Dynamics of the Segre varieties of a real submanifold in complex space

Abstract

For a smooth (or formal) generic submanifold M M of real codimension d d in complex space C N \mathbb {C}^N with 0 ∈ M 0\in M , we introduce the notion of a formal Segre variety mapping γ : ( C N × C N − d , 0 ) → ( C N , 0 ) \gamma : (\mathbb {C}^N\times \mathbb {C} ^{N-d},0)\to (\mathbb {C}^N,0) and its iterated Segre mappings at 0 0 , v j : ( C ( N − d ) j , 0 ) → ( C N , 0 ) v^j:(\mathbb {C}^{(N-d)j},0) \to (\mathbb {C}^N,0) , j ≥ 1 j\ge 1 . The Segre variety mapping γ \gamma extends the notion of Segre varieties of a real-analytic generic submanifold to the setting of smooth (or formal) submanifolds. One of the main results in this paper is that M M is of finite type (in the sense of Kohn and Bloom–Graham) at 0 0 if and only if there exists k 0 ≤ d + 1 k_0\le d+1 such that the (generic) rank of v k 0 v^{k_0} is N N . More generally, we prove that v k 0 v^{k_0} parameterizes the local CR orbit of M M at 0 0 .

Reflection Ideals and mappings between generic submanifolds in complex space

“…If the generic submanifolds M and M ′ are moreover real-algebraic, the map H is convergent, and the connected source manifold M is of finite type at some point, we prove (Theorem 2.7 below) that the reflection ideal has a set of algebraic generators. An important ingredient for the proofs of the above three theorems is the use of iterated Segre mappings, introduced in [BER96] (see also [BER00c]), which has already been applied to various mapping problems. Another important tool in the proofs is Artin's approximation theorem [A68] and an algebraic version of the latter in [A69].…”

Untitled

“…, (Z 2l+1 , ζ 2l+1 )) ∈ M (l) : Z 2l+1 = 0}. The reader can check that the map µ l | D l (0) coincides, up to a parametrization of M (l) , with a suitable iterated Segre mapping v 2l+1 at 0 as defined in [BER99b,BER00b]. Therefore, in view of the minimality criterion of [BER99a,BER00b] (see also [BER96]), the pair (λ, µ) is of finite type at 0 ∈ M if and only if M is minimal at 0.…”

On some rigidity properties of mappings between CR-submanifolds in complex space

“…It is the so-called Segre sets technique introduced by Baouendi, Ebenfelt and Rothschild [BER96,BER99a,BER03] that has been very successful for studying real-analytic and formal CR-maps between real-analytic CR-manifolds (see e.g. [BER96, BER97, Z97, BER98, BER99b, BER99a, Z99, M00, BRZ01a, BRZ01b, KZ01, La01, BMR02, M02a, M02b, MMZ03, BER03]).…”

“…After the pioneering work of Webster [W77], Segre varieties became a basic tool for mapping problems. For real-analytic generic submanifolds of higher codimension, Baouendi, Ebenfelt and Rothschild [BER96] (see also [BER99a,BER03]) introduced the Segre sets Q …”

“…It is often useful to parametrize the Segre sets by the so-called iterated Segre sets mappings (see [BER99a,BER03]). Since M is generic, we may assume, after possibly renumbering the coordinates, that the…”