aa r X i v : . [ m a t h . DG ] O c t PROJECTIVE INVARIANTS OF CR–HYPERSURFACES
C. HAMMOND AND C. ROBLES ∗ Abstract.
We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invari-ants for analytic hypersurfaces is given. The self-dual strongly C -linearly convex hyper-surfaces are characterized. Introduction
History.
Before stating the equivalence problem addressed in this paper, we describetwo related problems previously addressed in the literature.Cartan studied the equivalence of real-analytic CR-hypersurfaces in complex 2-manifoldsup to local biholomorphism [4, 3]. Chern and Moser generalized Cartan’s result [6, 7], solvingthe equivalence problem for nondegenerate, real-analytic CR-hypersurfaces in complex n -manifolds modulo local biholomorphism [6, Theorem 4.6].Following Chern and Moser, Jensen considered the equivalence problem for nondegenerateCR-hypersurfaces of complex projective space P W up to (local) projective deformation[11, 12, 13]. He showed that two nondegenerate CR-hypersurfaces in P W are locally first-order projective deformations of each other if and only if they locally biholomorphicallyequivalent [13, Theorem 7.2], and that two smooth CR–hypersurfaces are locally projectivelyequivalent if and only if they are locally second-order projective deformations of each other[12].1.2. Statement of the problem.
In this paper we apply techniques of E. Cartan [5, 9] tocharacterize CR-hypersurfaces in complex projective space up to projective transformation.While a projective transformation is a local biholomorphism of complex projective space P W , it is not the case that every local biholomorphism of P W is a projective transformation:in particular we consider the equivalence problem under a smaller transformation group thanCartan and Chern–Moser. Likewise, Jensen’s projective deformation is a more also a moregeneral equivalence than projective congruence. (See Remark 3.8.)Let W be a complex vector space, let P W denote the associated complex projective space.Given w ∈ W , let [ w ] denote the corresponding point in P W . Let GL( W ) denote the spaceof complex linear automorphisms of W . Date : November 7, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Equivalence problem, CR-hypersurface. ∗ Robles is partially supported by NSF-DMS 0805782 & 1006353.
Definition . Every A ∈ GL( W ) naturally induces a projective linear transformation A : P W → P W by [ w ] [ Aw ]. Two submanifolds S, e S ∈ P W are projectively equivalent ifthere exists a projective linear transformation A such that A ( S ) = e S .1.3. Contents.
The equivalence problem is lifted from CR-hypersurfaces S ⊂ P W to framebundles F S ⊂ GL( W ) over S in Section 2.1. Projective differential invariants (of all or-ders) are defined in Section 3.1, and in Section 3.2 it is shown that set of invariants iscomplete (in the analytic category): they characterize the analytic CR-hypersurfaces up toprojective equivalence (Proposition 3.11). Modulo normalizations (in Section 5), the setof second-order projective invariants is precisely the CR–second fundamental form. This isillustrated in Section 4 which presents the invariants from a slightly different, but sometimescomputationally more convenient, perspective.The self-dual strongly C –linearly convex hypersurfaces are characterized in Section 6, seeTheorem 6.8.1.4. Notation.
Fix the index ranges0 ≤ j, k ≤ n + 1 , ≤ a, b ≤ m , ≤ s, t ≤ n < α, β < m , < σ, τ < n . Let V be a real vector space of dimension n + 2 = 2 m + 2 with a complex structure J : V → V , J = − Id. Give ( V, J ) the structure of a complex vector space W by definingi v := J v , where i = √−
1. Define an equivalence relation on V \{ } by v ∼ ( xv + y J v ) forany (0 , = ( x, y ) ∈ R . Then the quotient space is naturally identified with P W ≃ CP m .Fix a basis e = ( e , . . . , e n +1 ) of V with the property that(1.2) J e a = e a +1 and J e a +1 = − e a ∀ ≤ a ≤ m . The f a := e a , ≤ a ≤ m , form a basis of of the complex vector space W . Remark . At times we will find it most convenient to work with the frame e , while atother times the frame f is better suited to the computation at hand. For this reason wewill develop both perspectives in the sections below. See also Remark 2.10.Define gl ( V, J ) := { X ∈ gl ( V ) | X J = J X } ≃ gl ( W ). Let X ∈ gl ( V ) be given by a matrix( X jk ) with respect to the basis ( e , . . . , e n +1 ). Then X ∈ gl ( V, J ) if and only if(1.4) X a b = X a +12 b +1 and X a b +1 = − X a +12 b . In this case, as an element of gl ( W ), X is given by the matrix ( Y ab ),(1.5) Y ab = X a b + i X a +12 b with respect to the basis ( f , . . . , f m ) of W . ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 3 Frame bundles
Frame bundle over P W . Let F denote the set of frames (or bases) e = ( e , . . . , e n +1 )of V satisfying (1.2). If we fix a frame, then F may be identified with GL( V, J ) ≃ GL( W ).Given A ∈ GL( W ), let L A : GL( W ) → GL( W ) denote left-multiplication by A . Let π : GL( W ) → P W denote the projection π ( e ) = [ e ]. Then the diagram below commutes. P W P W ✲ A GL( W ) GL( W ) ✲ L A ❄ π ❄ π The Maurer-Cartan form.
The gl ( V, J )–valued Maurer-Cartan 1-form ω = ( ω jk ) on F is defined by(2.1) d e j = ω kj e k . It is straight-forward to check that ω is left-invariant:(2.2) L ∗ A ω = ω , for all A ∈ GL( W ). We also have the Maurer-Cartan equation (2.3) d ω jk = − ω jℓ ∧ ω ℓk , and, by (1.2),(2.4) ω a b = ω a +12 b +1 and ω a b +1 = − ω a +12 b . If(2.5) Ω ba := ω b a + i ω b +12 a , then(2.6) d f a = Ω ba f b and we have the Maurer-Cartan equation (2.7) dΩ ba = − Ω bc ∧ Ω ca . In particular, Ω = (Ω ab ) is the gl ( W )–valued Maurer-Cartan form. In analogy with (2.2) wehave(2.8) L ∗ A Ω = Ω . For later convenience we note that(2.9) ω b a = (Ω ba + ¯Ω ba ) , ω b +12 a = − i2 (Ω ba − ¯Ω ba ) . Remark . As a follow up to Remark 1.3 we note that generally the gl ( W )–valuedMaurer-Cartan form Ω is more convenient in computations. However, we found that the gl ( V, J )–valued ω has the advantage of yielding a general formula for the k -th order dif-ferential invariants (Proposition 3.6). So we will continue to work with both, favoring themore convenient form for the computation at hand, and often giving equivalent statementsfor each. C. HAMMOND AND C. ROBLES
Change of frame.
