The finite jet determination problem for CR maps of positive codimension into Nash manifolds

Abstract

We prove the first general finite jet determination result in positive codimension for CR maps from real-analytic minimal submanifolds M ⊂ C N $M\subset \mathbb {C}^N$ into Nash (real) submanifolds M ′ ⊂ C N ′ $M^{\prime }\subset \mathbb {C}^{N^{\prime }}$ . For a sheaf S $\mathcal {S}$ of C ∞ $\mathcal {C}^\infty$ -smooth CR maps from M $M$ into M ′ $M^{\prime }$ , we show that the non-existence of so-called 2-approximate CR S $\mathcal {S}$ -deformations from M $M$ into M ′ $M^{\prime }$ implies the following strong finite jet determination property: There exists a map ℓ : M → Z + $\ell \colon M\rightarrow {\mathbb {Z}}_+$ , bounded on compact subsets of M $M$ , such that for every point p ∈ M $p\in M$ , whenever f , g $f,g$ are two elements of S p $\mathcal {S}_p$ with j p ℓ ( p ) f = j p ℓ ( p ) g $j^{\ell (p)}_pf=j^{\ell (p)}_pg$ , then f = g $f=g$ . Applying the deformation point of view allows a unified treatment of a number of classes of target manifolds, which includes, among others, strictly pseudoconvex, Levi–non-degenerate, but also some particularly important Levi-degenerate targets, such as boundaries of classical domains.