Big denominators and analytic normal forms with an appendix by M. Zhitomirskii

Abstract

We study the {\it regular} action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object $F_0$ and we are interested in an analytic normal form for the whole affine space $\{F_0 + h.o.t.\}$. We prove that if the cohomological operator defined by $F_0$ has the big denominators property and if a formal normal form is well chosen then this formal normal form holds in analytic category. We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. %This theorem, which a priori has nothing to do with local classification problems,generalizes the theorem on analytic normal forms. Moreover, we prove that if the denominators grow "relatively fast", but not fast enough to satisfy the big denominator property, then we have a normal form, respectively local solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by explanation of known results and by new results in the problems of local classification of singularities of vector fields, non-isolated singularities of functions, tuples of germs of vector fields, local Riemannian metrics and conformal structures.