It will be necessary to understand how Ω varies under a changeof frame. Consider a smooth map g = ( g ab ) : F → GL m +1 C . Let ˜ f a = g ba ( f ) f b . This mapinduces G : F → F by mapping the frame f = ( f a ) to the frame G ( f ) = ( g ba ( f ) f b ) = ˜ f .Let e Ω denote the pull-back G ∗ (Ω ˜ f ). Then (2.6) yields(2.11) e Ω ba = ( g − ) bc d g ca + ( g − ) bc Ω cd g da . Frame bundle over a CR–hypersurface in P W . Let S ⊂ P W be a CR–hypersurface.Let b S := { v ∈ V \{ } | [ v ] ∈ S } denote the cone over S . Then b S is a CR–hypersurface in( V, J ). If z = [ e ] ∈ S , let b T z S ⊂ V be the ( n + 1)–plane tangent to b S at e . (We distinguishthe linear subspace b T z S from the intrinsic tangent space T e b S .) Define F S := { ( e , . . . , e n +1 ) ∈ F | z = [ e ] ∈ S , span { e , . . . , e n } = b T z S } . Remark . Given A ∈ GL( W ), notice that L A ( F S ) = F e S where e S = A ( S ) = π ◦ L A ( F S ).In particular, two CR-hypersurfaces S, e S ⊂ P W are projectively equivalent if and only ifthere exists A ∈ GL( W ) such that F e S = L A ( F S ).The key ingredient in the solution of the projective equivalence problem is the following. Proposition 2.13.
Two CR-hypersurface S, e S ⊂ P W are projectively equivalent if andonly if there exists a smooth map φ : F S → F e S such that φ ∗ (Ω |F e S ) = Ω |F S . (Equivalently, φ ∗ ( ω |F e S ) = ω |F S .) The proof of the proposition makes use of the following well-known theorem of E. Cartan.
Theorem 2.14 (Cartan [10, Theorem 1.6.10]) . let G be a Lie group with Lie algebra g and Maurer-Cartan form ϑ . Let M be a manifold with a g –valued 1-form η satisfying theMaurer-Cartan equation d η = − [ η, η ] . Then for any p ∈ M there exists a neighborhood U and a map f : U → G such that f ∗ ϑ = η . Moreover any two such maps f , f : U → G arerelated by f = L g ◦ f for some g ∈ G .Remark. In the case that g is a matrix Lie algebra, d η = − [ η, η ] is (2.3). Proof of Proposition 2.13.
The proposition follows directly from Theorem 2.14 by taking G = GL( W ) and ϑ = Ω with M = F S and η = Ω |F S , f the inclusion map and f = φ . (cid:3) Consider the pull-back of the Maurer-Cartan form to F S . By construction { d e ( ξ ) | ξ ∈ T v F S } = span { e , . . . , e n } so that the ω , ω , . . . , ω n are linearly independent on F S , and(2.15) ω n +10 = 0 (2.9) ⇐⇒ Ω m − ¯Ω m = 0 . Definition.
A tangent vector v ∈ T F S is vertical if π ∗ ( v ) = 0, where π : F S → S is thenatural projection. A 1-form ̺ on F S is semi-basic if ̺ ( v ) = 0 for all vertical v ∈ T F S . Remark . The equation d e = ω j e j implies that the linearly independent ω , . . . , ω n span the semi-basic 1-forms. Equivalently, 1-forms Ω α , ¯Ω α and Ω m are linearly independenton F S and span the semi-basic 1-forms (as a real vector space). ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 5 Projective differential invariants w.r.t ω Definition and construction.
An application of the Maurer-Cartan equation (2.3)to (2.15) yields(3.1) 0 = − d ω n +10 = ω n +11 ∧ ω + ω n +12 ∧ ω + · · · + ω n +1 n ∧ ω n . Recall that the ω , . . . , ω n are linearly independent on F S , by Remark 2.16. The followinglemma is well-known. Lemma 3.2 (Cartan’s Lemma [2, 10] for ω ) . Suppose there exist real-valued 1-forms ξ , . . . , ξ n on F S such that ξ s ∧ ω s = 0 . Then there exist functions H st = H ts : F S → R , ≤ s, t ≤ n , such that ξ s = H st ω t . Lemma 3.2 and (3.1) imply there exist functions h st = h ts : F S → R , 1 ≤ s, t ≤ n , suchthat(3.3) ω n +1 s = h st ω t , ≤ s ≤ n . Note that ω n = ω n +11 = h s ω s . This forces(3.4) h s = h s = 0 , ∀ s < n , and h n = h n = 1 . The h (2) = { h st } are second-order projective differential invariants: they describe thesecond-order differential geometry of S at z = [ e ]; see Section 3.2 for more detail. The { h στ } ⊂ h (2) are the coefficients of the CR second fundamental form; see Section 4.3.We repeat the process above to obtain third-order invariants: apply the Maurer-Cartanequation (2.3) to differentiate (3.3). An application of Cartan’s Lemma 3.2 to the resulting2-form yields functions h rst : F S → R that are fully symmetric in the indices such that(3.5) d h rs + h rs ( ω + ω n +1 n +1 ) − h rt ω ts − h ts ω tr = h rst ω t . The together the coefficients h (2) = { h st } and h (3) = { h rst } describe the geometry of S to third order. See § p +1)-st order invariants h s ··· s p t . First some notation: given two tensors T s ...s p and U s p +1 ...s p + q ,let S p + q denote the symmetric group on p + q letters. Let T ( s ...s p U s p +1 ...s p + q ) = 1( p + q )! X σ ∈ S p + q T σ ( a ) ...σ ( s p ) U σ ( s p +1 ) ...σ ( s p + q ) denote the symmetrization of their product. For example, T ( s U s ) = ( T s U s + T s U s ).We exclude from the symmetrization operation any index that is outside the parentheses.For example, in h t ( s ...s p − ω ts p ) we symmetrize over only the s i , excluding the t index. C. HAMMOND AND C. ROBLES
Proposition 3.6.
Set h s = 0 and assume p > . There exist functions h s ··· s p t : F S → R ,fully symmetric in their indices, such that h s ...s p t ω t = d h s ...s p + ( p − h s ...s p ω + h s ...s p ω n +1 n +1 + p n ( p − h ( s ...s p − ω s p ) − h t ( s ...s p − ω ts p ) o + p − X j =1 (cid:0) pj (cid:1) n ( j − h ( s ...s j h s j +1 ...s p ) ω n +1 − h t ( s ...s j h s j +1 ...s p ) ω tn +1 o . Definition.
We let h ( p ) = { h s ··· s p } denote the p -th order coefficients; and h = ∪ h ( p ) denotethe complete set of coefficients. Remark . The h are projective invariants. To be precise, suppose that e ∈ F S . Let A ∈ GL( W ) and set e S = A ( S ). Then ˜ e = L A ( e ) ∈ F e S by Remark 2.12. From Proposition 3.6 andthe left-invariance (2.2) of ω , we see that h ( e ) = h (˜ e ). Thus h is invariant under projectivetransformation. In Proposition 3.11 we show that, for fixed e ∈ F S , the coefficients h ( e )characterize an analytic CR-hypersurface up to projective transformation. Remark.
The coefficients h are the CR–analogs of the coefficients of the Fubini forms of acomplex subvariety of P W , cf. [10, Chapter 3]. Definition.
In this setting, we say two CR-hypersurfaces S, e S ⊂ P W agree to order p ≥ φ : F S → F e S with the property that ω σ = φ ∗ e σ σ , ω n = φ ∗ e ω n (first-order agreement) and φ ∗ e h ( q ) = h ( q ) for all 2 ≤ q ≤ p . Remark . This is a considerably weaker condition than Jensen’s notion of p -th orderprojective deformation when p ≥
2. For example, in our case second order agreement isequivalent to ω σ = φ ∗ e ω σ , ω n = φ ∗ e ω n and ω n +1 s = φ ∗ e ω n +1 s . Jensen’s condition that S and e S be second-order projective deformations of each other includes the additional relationsΩ ab = φ ∗ e Ω ab , with 1 ≤ a, b ≤ m . Differentiating Jensen’s second-order equations yields ω = φ ∗ e ω which implies that S and e S are projectively equivalent. See [12]. Proof of Proposition 3.6.
