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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

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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

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Preliminary

Question When a real hypersurface M ⊂ Cn admits a holomorphic transversal embedding into a hyperquadric H2N−1

l

⊂ CN of possibly larger dimension?

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Preliminary

Hyperquadrics: H2N−1

l

:= {−

l

|zi|2 +

N

i=l+1

|zi|2 = 1} ⊂ CN. Transversal map F : dF does not map TpCn to TF(p)H2N−1

l

at p ∈ M.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Preliminary

Chern-Moser thoery

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Preliminary

Chern-Moser thoery Various embedding theorems in geometry

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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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

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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

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Preliminary

Webster, 1978 Theorem (Webster, 1978, Duke Math. J.) Every real-algebraic Levi-nondegenerate real hypersurface M ⊂ Cn is transversally holomorphically embeddable into a hyperquadric of suitable dimension and signature.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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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

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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

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Preliminary

Explicit Example: Theorem (Zaitsev, 2008, Math. Ann.)The hypersurface in C2 given by Imw = |z|2 + Re

k≥2

zkz(k+2)!, (z, w) ∈ C2, |z| < ǫ. 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

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Preliminary

Motivated by Webster’s theorem and embedding theorems in geometry: Equivalently, Is there a uniform bound for the minimal embedding dimension

f M in terms of n :

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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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 ⊂ Cn of degree k ≥ µ is not transversally holomorphically embeddable into any hyperquadric H2N−1

l

⊂ CN.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

We can give an explicit bound for µ(n, N) : µ(n, N) = 2 + N − n + N(N + 1)/2 + p(n, N) p(n, N)

where p(n, N) = n − 1 + (n−1)n

2

N − 1 n − 1

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

We can give an explicit bound for µ(n, N) : µ(n, N) = 2 + N − n + N(N + 1)/2 + p(n, N) p(n, N)

where p(n, N) = n − 1 + (n−1)n

2

N − 1 n − 1

When n = 2, N = 3, we have µ(2, 3) = 18.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Preliminary

We now concentrate on the strongly pseudoconvex case: Question Is every compact real-algebraic strongly pseudoconvex real hypersuraface in Cn holomorphically embeddable into a sphere

f sufficiently large dimension?

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces Mǫ ⊂ C2 are constructed: Mǫ := {(z, w) ∈ C2 : ε0(|z|8+cRe|z|2z6)+|w|2+|z|10+ε|z|2−1 = 0}, where 0 < ε < 1, 0 < ε0 << 1, 2 < c < 16

7 .

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces Mǫ ⊂ C2 are constructed: Mǫ := {(z, w) ∈ C2 : ε0(|z|8+cRe|z|2z6)+|w|2+|z|10+ε|z|2−1 = 0}, where 0 < ε < 1, 0 < ε0 << 1, 2 < c < 16

7 .

Mǫ is a real algebraic hypersurface.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces Mǫ ⊂ C2 are constructed: Mǫ := {(z, w) ∈ C2 : ε0(|z|8+cRe|z|2z6)+|w|2+|z|10+ε|z|2−1 = 0}, where 0 < ε < 1, 0 < ε0 << 1, 2 < c < 16

7 .

Mǫ is a real algebraic hypersurface. For small ε, ε0, Mε is diffeomorphic to S3 ⊂ C2.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces Mǫ ⊂ C2 are constructed: Mǫ := {(z, w) ∈ C2 : ε0(|z|8+cRe|z|2z6)+|w|2+|z|10+ε|z|2−1 = 0}, where 0 < ε < 1, 0 < ε0 << 1, 2 < c < 16

7 .

Mǫ is a real algebraic hypersurface. For small ε, ε0, Mε is diffeomorphic to S3 ⊂ C2. For 0 < ε < 1, Mε is strongly pseudoconvex.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

In Huang-Li-X., 2015, I.M.R.N, the hypersurfaces Mǫ ⊂ C2 are constructed: Mǫ := {(z, w) ∈ C2 : ε0(|z|8+cRe|z|2z6)+|w|2+|z|10+ε|z|2−1 = 0}, where 0 < ε < 1, 0 < ε0 << 1, 2 < c < 16

7 .

Mǫ is a real algebraic hypersurface. For small ε, ε0, Mε is diffeomorphic to S3 ⊂ C2. For 0 < ε < 1, Mε is strongly pseudoconvex. M0 has a Kohn-Nirenberg point at (0, 1).

