Hyperbolic CR singularities Laurent Stolovitch Zhiyan Zhao Laboratoire J.A. Dieudonn´ e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 Virtual Conference on Several Complex Variables, August 2020 Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 1 / 21

Surfaces with CR singularity Surface with CR singularity : real analytic surface M ⊂ ( C 2 , 0): z 1 + γ ( z 2 z 2 1 ) + O 3 ( z 1 , ¯ M : z 2 = z 1 ¯ 1 + ¯ z 1 ) , γ ≥ 0 . z 1 + γ ( z 2 z 2 r.a. perturbation of the Bishop quadric Q γ : z 2 = z 1 ¯ 1 + ¯ 1 ) γ ∈ R + — Bishop invariant If γ � = 1 2 , the origin is an isolated Cauchy-Riemann singularity : ∀ p � = 0, ” C �⊂ T p M ” (ie. totally real at p � = 0) T 0 M = { z 2 = 0 } M is said to be : elliptic si 0 ≤ γ < 1 2 hyperbolic if γ > 1 2 parabolic if γ = 1 2 Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 2 / 21

Geometry near an elliptic CR singularity Questions • Holomorphic Flattening : is φ ( M ) ⊂ Im( z 2 ) = 0 ? • What is the local hull of holomorphy ? Answers throught : Normal form of M with respect to holomorphic change of coordinates near the origin. Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 3 / 21

Normalization near an elliptic CR singularity Theorem (Moser-Webster 1983) If 0 < γ < 1 2 , there exists a holomorphic change of variables near the origin such that M reads z 1 + ( γ + δx s 2 )( z 2 z 2 x 2 = z 1 ¯ 1 + ¯ 1 ) , y 2 = 0 , z 2 = x 2 + i y 2 avec δ = ± 1 si s ∈ N ∗ ou δ = 0 si s = ∞ . Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 4 / 21

Complexification of M z 2 ) ← ( z 1 , z 2 , w 1 , w 2 ) =: ( z, w ) ∈ C 4 Complexification of M : ( z 1 , z 2 , ¯ z 1 , ¯ � z 2 = z 1 w 1 + γ ( z 2 1 + w 2 1 ) + H ( z 1 , w 1 ) M ⊂ C 4 : 1 ) + ¯ w 2 = z 1 w 1 + γ ( z 2 1 + w 2 H ( w 1 , z 1 ) Canonical projections : π 1 ( z, w ) = z et π 2 ( z, w ) = w for ( z, w ) ∈ M . According to Moser-Webster, π 1 et π 2 are 2-1 branched coverings: π 2 ( z, w ) = π 2 ( z ′ , w ′ ), ( z, w ) , ( z ′ , w ′ ) ∈ M ⇒ unique solution ( z ′ , w ′ ) =: τ 1 ( z, w ) with z ′ � = z = � ( z − z ′ ) ( w + γ ( z + z ′ )) + H ( z, w ) − H ( z ′ , w ) = 0 Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 5 / 21

Moser-Webster involutions � pair of holomorphic involutions: pour γ > 0 z ′ 1 = − z 1 − 1 γ w 1 + h 1 ( z 1 , w 1 ) � �� � τ 1 : − − − − − − τ 1 ◦ τ 1 = Id ord 0 ≥ 2 w ′ 1 = w 1 � z ′ 1 = z 1 τ 2 : − − − − − − τ 2 ◦ τ 2 = Id w ′ 1 = − 1 γ z 1 − w 1 + h 2 ( z 1 , w 1 ) τ 2 = ρτ 1 ρ, ρ ( z, w ) := ( ¯ w, ¯ z ) Proposition (Moser-Webster 1983) Holomorphic classification of surface M ∈ C 4 � Holomorphic classification of ( τ 1 , τ 2 ) Remark. Normal form of M ⊂ C 2 � Normal form of ( τ 1 , τ 2 ). Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 6 / 21

Appropriate coordinates � ξ ′ = λη + h . o . t . � ξ ′ = λ − 1 η + h . o . t . τ 1 : η ′ = λ − 1 ξ + h . o . t . , τ 2 : , η ′ = λξ + h . o . t . � ξ ′ = λ 2 ξ + h . o . t . σ := τ 1 ◦ τ 2 : η ′ = λ − 2 η + h . o . t . , λ is a root of γλ 2 − λ + γ = 0 Remark ⇒ λ = ¯ elliptic surface M , 0 < γ < 1 2 = λ and | λ | � = 1 — origin is an hyperbolic fixed point of τ 1 , τ 2 et τ 1 ◦ τ 2 hyperbolic surface M , γ > 1 2 = ⇒ | λ | = 1 — origin is an elliptic fixed point of τ 1 , τ 2 et σ = τ 1 ◦ τ 2 Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 7 / 21