From (3.5) we see that Proposition 3.6 holds for p = 2. Differen-tiating (3.5) and applying Cartan’s Lemma 3.2 yields functions h rstu : F S → R , completelysymmetric in their indices, such that(3.9) h rstu ω u = d h rst + h rst (2 ω + ω n +1 n +1 ) − h ust ω ur − h rut ω us − h rsu ω ut + h rs ω t + h st ω r + h tr ω s − ( h rs h tu + h st h ru + h tr h su ) ω un +1 . This establishes Proposition 3.6 in the case p = 3. The general case may not be establishedby induction on p . Details are left to the reader. (cid:3) Local coordinate computation.
In this section we give a local coordinate com-putation of h st at a point w o ∈ S . Given w o = [1 : 0 : · · · : 0] ∈ S fix coordinates z = ( z , . . . , z m ) [1 : z : · · · : z m ] on P W in a neighborhood of w o so that S is locallygiven as a graph y m = f ( z , ¯ z , . . . , z m − , ¯ z m − , x m ) = f ( x , y , . . . , x m − , y m − , x m ) , z j = x j + i y j , ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 7 over its (real) tangent space T p S = { y m = 0 } . Fix the index range 0 < α, β < m . Locallythe tangent space is spanned by e α = ∂ x α + f x α ∂ y m , e α +1 = ∂ y α + f y α ∂ y m , e m = ∂ x m + f x m ∂ y m . The vector e m +1 = Je m = ∂ y m − f x m ∂ x m completes the local framing of the tangent spaceto a framing of C m (as a real vector space). If we set e = ∂ x + x α ∂ x α + y α ∂ y α + x m ∂ x m + f ∂ y m e = Je = ∂ y − y α ∂ x α + x α ∂ y α + x m ∂ y m − f ∂ x m then we may regard e = ( e , . . . , e m +1 ) as framing of V . However this frame does notsatisfy (1.2). We modify e as follows. Set e = e , e = e , e m = e m , e m +1 = e m +1 . Let δ = 1 / (1 + f x m ) and define e α = e a + δ ( f y α − f x a f x m ) e m = ∂ x α + δ ( f y α − f x α f x m ) ∂ x m + δ ( f x α + f y α f x m ) ∂ y m ,e α +1 = e α +1 − δ ( f x α + f y a f x m ) e m = ∂ y α − δ ( f x α + f y α f x m ) ∂ x m + δ ( f y α − f x α f x m ) ∂ y m . Then e = ( e , . . . , e m +1 ) is a local section of F S . The dual coframe is e = d x , e = d x , e a = d x a − x α d x + y α d y , e a +1 = d y a − y α d x − x α d y ,e m = d x m − x m d x + f d y − δ ( f y α − f x α f x m ) d x α + δ ( f x α + f y α f x m ) d y α e m +1 = d y m − f d x − x m d y − δ ( f x α + f y α f x m ) d x α − δ ( f y α − f x α f x m ) d y α . Notice that, at z ( w o ) = 0 we have f = 0 = d f , where the last equality follows fromthe fact that the coordinates locally realize S as a graph over its tangent space at w o .Differentiating at w o yieldsd e = d x α e α + d y α e α +1 + d x m e n , d e = − d y α e α + d x α e α +1 + d x m e n +1 , d e α = d f y α e n + d f x α e n +1 , d e α +1 = − d f x α e n + d f y α e n +1 , d e m = d f x m e n +1 , d e m +1 = = − d f x m e n . From (2.1) we see that, under pulled-back by the section e , the only nonzero components ω jk of the Maurer-Cartan form are ω α = d x α , ω α = − d y α , ω α +10 = d y α , ω α +11 = d x α ,ω n = d x m , ω n β = d f y β , ω n β +1 = − d f x β , ω nn +1 = − d f x m ,ω n +11 = d x m , ω n +12 β = d f x β , ω n +12 β +1 = d f y β , ω n +1 n = d f x m . Equation (3.3) yields (again, at z ( w o ) = 0)(3.10) h α, β = f x α x β , h α, β +1 = f x α y β , h α,n = f x α x m ,h α +1 , β +1 = f y α y β , h α +1 ,n = f y α x m , h n,n = f x m x m . C. HAMMOND AND C. ROBLES
To be precise, the coefficients h on the left are evaluated at e ( w o ) ∈ F S , and the derivativeson the right are evaluated at z ( w o ) = 0. More generally, for p >
1, Proposition 3.6 yields h s ...s p t ω t = d h s ...s p − p h t ( s ...s p − ω ts p ) − p − X j =1 (cid:0) pj (cid:1) h n ( s ...s j h s j +1 ...s p ) d f x m . In the case that none of the indices are equal to 1, we see that h s ··· s p t is a ( p +1)-st derivativeof f , modulo terms involving lower over derivatives. This is why we refer to the h s ··· s p as p -th order invariants. Moreover, given the h s ··· s p we may recover the partial derivatives of f at 0 and therefore the hypersurface S (assuming S is real analytic). That is, a connected,analytic S is completely determined by h ( e ).The frame e ( w o ) over w o is not unique in this respect. Given second frame ˜ e over w o , itis always possible to find a local section of the form above through ˜ e . Proposition 3.11.
Let S, e S ⊂ P W be two connected, analytic CR-hypersurfaces. Thenthere exists a complex projective linear transformation A : P W → P W such that A ( S ) = e S if and only if there exist frames e ∈ F S and ˜ e ∈ F e S such that h ( e ) = e h (˜ e ) .Remark. In Section § h s ··· s p with some s i = 1 are redundant. Proof.
Let A : P W → P W be an invertible projective linear transformation, and let L A :GL( W ) → GL( W ) denote the lift (left multiplication) to GL( W ). If A maps S to e S , thenthe lift L A maps F S to F e S . Given e ∈ F S , let ˜ e = e A ( e ). Then (2.2) and Proposition 3.6yield h ( e ) = h (˜ e ).Conversely, suppose that h ( e ) = h (˜ e ) for some frames e ∈ F S and ˜ e ∈ F e S . Define A ∈ GL( W ) by Ae = ˜ e . Then A ( S ) and e S agree to infinite order at ˜ e . It follows from thediscussion preceding the lemma that A ( S ) = e S . (cid:3) Redundancy of the index 1 in h . The alert reader will have noticed that (2.4)allows us to write (3.1) as0 = ω n +12 ∧ ω + · · · + ω n +1 n − ∧ ω n − + ( ω n +1 n − ω ) ∧ ω n Cartan’s Lemma then yields a smaller set of second-order invariants { h st | ≤ s, t ≤ n } ,defined by ω n +1 σ = h σt ω τ and ω n +1 n − ω = h nt ω t , 2 ≤ t ≤ n . That is, the coefficients of h (2) with some index equal to 1 are redundant.This pattern continues. First define J α β := 0 =: J α +12 β +1 , J α +12 β := δ αβ and J α β +1 := − δ αβ . Then J σν J ντ = − δ στ , and (2.4) and Corollary 5.5 yield(3.12) ω σ = J στ ω τ , ω σ = − J τσ ω τ , ω σ = J τσ ω τ ,ω nσ = J νσ h ντ ω τ , ω σn +1 = J στ ω τn , ω στ = − J σµ ω µν J ντ . ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 9
To see that the third-order invariants h rst with some index equal to 1 are redundant differ-entiate 0 = h = h σ and h n = 1 to obtain0 = h s = h nn ,h σn = − J τσ h τn , (3.13) h στ = − J νσ h ντ − J ντ h σν . The general observation may be established by induction: Define1 a σ · · · σ p n b := 1 · · · | {z } a ′ s σ · · · σ p n · · · n | {z } b n ′ s . Assume that there exists r o ∈ Z such that every h a σ ··· σ p n b ∈ { h ( r ) | r ≤ r o } with a > { h τ ··· τ p n b } . Above we saw that this is the case for r o = 2 , h a σ ··· σ p n b ∈ h ( r o +1) with a > { h τ ··· τ p n b } . Details areleft to the reader. As a corollary to Lemma 3.11 we have the following. Corollary 3.14.
