Embeddability of real hyersurfaces into hyperquadrics and spheres Ming Xiao University of California San Diego Midwestern Workshop on Asymptotic Analysis IUPUI, Indianapolis October 7th, 2017 Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Question When a real hypersurface M ⊂ C n admits a holomorphic ⊂ C N of transversal embedding into a hyperquadric H 2 N − 1 l possibly larger dimension? Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Hyperquadrics: l N � � | z i | 2 + | z i | 2 = 1 } ⊂ C N . H 2 N − 1 := {− l i = 1 i = l + 1 Transversal map F : dF does not map T p C n to T F ( p ) H 2 N − 1 at p ∈ M . l Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Chern-Moser thoery Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Chern-Moser thoery Various embedding theorems in geometry Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Various embedding theorems in geometry Differential Geometry: The Whitney embedding theorem Embedding of general smooth manifolds into their models (real Euclidean spaces) Riemannian Geometry: The Nash embedding theorem Embedding of general Riemannian manifolds into their models (real Euclidean spaces) Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Stein Space theory: The Remmert embedding theorem Embedding of Stein manifolds into their models (complex Euclidean spaces) Pseudoconformal geometry: One may ask whether there is such an analogue. Embedding of hypersurfaces into their models (hyperquadrics) Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Webster, 1978 Theorem (Webster, 1978, Duke Math. J.) Every real-algebraic Levi-nondegenerate real hypersurface M ⊂ C n is transversally holomorphically embeddable into a hyperquadric of suitable dimension and signature. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary However, not every real analytic Levi-nondegenerate hypersurface can be transversally holomorphically embedded into a hyperquadric of sufficient large dimension. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Forstneri´ c 1986, Faran 1988 Most real analytic strongly pseudoconvex hypersurface cannot be holomorphically embedded into any sphere. Forstner´ c 2004 Most real-analytic hypersurfaces do not admit a transversal holomorphic embedding into any real algebraic hypersurface, in particular, any hyperquadrics. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Explicit Example: Theorem (Zaitsev, 2008, Math. Ann.)The hypersurface in C 2 given by � Im w = | z | 2 + Re z k z ( k + 2 )! , ( z , w ) ∈ C 2 , | z | < ǫ. k ≥ 2 for any 0 < ǫ ≤ 1 is not transversally holomorphically embeddable into a hyperquadric of any dimension. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary Motivated by Webster’s theorem and embedding theorems in geometry: Equivalently, Is there a uniform bound for the minimal embedding dimension of M in terms of n : Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results Theorem (Kossovskiy-X., to appear in Advances in Math.) For any integers N > n > 1 , there exists µ = µ ( n , N ) such tha a Zariski generic real-algebraic hypersurface M ⊂ C n of degree k ≥ µ is not transversally holomorphically embeddable into any hyperquadric H 2 N − 1 ⊂ C N . l Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results We can give an explicit bound for µ ( n , N ) : � N ( N + 1 ) / 2 + p ( n , N ) � µ ( n , N ) = 2 + N − n + , p ( n , N ) � N − 1 � where p ( n , N ) = n − 1 + ( n − 1 ) n . 2 n − 1 Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results We can give an explicit bound for µ ( n , N ) : � N ( N + 1 ) / 2 + p ( n , N ) � µ ( n , N ) = 2 + N − n + , p ( n , N ) � N − 1 � where p ( n , N ) = n − 1 + ( n − 1 ) n . 2 n − 1 When n = 2 , N = 3 , we have µ ( 2 , 3 ) = 18 . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Preliminary We now concentrate on the strongly pseudoconvex case: Question Is every compact real-algebraic strongly pseudoconvex real hypersuraface in C n holomorphically embeddable into a sphere of sufficiently large dimension? Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces M ǫ ⊂ C 2 are constructed: M ǫ := { ( z , w ) ∈ C 2 : ε 0 ( | z | 8 + c Re | z | 2 z 6 )+ | w | 2 + | z | 10 + ε | z | 2 − 1 = 0 } , where 0 < ε < 1 , 0 < ε 0 << 1 , 2 < c < 16 7 . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces M ǫ ⊂ C 2 are constructed: M ǫ := { ( z , w ) ∈ C 2 : ε 0 ( | z | 8 + c Re | z | 2 z 6 )+ | w | 2 + | z | 10 + ε | z | 2 − 1 = 0 } , where 0 < ε < 1 , 0 < ε 0 << 1 , 2 < c < 16 7 . M ǫ is a real algebraic hypersurface. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces M ǫ ⊂ C 2 are constructed: M ǫ := { ( z , w ) ∈ C 2 : ε 0 ( | z | 8 + c Re | z | 2 z 6 )+ | w | 2 + | z | 10 + ε | z | 2 − 1 = 0 } , where 0 < ε < 1 , 0 < ε 0 << 1 , 2 < c < 16 7 . M ǫ is a real algebraic hypersurface. For small ε, ε 0 , M ε is diffeomorphic to S 3 ⊂ C 2 . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces M ǫ ⊂ C 2 are constructed: M ǫ := { ( z , w ) ∈ C 2 : ε 0 ( | z | 8 + c Re | z | 2 z 6 )+ | w | 2 + | z | 10 + ε | z | 2 − 1 = 0 } , where 0 < ε < 1 , 0 < ε 0 << 1 , 2 < c < 16 7 . M ǫ is a real algebraic hypersurface. For small ε, ε 0 , M ε is diffeomorphic to S 3 ⊂ C 2 . For 0 < ε < 1 , M ε is strongly pseudoconvex. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces M ǫ ⊂ C 2 are constructed: M ǫ := { ( z , w ) ∈ C 2 : ε 0 ( | z | 8 + c Re | z | 2 z 6 )+ | w | 2 + | z | 10 + ε | z | 2 − 1 = 0 } , where 0 < ε < 1 , 0 < ε 0 << 1 , 2 < c < 16 7 . M ǫ is a real algebraic hypersurface. For small ε, ε 0 , M ε is diffeomorphic to S 3 ⊂ C 2 . For 0 < ε < 1 , M ε is strongly pseudoconvex. M 0 has a Kohn-Nirenberg point at ( 0 , 1 ) . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results Theorem (Huang-X., 2016) For sufficient small ε, ε 0 , M ε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of M ε to a unit sphere must be constant. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results Theorem (Huang-X., 2016) For sufficient small ε, ε 0 , M ε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of M ε to a unit sphere must be constant. We thus give a negative answer to the question: Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Main Results Theorem (Huang-X., 2016) For sufficient small ε, ε 0 , M ε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of M ε to a unit sphere must be constant. We thus give a negative answer to the question: There exist compact, real algebraic, strongly pseudoconvex hypersurfaces that cannot be locally holomorphically embedded into any sphere. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Sketch of proof Proof: Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Sketch of proof Proof: Let F be a holomorphic map sending an open piece of M ε to some unit sphere S 2 N − 1 . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Sketch of proof Proof: Let F be a holomorphic map sending an open piece of M ε to some unit sphere S 2 N − 1 . Step 1: Rationality of F . Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Sketch of proof Proof: Let F be a holomorphic map sending an open piece of M ε to some unit sphere S 2 N − 1 . Step 1: Rationality of F . Step 1 (a): Algebraicity of F . Huang’s algebraicity theorem. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Sketch of proof Proof: Let F be a holomorphic map sending an open piece of M ε to some unit sphere S 2 N − 1 . Step 1: Rationality of F . Step 1 (a): Algebraicity of F . Huang’s algebraicity theorem. Step 1 (b): Single-valueness of F . monodromy argument. Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres
Recommend
More recommend