Normal forms of involutions Theorem (Moser-Webster 1983, formal normal form) Assume: λ not a root of unity Conclusion : exists a unique formal normalized transformation ψ s.t. � ξ ′ = Λ( ξη ) η � ξ ′ = Λ − 1 ( ξη ) η ψ − 1 ◦ τ 1 ◦ ψ : ψ − 1 ◦ τ 2 ◦ ψ : , , η ′ = Λ − 1 ( ξη ) ξ η ′ = Λ( ξη ) ξ where Λ( t ) ∈ C [[ t ]] . s.t. Λ( t ) = ¯ Λ( t ) (elliptic case) ou Λ( t ) · ¯ Λ( t ) = 1 (hyperbolic case). Theorem (Moser-Webster 1983, Convergence in elliptic case) If λ = ¯ λ and | λ | � = 1 , then Λ and ψ are holomorphic on a neighborhood of the origin. = ⇒ Holomorphic equivalence of inital manifold M to NF manifold Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 8 / 21

Non exceptional hyperbolic CR singularity | λ | = 1 not a root of unity ( non exceptionnal ). Moser-Webster � normalizing transformation ψ might not converge at the origin: no holomorphic equivalence to a normal form and even, no holomorphic flattening. Theorem (Gong 1994: non exceptional degenerate case) Assumptions: 1 | λ | = 1 and λ satisfies diophantine condition: | λ n − 1 | > c n δ 2 τ 1 et τ 2 are formally linearizable (i.e. Λ( ξη ) = λ ; i.e. M formally equivalent to the quadric), Then, ψ is holomorphic in a neighborhood of the origin : M is holomorphically equivalent to the quadric. Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 9 / 21

Non degenerate hyperbolic CR singularity surface � ξ ′ = λη + h . o . t . � ξ ′ = λ − 1 η + h . o . t . τ 1 : η ′ = λ − 1 ξ + h . o . t . , τ 2 : η ′ = λξ + h . o . t . � i 2 α , α c n ( ξη ) n . λ := e π ∈ R \ Q , Λ( ξη ) = λ + ˜ n ≥ 1 Theorem (S.-Zhao 2020) Assume Λ( ξη ) � = λ . If r > 0 is small enough, there exists a ”asymptotic full measure” parameters set O r ⊂ ] − r 2 , r 2 [ s.t. ∀ ω ∈ O r , ∃ µ ω ∈ R and an holomorphic transformation Ψ ω on C r ω := { ξη = ω, | ξ | , | η | < r } with Ψ ω ◦ ρ = ρ ◦ Ψ ω and s.t. , on C r ω , � ξ ′ = e � ξ ′ = e − i i 2 µ ω η 2 µ ω η Ψ − 1 Ψ − 1 ω ◦ τ 1 ◦ Ψ ω : ω ◦ τ 2 ◦ Ψ ω : , , η ′ = e − i η ′ = e i 2 µ ω ξ 2 µ ω ξ Remark Ψ ω ( C r ω ) is an holomorphic invariant set of τ i ’s and their restriction is conjugated to a linear map . ”Asymptotic full measure”= |O r | r → 0 − → 1. 2 r 2 Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 10 / 21

Geometric consequences Theorem (S.-Zhao 2020) Let M be a surface with an hyberbolic CR singularity at the origin which is non exceptionnal and not formally equivalent to a quadric. Then: there exist a neighborhood of the origin and a family of holomorphic curves {S ω } ω ∈O which intersects M along holomorphic hyperbolas : 2 real curves which are simultaneously holomorphically mapped to the two branches of the hyperbolas ξη = ω , ω � = 0 . Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 11 / 21

Intersection at the origin of M by an holomorphic curve M S ω Ψ − 1 ω ξη = ω Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 12 / 21

Intersection of M along 2 real lines at the origin Theorem (Klingenberg 1985) Let M be a surface with an hyberbolic CR singularity at the origin with 2 α satisfiying the diophantine condition above. Then , there exists a i λ = e unique holomorphic curve intersecting M along 2 totally real curves interesecting transversally at the origin. Remarque. These are the ”traces” of 2 lines ξη = 0. Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 13 / 21

Idea of the proof : KAM (Kolmogorv-Arnold-Moser) scheme Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 14 / 21

KAM (Kolmogorv-Arnold-Moser) scheme — formulation Pair of holomorphic involutions � ξ ′ = e i 2 α ( ξη ) η + p ( ξ, η ) τ 2 = ρ ◦ τ 1 ◦ ρ, ρ : ( ξ, η ) �→ (¯ τ 1 : 2 α ( ξη ) ξ + q ( ξ, η ) , ξ, ¯ η ) η ′ = e − i restricted to a “crown” C r ω,β := {| ξη − ω | < β, | ξ | , | η | < r } , ω ∈ O ⊂ ] − r 2 , r 2 [. Non degeneracy: ∃ s ∈ N ∗ , ∀ ω ∈ O , | α ( s ) ( ω ) | > 1 2 , Laurent Stolovitch, Zhiyan Zhao (Laboratoire J.A. Dieudonn´ Hyperbolic CR singularities e, Universit´ e Cˆ ote d’Azur, Nice, France August 18, 2020 15 / 21

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