The h o = { h σ ··· σ p n b } form a complete set of projective differential invari-ants for an analytic CR-hypersurface. That is, two hypersurfaces S and e S are projectivelyequivalent if and only if there exists e ∈ F S and ˜ e ∈ F e S such that h o ( e ) = h o (˜ e ) . We close the section by pointing out that, while the h a τ ··· τ p n b with a > Projective differential invariants w.r.t.
Ω4.1.
Second-order invariants.
By Remark 2.16 the { Ω αm , ¯Ω αm , Ω m } span the semi-basicforms on F S . The Ω-version of Cartan’s Lemma is as follows. Lemma 4.1 (Cartan’s Lemma [2, 10] for Ω) . Suppose there exist complex-valued 1-forms ζ α , ζ ¯ α and ζ m on F S such that ζ α ∧ Ω α + ζ ¯ α ∧ ¯Ω α + ζ m ∧ Ω m = 0 . Then there exist functions Z αβ , Z α ¯ β , Z ¯ αβ , Z ¯ α ¯ β , Z α , Z ¯ α , Z : F S → C such that ζ α = Z αβ Ω β + Z α ¯ β ¯Ω β + Z α Ω m ,ζ ¯ α = Z ¯ αβ Ω β + Z ¯ α ¯ β ¯Ω β + Z ¯ α Ω m ,ζ m = Z β Ω β + Z ¯ β ¯Ω β + Z Ω m , and Z αβ = Z βα , Z α ¯ β = Z ¯ βα , Z ¯ α ¯ β = Z ¯ β ¯ α . Recall that (2.15) holds on F S . Differentiating with the Maurer-Cartan equation (2.7),we obtain(4.2) 0 = d (Ω m − ¯Ω m ) = (Ω + ¯Ω mm − ¯Ω − Ω mm ) ∧ Ω m − Ω mα ∧ Ω α + ¯Ω mα ∧ ¯Ω α . Note that Ω + ¯Ω mm − ¯Ω − Ω mm takes values in i R . So Lemma 4.1 yields functions P αβ = P βα , ¯ P α ¯ β = − P β ¯ α , P αm , P mm = − ¯ P mm : F S → C such thatΩ mα = P αβ Ω β + P α ¯ β ¯Ω β + P αm Ω m , (4.3a) (Ω + ¯Ω mm ) − ( ¯Ω + Ω mm ) = − P αm Ω α + ¯ P αm ¯Ω α + P mm Ω m . (4.3b)Making use of (2.4), (2.5), (2.9), (3.3) and (4.3) we see that(4.4) P αβ = ( h α, β +1 + h α +1 , β ) + i2 ( h α, β − h α +1 , β +1 ) ,P α ¯ β = ( h α +1 , β − h α, β +1 ) + i2 ( h α, β + h α +1 , β +1 ) ,P αm = h n, α +1 + i h n, α ,P mm = − h nn . In particular, the coefficients on the right-hand side of (4.3) are the second-order differen-tial invariants on F S . Indeed, the − i2 P α ¯ β are the coefficients of the Levi form in a localcoordinate frame. See Section 4.2. (In an abuse of language we will often refer to P α ¯ β asthe coefficients of the Levi form.) The frame bundle F S admits a sub-bundle P S on whichthe P αm and P mm vanish; see Lemma 5.1. Remark.
Together the P αβ and P α ¯ β may be identified with the derivative of a Gauss map(see Section 6.1). The are, respectively, the anti-hermitian and hermitian parts of the CRsecond fundamental form.4.2. Local coordinate computation.
Here we convert the expressions of § z and ¯ z derivatives. Recall ∂ z = ( ∂ x − i ∂ y ) and ∂ ¯ z = ( ∂ x + i ∂ y ), so that (3.10) and (4.4) yield f z α z β = ( f x α x β − f y α y β ) − i ( f x α y β + f y α x β ) = − i2 P αβ ,f z α ¯ z β = ( f x α x β + f y α y β ) + i ( f x α y β − f y α x β ) = − i2 P α ¯ β . (4.5)So we have the power series representation for y m = f at p ,(4.6) y m = h α, β x α x β + 2 h α, β +1 x α y β + h α +1 , β +1 y α y β + ∆= ( h α, β − h α +1 , β +1 − h α, β +1 ) z α z β + ( h α, β + h α +1 , β +1 + i ( h α, β +1 − h α +1 , β )) z α ¯ z β + ( h α, β − h α +1 , β +1 + 2 i h α, β +1 ) ¯ z α ¯ z β + ∆= − i2 P αβ z α z β − i P α ¯ β z α ¯ z β + i2 ¯ P αβ ¯ z α ¯ z β + ∆ , where ∆ denotes the terms of degree greater than two, as well as the quadratic termsinvolving x m . Definition.
From (4.5) we see that − i2 P α ¯ β is the Levi form . Remark.
