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

• Published in: Proceedings of the London Mathematical Society 2022-06, Vol.124 (6), p.737-771
• Main Authors: Lamel, Bernhard, Mir, Nordine
• Format: Article
• Language: English
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms.12439

We prove the first general finite jet determination result in positive codimension for CR maps from real‐analytic minimal submanifolds M⊂CN$M\subset \mathbb {C}^N$ into Nash (real) submanifolds M′⊂CN′$M^{\prime }\subset \mathbb {C}^{N^{\prime }}$. 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 Sp$\mathcal {S}_p$ with jpℓ(p)f=jpℓ(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.