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

Theorem (Huang-X., 2016) For sufficient small ε, ε0, Mε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of Mε to a unit sphere must be constant.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Main Results

Theorem (Huang-X., 2016) For sufficient small ε, ε0, Mε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of Mε 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

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Main Results

Theorem (Huang-X., 2016) For sufficient small ε, ε0, Mε cannot be locally holomorphically embedded into any sphere. More precisely, any holomorphic map that sends an open piece of Mε 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

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Sketch of proof

Proof:

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−1.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−1. Step 1: Rationality of F.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−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

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−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

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−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. A lemma of Huang-Zaitsev: Topology of the complement of the branching variety

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Proof: Let F be a holomorphic map sending an open piece of Mε to some unit sphere S2N−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. A lemma of Huang-Zaitsev: Topology of the complement of the branching variety Step 1 (c): A theorem of Chiappari ⇒ F extends to a holomorphic map in a neighborhood of Dε.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 2: Study Qp0 ∩ Mε, where p0 = (0, 1) is the Kohn-Nirenberg point of M0.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 2: Study Qp0 ∩ Mε, where p0 = (0, 1) is the Kohn-Nirenberg point of M0. Note p0 is on Mε for every 0 ≤ ε < 1. Moreover, Qp0 = {w = 1}. Recall Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 2: Study Qp0 ∩ Mε, where p0 = (0, 1) is the Kohn-Nirenberg point of M0. Note p0 is on Mε for every 0 ≤ ε < 1. Moreover, Qp0 = {w = 1}. Recall Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0. There is p ∈ Qp0 such that p ∈ D0.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 2: Study Qp0 ∩ Mε, where p0 = (0, 1) is the Kohn-Nirenberg point of M0. Note p0 is on Mε for every 0 ≤ ε < 1. Moreover, Qp0 = {w = 1}. Recall Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0. There is p ∈ Qp0 such that p ∈ D0. For 0 < ε << 1, p ∈ Dε.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 2: Study Qp0 ∩ Mε, where p0 = (0, 1) is the Kohn-Nirenberg point of M0. Note p0 is on Mε for every 0 ≤ ε < 1. Moreover, Qp0 = {w = 1}. Recall Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0. There is p ∈ Qp0 such that p ∈ D0. For 0 < ε << 1, p ∈ Dε. Qp0 ∩ Mε is of real dimension one ⇒ Qp0 ∩ Mε has an accumulation point q0.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 3: F is a constant map. F preserves the Segre varieties: F(Qp0 ∩ U) ⊂ QF(p0).

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 3: F is a constant map. F preserves the Segre varieties: F(Qp0 ∩ U) ⊂ QF(p0). By unique continuation, if q ∈ Qp0 ∩ Mε, then F(q) ∈ QF(p0) ∩ S2N−1 = {F(p0)}. ⇒ F is not one-to-one near q0, where q0 is an accumulation point of Qp0 ∩ Mǫ.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 3: F is a constant map. F preserves the Segre varieties: F(Qp0 ∩ U) ⊂ QF(p0). By unique continuation, if q ∈ Qp0 ∩ Mε, then F(q) ∈ QF(p0) ∩ S2N−1 = {F(p0)}. ⇒ F is not one-to-one near q0, where q0 is an accumulation point of Qp0 ∩ Mǫ. F is a constant map.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 3: F is a constant map. F preserves the Segre varieties: F(Qp0 ∩ U) ⊂ QF(p0). By unique continuation, if q ∈ Qp0 ∩ Mε, then F(q) ∈ QF(p0) ∩ S2N−1 = {F(p0)}. ⇒ F is not one-to-one near q0, where q0 is an accumulation point of Qp0 ∩ Mǫ. F is a constant map. Suppose not. Hopf lemma type argument ⇒ F is local embedding at q0.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Sketch of proof

Step 3: F is a constant map. F preserves the Segre varieties: F(Qp0 ∩ U) ⊂ QF(p0). By unique continuation, if q ∈ Qp0 ∩ Mε, then F(q) ∈ QF(p0) ∩ S2N−1 = {F(p0)}. ⇒ F is not one-to-one near q0, where q0 is an accumulation point of Qp0 ∩ Mǫ. F is a constant map. Suppose not. Hopf lemma type argument ⇒ F is local embedding at q0. This is a contradiction.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Remarks

Each Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0 can be transversally holomorphically embedded into the hyperquadric in C6 with one negative Levi eigenvalue: H11

1 =

(z1, ..., z6) ∈ C6 :

5

|zi|2 − |z6|2 = 1

F(z, w) = √ε0z4, z5, √εz, w, 1 2 √ε0c(z7 + z), 1 2 √ε0c(z7 − z)

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Remarks

Each Mε : ε0(|z|8 + cRe|z|2z6) + |w|2 + |z|10 + ε|z|2 − 1 = 0 can be transversally holomorphically embedded into the hyperquadric in C6 with one negative Levi eigenvalue: H11

1 =

(z1, ..., z6) ∈ C6 :

5

|zi|2 − |z6|2 = 1

F(z, w) = √ε0z4, z5, √εz, w, 1 2 √ε0c(z7 + z), 1 2 √ε0c(z7 − z)

A lot more examples can be constructed.

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres

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Thank you very much for your attention!

Ming Xiao Embeddability of real hyersurfaces into hyperquadrics and spheres