Classically, strong pseudoconvexity is the condition that the Levi form be negativedefinite. If we replace e with − e , then the coefficients P α ¯ β change sign. So when workingon the frame bundle F S we will define S to be strongly pseudoconvex if the coefficients P α ¯ β : F S → C define a definite form. Definition . A strongly pseudoconvex hypersurface S is strongly C -linearly convex (SCLC)if no real line tangent to b H z S makes second order contact with b S at w ∈ ˆ z . ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 11
Let z = ( z , . . . , z m − ) ∈ C m − . Define P ( z , z ) = P αβ z α z β and L ( z , ¯ z ) = P α ¯ β z α ¯ z β . From(4.6) we see that the surface S is strongly C –linearly convex if and only if(4.8) 0 = i2 P ( z , z ) + i L ( z , ¯ z ) + i2 P ( z , z ) = − Im P ( z , z ) + i L ( z , ¯ z )for all 0 = z ∈ C m − . Equivalently,(4.9) | Im P ( z , z ) | < | L ( z , ¯ z ) | ∀ = z ∈ C m − . Equation (4.8) can be expressed as(4.10) 0 = i (cid:0) z t ¯ z t (cid:1) (cid:18) P LL t − ¯ P (cid:19) (cid:18) z ¯ z (cid:19) , for all 0 = z ∈ C m − , where P = ( P αβ ) and L = ( P α ¯ β ). In particular the matrix abovemust be invertible. Indeed,(4.11) (cid:18) P LL t − ¯ P (cid:19) − = (cid:18) Q MM t − ¯ Q (cid:19) where Q = L − P ( P L − P − L ) − ,M = ( ¯ P L − P − ¯ L ) − . Note that Q t = Q and ¯ M t = − M .4.3. Second order projectively invariant tensors on S . In this section we illustratethe frame bundle construction of the second fundamental form of S .Let J z : T z P W → T z P W denote the complex structure on P W . Given z ∈ S , let H z = T z S ∩ J z ( T z S ) denote the maximal complex subspace of T z S . Analogously define b H z S = b T z S ∩ J ( b T z S ) ⊂ W . Given z ∈ P W , let ˆ z = L z ∈ W denote the corresponding(complex) line through the origin. Then H z S = L ∗ z ⊗ ( b H z S/L z ) , and HS → S defines a rank m − S . Similarly, define thenormal (complex) line bundle N S → S by N z S = T z P W/H z S = L ∗ z ⊗ ( W/ b H z S ) . Given f = ( f , . . . , f n ) ∈ F S , set z = [ f ] ∈ S and let ( f , . . . , f n ) denote the dual basisof W ∗ . Then f α = f ⊗ ( f α mod L z ) defines a basis of H z S , and f m = f ⊗ ( f m mod b H z S )spans N z S . Let ( f , . . . , f m − ) denote the dual basis of H ∗ z S = ( H z S ) ∗ . Define P f = P αβ f α f β ⊗ f m ∈ (Sym H ∗ z S ) ⊗ N z S , and L f = P α ¯ β f α ¯ f β ⊗ f m ∈ H ∗ z S ⊗ ¯ H ∗ z S ⊗ N z S .
We claim that P and L descend to well-defined sections of (Sym H ∗ ) ⊗ N S and H ∗ S ⊗ ¯ H ∗ S ⊗ N S , respectively, over S . (The tensors P and L are respectively the anti-hermitianand hermitian parts of the second fundamental form of S .) To establish the claim it sufficesto show that P and L are constant on fibres of F S . This is seen as follows.First consider a change of frame ( § f = g f , ˜ f α = g βα f β , ˜ f m = g mm f m . Such a change of frame is called a block fibre motion on F S . Computing with (2.11) and(4.3a) we see that the change in P and L is given by(4.13) e P αβ = P γε g γα g εβ g g mm and e P α ¯ β = P γ ¯ ε g γα ¯ g εβ ¯ g g mm . The transformation in the coefficients P αβ and P α ¯ β precisely cancels transformation in f sothat P f = P ˜ f and L f = L ˜ f .Next, consider a change of frame ( § f = f , ˜ f α = f α + g α f , ˜ f m = f m + g m f + g αm f α . These changes of frame are shear fibre motions . The entire group of fibre motions on F S is generated by block and shear transformations. Computing with (2.11) and (4.3a) we seethat P and L are unchanged by shear fibre motions:(4.15) e P αβ = P αβ and e P α ¯ β = P αβ . As a consequence of (4.13) and (4.15) we see that P f and L f are constant under the fibremotions. Our claim follows.The tensors P and L are projectively invariant. To be precise, suppose that e S is a secondhypersurface that is projectively equivalent to S via A ∈ GL( W ). Note that A naturallyidentifies (Sym H ∗ z S ) ⊗ N z S with (Sym H ∗ Az e S ) ⊗ N Az e S , and H ∗ z S ⊗ ¯ H ∗ z S ⊗ N z S with H ∗ Az e S ⊗ ¯ H ∗ Az e S ⊗ N Az e S . Thus we may define the pull-backs A ∗ e P and A ∗ e L . From (2.8)we have P = e P ◦ A and Q = e Q ◦ A . Thus A ∗ e P = P and A ∗ e L = L . Whence projectiveequivalence. Remark.
G. Jensen [13] proved that a nondegenerate hypersurface S has P ≡ S is projectively equivalent to a quadric hypersurface (Section 5.2). In related work Detrazand Tr´epreu [8] showed that the hyperquadric appears as one of two types of hypersurfacesin C n that are characterized by an elliptic system. Remark.
In the case that S is strongly C -linearly convex the inverse matrix (4.11) exists.If Q = ( Q αβ ) and M = ( Q α ¯ β ), then Q f = Q αβ f α f β ⊗ f m ∈ (Sym H z S ) ⊗ N ∗ z S , and M f = − Q α ¯ β f α ¯ f β ⊗ f m ∈ H z S ⊗ ¯ H z S ⊗ N ∗ z S . similarly define projectively invariant tensors Q and M on S . Here f m ∈ N ∗ z S is dual to f m ∈ N z S . As we will see in (6.6), Q and M are pull-backs (under a ‘lifted’ Gauss map) ofthe second fundamental form of the dual-hypersurface.5. A reduction of the bundle F S In this section we will show that the second order coefficients h σn and h nn (equivalently,for P αm and P mm ) may be normalized to zero. That is, F S admits a sub-bundle P S ofadapted frames over S on which the coefficients vanish. ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 13
Normalizations.Lemma 5.1.
Let S ⊂ P W be a CR-hypersurface. There exists a sub-bundle P S ⊂ F S onwhich the coefficients P αm and P mm vanish. In particular the restriction of the Maurer-Cartan form Ω to P S satisfies ¯Ω m = Ω m (5.2a) Ω mα = P αβ Ω β + P α ¯ β ¯Ω α , (5.2b) Ω + ¯Ω mm = ¯Ω + Ω mm , (5.2c) with P βα = P αβ and ¯ P α ¯ β = − P β ¯ α .Proof. Since (2.15) holds on F S , equation (5.2a) is immediate. By (4.3) it remains to showthat P αm and P mm can be normalized to zero. Consider a change of frame ( § h m = g α g αm − g m = ( g − ) m . Then (2.11) yields e Ω = Ω − g α Ω α + h m Ω m , e Ω α = Ω α − g αm Ω m , e Ω m = Ω m , e Ω mα = Ω mα + g α Ω m , e Ω mm = Ω mm + g αm Ω mα + g m Ω m . The coefficients P αm transform as e P αm = P αm + g α + P αβ g βm + P α ¯ β ¯ g βm . By selecting g α appropriately we may may normalize to(5.3) P αm = 0 . Similarly, (4.3b) yields e P mm = P mm + ( g α g αm − ¯ g α ¯ g αm ) − g m − ¯ g m ). So we may use g m tonormalize Im( P mm ) = 0. Since P mm takes values in i R , we have(5.4) P mm = 0 . Let P S ⊂ F S denote the sub-bundle on which (5.3) and (5.4) hold. (cid:3) Corollary 5.5.
The second-order invariants h sn , ≤ s ≤ n , vanish on P S . In particular,the following relations hold on P S : ω n +10 = ω n , ω n +11 = ω n , ω n +1 σ = h στ ω τ , ω n +1 n = ω = − ω nn +1 . Proof.
