problem 1 can be seen as a mixture of the real and complex version of Artin's approximation theorem. Indeed, the manifolds one has to deal with are real-analytic, and one tries to get complex-analytic convergent maps between them. The fact that one definitely needs to impose conditions on $(M,0)$ and $(M',0)$ follows from a famous example found by Moser and Webster [10] in 1983: There exist real-analytic submanifolds $(M,0)$ and $(M',0)$ of $C^2$ of (real) codimension $2$ which are formally equivalent but there does not exist a convergent (holomorphic) transformation taking $(M,0)$ into $(M',0)$. That is, there is a formal biholomorphic transformation $Hcolon (M,0) o(M',0)$, but no convergent holomorphic invertible map takes $(M,0)$ into $(M',0)$---in particular, the approximation property does not hold for this pair of manifolds. In Moser and Webster's example, the point is that the manifolds are totally real at all points except for the point $0$, where they have a so-called complex tangent: If we denote the maximal complex subspace of $T_p M$ by $T_p^cM$, then $dim T_p^c M = 0$ for $ M i p eq 0$, and $dim T_0^c M = 0$. So for the approximation property to hold, it seems reasonable (though not necessary) to require at least that the dimension of the complex tangent spaces is constant; this condition is denoted by saying that $M$ is CR or Cauchy-Riemann. The common dimension of the complex tangent spaces is the CR-dimension of $M$. This brings us to a very general question asked, among others, by F. Treves: Does every pair of CR-manifolds $(M,0)$ and $(M',0)$ have the approximation property? This second problem is the heart of the present project.