The corollary follows from (2.4), (2.15), (3.4), (3.3), (4.4) and Lemma 5.1. (cid:3)
Remark . The bundle P S is of (real) dimension 2 m + 3 and is preserved by the shearfibre motions (4.14) satisfying g α + P αβ g βm + P α ¯ β ¯ g βm = 0 , and(5.7) 4i Im( g m ) = 2 ( g m − ¯ g m ) = ( g α g αm − ¯ g α ¯ g αm ) = 2i Im( g α g αm )= − P αβ g αm g βm + P αβ g αm g βm − P α ¯ β g αm ¯ g βm . Lemma 5.8.
Let S ⊂ P W be a CR-hypersurface and A ∈ GL( W ) . Then L A ( P S ) = P e S ,where e S = A ( S ) ⊂ P W .Proof. In Remark 2.12 we observed that L A ( F S ) = F e S . Thus L A ( P S ) ⊂ F e S . From left-invariance (2.8) of Ω we see that (5.2) holds on L A ( P S ). Hence L A ( P S ) = P e S . (cid:3) Proposition 5.9.
Two CR-hypersurfaces S, e S ⊂ P W are projectively equivalent if and onlyif there exists a smooth map φ : P S → P e S such that φ ∗ (Ω |P e S ) = Ω |P S . (Equivalently, φ ∗ ( ω |P e S ) = ω |P S .) The proof of Proposition 5.9 is identical to that of Proposition 2.13.5.2.
Example: the hyperquadric.
The homogenous model of a strongly pseudoconvexhypersurface in CP n is the hyperquadric. Fix a frame e = ( e , . . . , e n +1 ) of V in F and let f = ( f , . . . , f m ) be the corresponding basis of W . Define linear coordinates z = z a f a on W .Let [ z ] = [ z : · · · : z m ] be the corresponding complex homogeneous coordinates on P W .Define q ( z ) = ¯ z t Qz = i( z ¯ z m − z m ¯ z ) + X α z α ¯ z α . Let SU(1 , m ) = SU(
W, q ) ⊂ GL( W ) be the subgroup stabilizing q . Then the Lie algebra su (1 , m ) is given by matrices X = ( X ab ) ∈ gl ( W ) satisfying the following:(5.10) 0 = ¯ X + X mm = ¯ X m − X m = ¯ X m − X m = ¯ X mα − i X α = ¯ X αm + i X α , ( X αβ ) ∈ u ( m − , and Tr( X ) = 0 . Let S = { q = 0 } ⊂ P W . Note that e ∈ F S . Define G = SU(1 , m ) · e ⊂ F S . Then G isa sub-bundle of the adapted frames over S , and is naturally identified with SU(1 , m ). TheMaurer-Cartan form, when restricted to G , takes values in su (1 , m ). In particular,Ω mα = − i ¯Ω α and ¯Ω + Ω mm = 0 . We see that
G ⊂ P S , and P αβ = 0 and P α ¯ β = − i δ αβ on G .5.3. An ω –coframe on P S . By Corollary 5.5 we have 0 = h σn = h nn on the sub-bundle P S . Differentiating these expressions and applying (3.4), (3.5), (3.13) and (2.4) yields0 = h sn and ω σ + h στ ω τn = − h στn ω τ − h σnn ω n , (5.11a) ω σ + J νσ h ντ ω τn = − J νσ h νnτ ω τ − J νσ h νnn ω n , (5.11b) 2 ω n +1 (2.4) = − ω n = h nnτ ω τ + h nnn ω n . (5.11c)Note that (5.11b) is a consequence of (3.12) and (5.11a). Lemma 5.12.
The E ω ( P S ) := { ω , ω t , ω α β , ω α +12 β , ω n , ω τn , ω nn } form a coframing of P S . The remaining components of ω are given by (2.4) , Corollary 5.5, (5.11a) and (5.11c) . The claim that E ω ( P S ) is a coframing on P S follows by dimension count. See Remark 5.6. Corollary 5.13.
Two CR-hypersurfaces S, e S ∈ P W are projectively equivalent if and onlyif there exists a smooth map φ : P S → P e S such that (5.14) φ ∗ E ω ( P e S ) = E ω ( P S ) , ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 15 and (5.15) h στ = e h στ ◦ φ , h στn = e h στn ◦ φ ,h σnn = e h σnn ◦ φ , h nnn = e h nnn ◦ φ . Equivalently, φ ∗ ( ω |P e S ) = ω |P S .Proof. By the relations (2.15), (3.3), (5.11b) and (5.11c) the equations (5.14) and (5.15)hold if and only if φ ∗ ( ω |P e S ) = ω |P S . The corollary then follows from Proposition 5.9. (cid:3) In order to establish the Ω–versions of Lemma 5.12 and Corollary 5.13 (in Section 5.6)we must first compute two derivatives in Sections 5.4 and 5.5.5.4.
Differentiate (5.2b) . Differentiating (5.2b) with (2.7) produces0 = P αβ ∧ Ω β + P α ¯ β ∧ ¯Ω β − (cid:0) Ω α + P αβ Ω βm + P α ¯ β ¯Ω βm (cid:1) ∧ Ω m , where P αβ = d P αβ + P αβ (Ω + Ω mm ) − P γβ Ω γα − P αγ Ω γβ , P α ¯ β = d P α ¯ β + P α ¯ β ( ¯Ω + Ω mm ) − P γ ¯ β Ω γα − P α ¯ γ ¯Ω γβ . Lemma 5.1 implies(5.16) P αβ = P βα and ¯ P α ¯ β = −P β ¯ α . Cartan’s Lemma 4.1 and (5.16) yield functions P αβγ , P αβ ¯ γ , P αβm , P α ¯ βγ , P α ¯ β ¯ γ , P α ¯ βm , P αmm : P S → C such that P αβ = P αβγ Ω γ + P αβ ¯ γ ¯Ω γ + P αβm Ω m , P α ¯ β = P α ¯ βγ Ω γ + P α ¯ β ¯ γ ¯Ω γ + P α ¯ βm Ω m , Ω α + P αβ Ω βm + P α ¯ β ¯Ω βm = − P αβm Ω β − P α ¯ βm ¯Ω β − P αmm Ω m , (5.17)with P αβγ = P βαγ = P αγβ , P αβ ¯ γ = P βα ¯ γ = P α ¯ γβ , P α ¯ β ¯ γ = P α ¯ γ ¯ β ,P αβm = P βαm , ¯ P α ¯ βγ = − P β ¯ α ¯ γ , ¯ P α ¯ βm = − P β ¯ αm . Note that (5.17) is the Ω–version of (5.11b), and it is straight forward to check that P αβm = ( h α, β +1 ,n − h β, α +1 ,n ) − i2 ( h α, β,n − h α +1 , β +1 ,n ) ,P α ¯ βm = ( h α, β +1 ,n − h β, α +1 ) − i2 ( h α, β,n + h α +1 , β +1 ,n ) ,P αmm = h α +1 ,nn + i h α,nn . Differentiate (5.2c) . Differentiating (5.2c) produces0 = (cid:0) − Ω α − P αβ Ω βm − P α ¯ β ¯Ω βm (cid:1) ∧ Ω α + (cid:0) ¯Ω α + ¯ P αβ ¯Ω βm + ¯ P α ¯ β Ω βm (cid:1) ∧ ¯Ω α + 2 (cid:0) ¯Ω m − Ω m (cid:1) ∧ Ω m . Cartan’s Lemma 4.1 yields P m : P S → C such that(5.18) − m − ¯Ω m ) = P αmm Ω α − ¯ P αmm ¯Ω α − P m Ω m with ¯ P m = − P m . This expression is the Ω–version of (5.11c). It is straight-forward to check that P mmm = − h nnn . Remark.
The coefficient functions P abc in this section and Section 5.4 are the third-orderinvariants of S with respect to Ω. Unlike P αβ and P α ¯ β they do not yield well-defined tensorson S .5.6. A Ω–coframe on P S . In analogy with Lemma 5.12 we have the following.
Lemma 5.19.
The 1-forms E Ω ( P S ) := { Ω , ¯Ω , Ω α , ¯Ω α , Ω m , Ω αβ , ¯Ω αβ , Ω mm , Ω αm , ¯Ω αm , Ω m } form a coframing of P S (over R ) and the remaining components of Ω are given by (5.2) , (5.17) and (5.18) . The lemma follows by dimension count, see Remark 5.6.
Remark.
In the case that S is strongly C -linearly convex, the { Ω α , Ω αm } in the coframingmay be replaced with the { Ω mα , Ω α } . See Section 5.7. Corollary 5.20.
Two CR-hypersurfaces S, e S ∈ P W are projectively equivalent if and onlyif there exists a smooth map φ : P S → P e S such that φ ∗ E Ω ( P e S ) = E Ω ( P S ) and P αβ = e P αβ ◦ φ , P α ¯ β = e P α ¯ β ◦ φ ,P αβm = e P αβm ◦ φ , P α ¯ βm = e P α ¯ βm ◦ φ ,P αmm = e P αmm ◦ φ , P mmm = e P mmm ◦ φ . Equivalently, φ ∗ (Ω |P e S ) = Ω |P S . The proof is identical to that of Corollary 5.13, and is left to the reader.5.7.
The case that S is SCLC. In the case that S is strongly C –linearly convex (Defi-nition 4.7) equations (5.2b), (5.17) and (5.18) have alternate formulations. Let Q = ( Q αβ )and M = ( Q α ¯ β ) be given by (4.11). ThenΩ α = Q αβ Ω mβ − Q α ¯ β ¯Ω mβ , (5.21a) Ω αm + Q αβ Ω β − Q α ¯ β ¯Ω β = − Q αβm Ω mβ + Q α ¯ βm ¯Ω mβ − Q αmm Ω m , (5.21b) − m − ¯Ω m ) = Q αmm Ω mα − ¯ Q αmm ¯Ω mα − P m Ω m , (5.21c)where Q αβm = Q αγ P γεm Q εβ + Q αγ P γ ¯ εm Q β ¯ ε − Q α ¯ γ ¯ P γεm Q β ¯ ε + Q α ¯ γ P ε ¯ γm Q εβ ,Q α ¯ βm = Q αγ P γεm Q ε ¯ β − Q αγ P γ ¯ εm ¯ Q εβ + Q α ¯ γ ¯ P γεm ¯ Q εβ + Q α ¯ γ P ε ¯ γm Q ε ¯ β ,Q αmm = Q αγ P γmm − Q α ¯ γ ¯ P γmm . As we will see in Section 6, the coefficients Q above may be identified with the P coefficientsof the dual hypersurface. ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 17 Dual hypersurfaces
In this section we define the Gauss map of a CR-hypersurface S ⊂ P W and characterizethe self-dual strongly C –linearly convex (SCLC) hypersurfaces (Theorem 6.8).6.1. Gauss map.
Let F S be the adapted frame bundle over a SCLC hypersurface S . Given f = ( f , . . . , f m ) ∈ F S , let z = [ f ] ∈ S , and let b H z S = b T z S ∩ J ( b T z S ) be the maximalcomplex subspace of b T z S ⊂ W . Note that b H z S = span { f , . . . , f m − } (see § f , . . . , f m ) ∈ W ∗ be the basis dual to f . Note that f m vanishes when restricted to b H .Since b H depends only on z ∈ S , this implies that, modulo rescaling, f m depends only on z ∈ S . In particular, the map F S → P W ∗ sending f [ f m ] ∈ P W ∗ descends to S where itdefines the Gauss map γ : S → P W ∗ . Under the identification of P W ∗ with the Grassmannian Gr( m, m + 1) of m -dimensional C –planes in W , γ is the map sending z ∈ S to b H z S ∈ Gr( m, m + 1).
Definition.
The image S ∗ = γ ( S ) is the dual of S . Let ζ = ( ζ , . . . , ζ m ) ∈ GL( W ∗ ) denote a basis of W ∗ . The Maurer-Cartan form Λ onGL( W ∗ ) is defined by d ζ a = Λ ba ζ b . Define a map Γ : GL( W ) → GL( W ∗ ) by Γ( f ) =( f m , f α , f ). Note that Γ = Id. From (2.6) we see that(6.1) d f a = − Ω ab f b . Thus,(6.2) Γ ∗ Λ Λ β Λ m Λ α Λ αβ Λ αm Λ m Λ mβ Λ mm = − Ω mm Ω βm Ω m Ω mα Ω βα Ω α Ω m Ω β Ω . The following lemma is well-known; see [1, § Lemma 6.3. If S ⊂ P W is a strongly C –linearly convex hypersurface, then S ∗ ⊂ P W ∗ isalso a strongly C –linearly convex hypersurface and Γ( P S ) = P S ∗ .Proof. From (6.1) we have(6.4) d f m = − Ω m f − Ω mβ f β − Ω mm f m . It is a consequence of strong C –linear convexity that { Ω m , Ω mα , ¯Ω mα , Ω mm , ¯Ω mm } are linearlyindependent over R on F S , see (5.21a). Thus, b T γ ( z ) S ∗ = d γ ( T f F S ) is of real dimension n + 1 = 2 m + 1. In particular, S ∗ is a hypersurface in P W ∗ .From (6.1) we see that Γ maps F S to F S ∗ . If we restrict Γ to P S , then (6.2) and Lemma5.1 yield Γ ∗ ( ¯Λ m ) = Γ ∗ (Λ m ) , Γ ∗ (Λ mα ) = − Ω α = − Q αβ Ω mβ + Q α ¯ β ¯Ω mβ = Q αβ Γ ∗ (Λ β ) − Q α ¯ β Γ ∗ ( ¯Λ β ) , Γ ∗ ( ¯Λ + Λ mm ) = Γ ∗ (Λ + ¯Λ mm ) . The coefficients Q = ( Q αβ ) and M = ( Q α ¯ β ) above are defined by (4.11). Since Γ : GL( W ) → GL( W ∗ ) is a diffeomorphism, we have(6.5) ¯Λ m = Λ m , Λ mα = Q αβ Λ β − Q α ¯ β ¯Λ β , ¯Λ + Λ mm = Λ + ¯Λ mm . This implies that Γ( P S ) = P S ∗ , and(6.6) Q αβ = Γ ∗ ( P ∗ αβ ) and − Q α ¯ β = Γ ∗ ( P ∗ α ¯ β ) , where P ∗ αβ and P ∗ α ¯ β are the coefficients of the second fundamental form on S ∗ .To see that S ∗ is strongly C –linearly convex it suffices, by (6.6) and (4.10), to show that(6.7) 0 = i (cid:0) w t ¯ w t (cid:1) (cid:18) Q − M ¯ M − ¯ Q (cid:19) (cid:18) w ¯ w (cid:19) , for all 0 = w ∈ C m − . Define 0 = z ∈ C m − by (cid:18) w ¯ w (cid:19) = (cid:18) P − L ¯ L − ¯ P (cid:19) (cid:18) z ¯ z (cid:19) . Then the right-hand side of (6.7) is equal toi (cid:0) z t ¯ z t (cid:1) (cid:18) P − L ¯ L − ¯ P (cid:19) (cid:18) z ¯ z (cid:19) = 2 Im P ( z , z ) − L ( z , ¯ z ) . Since S is SCLC this quantity is nonzero for all 0 = z ∈ C m − ; see (4.9). We conclude that(6.7) holds and S ∗ is SCLC. (cid:3) Self-dual hypersurfaces.
Definition.
The SCLC hypersurface S is self-dual if there exists an (complex) linear iso-morphism A : W ∗ → W such that A ( S ∗ ) = S . Theorem 6.8.
A strongly C –linearly convex CR-hypersurface S ⊂ P W is self-dual if andonly if there exists a smooth map φ : P S → P S such that (6.9) φ ∗ Ω Ω β Ω m Ω α Ω αβ Ω αm Ω m Ω mβ Ω mm = − Ω mm Ω βm Ω m Ω mα Ω βα Ω α Ω m Ω β Ω . In particular, the map φ must satisfy Q αβ = P αβ ◦ φ , − Q α ¯ β = P α ¯ β ◦ φ , (6.10a) Q αβm = P αβm ◦ φ , − Q α ¯ βm = P α ¯ βm ◦ φ , (6.10b) Q αmm = P αmm ◦ φ , P mmm = P mmm ◦ φ . (6.10c) Proof.
Suppose that S is self dual. Then there exists a linear isomorphism A : W ∗ → W suchthat A ( S ∗ ) = S . Given ζ = ( ζ , ζ α , ζ m ) ∈ GL( W ∗ ) define f = A ( ζ ) = ( f , f α , f m ) ∈ GL( W )by f a := A ( ζ a ). This defines an induced map L A : GL( W ∗ ) → GL( W ). We have(6.11) L ∗ A (Ω f ) = Λ ζ . ROJECTIVE INVARIANTS OF CR–HYPERSURFACES 19
This, together with (6.2), implies that φ = L A ◦ Γ satisfies (6.9).Conversely suppose that there exists a smooth map φ : P S → P S satisfying (6.9). Notethat the right-hand side of (6.9) is a gl ( W )–valued 1-form satisfying the Maurer-Cartanequation. It now follows from Theorem 2.14 that φ = L A ◦ Γ for some linear isomorphism A : W ∗ → W .It remains to establish (6.10). The first line (6.10a) is a consequence of (6.6). To establish(6.10b) consider the S ∗ version of (5.17). We haveΓ ∗ ( P ∗ αβm ) Ω mβ + Γ ∗ ( P ∗ α ¯ βm ) ¯Ω mβ + Γ ∗ ( P ∗ αmm ) Ω m = − Γ ∗ ( P ∗ αβm Λ β + P ∗ α ¯ βm ¯Λ β + P ∗ αmm Λ m ) (5.17) = Γ ∗ (Λ α + P ∗ αβ Λ βm + P ∗ α ¯ β ¯Λ βm ) (6.2) , (6.6) = − Ω αm − Q αβ Ω β + Q α ¯ β ¯Ω β (5.21b) = Q αβm Ω mβ + Q α ¯ βm ¯Ω mβ + Q αmm Ω m . Thus(6.12) Γ ∗ ( P ∗ αβm ) = Q αβm , Γ ∗ ( P ∗ α ¯ βm ) = − Q α ¯ βm and Γ ∗ ( P ∗ αmm ) = Q αmm . This yields (6.10b).To establish (6.10c) consider the S ∗ version of (5.18)Γ ∗ ( P ∗ αmm ) Ω mα − Γ ∗ ( ¯ P ∗ αmm ) ¯Ω mα − Γ ∗ ( P ∗ mmm ) Ω m = − Γ ∗ ( P ∗ αmm Λ α − ¯ P ∗ αmm ¯Λ α − P ∗ mmm Λ m ) (5.18) = 2 Γ ∗ (Λ m − ¯Λ m ) (6.2) = − m − ¯Ω m ) (5.21c) = Q αmm Ω mα − ¯ Q αmm ¯Ω mα − P m Ω m . In particular,(6.13) Γ ∗ ( P ∗ m ) = P m . This yields (6.10c). (cid:3)
Remark.
Note that (6.9) implies that ( φ ) ∗ Ω = Ω. Thus there exists A ∈ GL( W ) such that φ = L A |P S . Example . It is well-known that the hyperquadric (see Section 5.2) is self-dual. Followingthe notation of Section 5.2, by an argument analogous to the proof of Theorem 6.8, theself-duality of the hyperquadric is equivalent to the existence of a map φ : G → G suchthat (6.9) holds. Given f = g · f ∈ G , with g ∈ SU(1 , m − f = ¯ g · f . Then φ ( f , f α , f m ) = ( − ¯ f , i ¯ f α , ¯ f m ) defines a map G → G satisfying (6.9).6.3.
The ω –version. The equations (6.10) provide second and third-order conditions for aSCLC hypersurface to be self-dual. If we shift from the Ω perspective to the ω perspectivewe obtain p -th order conditions as follows.The equations (2.5), (4.4) and (5.21a) imply that we may solve (3.3) for ω s ; that is, thereexist functions k st = k ts : F S → R such that ω s = k st ω n +1 t . More generally, in analogy with Proposition 3.6, there exist functions k s ··· s p t : F S → R , p >
1, fully symmetric in theirindices, and inductively defined by k s ...s p t ω n +1 t = − d k s ...s p + ( p − k s ...s p ω n +1 n +1 + k s ...s p ω + p n ( p − k ( s ...s p − ω s p ) n +1 − k t ( s ...s p − ω s p ) t o + p − X j =1 (cid:0) pj (cid:1) n ( j − k ( s ...s j k s j +1 ...s p ) ω n +1 − k t ( s ...s j k s j +1 ...s p ) ω t o . (Our convention is that k s = 0.)Let η denote the Maurer-Cartan form on GL( V ∗ , J ). It is straight-forward to check thatthe ω –version of (6.2) is Γ ∗ η jk = − ω ν ( k ) ν ( j ) , where ν is the permutation of { , . . . , n } defined by ν (0) = n + 1, ν (1) = n , ν (2 α ) =2 α + 1 and ν = Id. In particular, if h ∗ denotes the differential invariants on S ∗ given byProposition 3.6, thenΓ ∗ ( h ∗ s ··· s p ) = h ∗ s ··· s p ◦ Γ = k ν ( s ) ··· ν ( s p ) =: k s ··· s p ν . This equation generalizes (with respect to ω ) the equations (6.6), (6.12) and (6.13). Inparticular, by working with respect to ω we may strengthen Theorem 6.8 to the following. Theorem 6.15.
A strongly C –linearly convex CR-hypersurface S ⊂ P W is self-dual if andonly if there exists a smooth map φ : F S → F S such that φ ∗ ω jk = − ω ν ( k ) ν ( j ) . In particular, the map φ must satisfy φ ∗ ( h s ··· s p ) = k s ··· s p ν . By Proposition 3.11 we have the following.
Theorem 6.16.
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