Hayama Symposium on Complex Analysis in Several Variables XVII p. 1/19 Projective space and twistor theory Michael Eastwood [ Toby Bailey Robin Graham Paul Baird Hubert Goldschmidt ] Australian National University
Hayama Symposium on Complex Analysis in Several Variables XVII p. 2/19 Topics CP 3 is the twistor space of S 4, } classical twistor theory Penrose transform on CP 3, & CR geometry Funk-Radon transform on RP 2, } with round metric X-ray transform on RP 3, X-ray transform on CP 2, Penrose transform on CP 2, with Fubini-Study metric X-ray transform on CP 3, BGG-like complexes on CP 3. Bernstein-Gelfand-Gelfand
Hayama Symposium on Complex Analysis in Several Variables XVII p. 3/19 Conformal foliations U= unit vector field on Ω open R 3. h C U is (transversally) conformal L U preserves the conformal metric orthogonal to its leaves isothermal coördinates h=f+ ig f, g =0 f = g conjugate functions
Hayama Symposium on Complex Analysis in Several Variables XVII p. 4/19 Conjugate functions on R 3 f= f(q,r,s) g= g(q,r,s) s.t.{ f, g =0 f = g f= r g= s f= q 2 r 2 s 2 g= 2q r 2 + s 2 f= r q2 + r 2 + s 2 r 2 + s 2 g= s q2 + r 2 + s 2 r 2 + s 2 f= (1 q2 r 2 s 2 )r+ 2qs r 2 + s 2 g= (1 q2 r 2 s 2 )s 2qr r 2 + s 2 R 3 S 3 Hopf R 2 S 2 { }
Hayama Symposium on Complex Analysis in Several Variables XVII p. 5/19 Almost Hermitian structures NB: J(p,q,r,s) R 4 R 4 satisfies J 2 = Id J SO(4) J= 0 u v w u 0 w v v w o u w v u 0 u 2 + v 2 + w 2 = 1, two-sphere Consider R 3 ={(p,q,r,s) R 4 p=0} R 4 NB: U (J p ) R 3 =(u q + v r + w s ) R 3 unit vector field also two-sphere
Hayama Symposium on Complex Analysis in Several Variables XVII p. 6/19 Sphere bundles bundle of unit vectors bundle of almost Hermitian structues section unit vector field Q Z τ R 3 R 4 section almost Hermitian structure
Hayama Symposium on Complex Analysis in Several Variables XVII p. 7/19 Hermitian structures Lemma J is integrable U (J p ) R 3 is conformal Conversely?? NB: J integrable J real-analytic Question: U conformal U real-analytic?? Answer: NO! However: U real-analytic and conformal U extends uniquely to an integrable J. WHY?
Hayama Symposium on Complex Analysis in Several Variables XVII p. 8/19 Twistor geometry Q Z R 3 R 4 compactify Q CP 3 τ S 3 S 4 Q = {[Z] CP 3 Z 1 2 + Z 2 2 = Z 3 2 + Z 4 2 } Levi-indefinite hyperquadric Q CP 3 (cf. saddle)
Hayama Symposium on Complex Analysis in Several Variables XVII p. 9/19 Twistor results Q CP 3 τ S 3 S 4 Q CP 3 τ Theorem A section S 4 open Ω J CP 3 of τ defines an integrable Hermitian structure if and only if M J(Ω) is a complex submanifold. Theorem A section S 3 open Ω U Q of τ Q S 3 defines a conformal foliation if and only if M U(Ω) is a CR submanifold.
Hayama Symposium on Complex Analysis in Several Variables XVII p. 10/19 CR submanifolds and functions M Q CP 3 is a CR submanifold? It means: TM JTQ is preserved by J. It does not mean: M={f= 0} where f is a CR function:(x+ ijx)f= 0 X Γ(TQ JTQ). Implicit function theorem is false in the CR category CR functions on Q are real-analytic. conformal foliations on S 3 need not be.
Hayama Symposium on Complex Analysis in Several Variables XVII p. 11/19 CR functions {[Z] CP 2 Z 1 2 + Z 2 2 = Z 3 2 }= three-sphere......... CP 2 Theorem (H. Lewy 1956) CR holomorphic extension {[Z] CP 3 Z 1 2 + Z 2 2 = Z 3 2 + Z 4 2 }=Q CP 3......... Corollary CR holomorphic extension Hence, a CR function on Q is real-analytic!
Hayama Symposium on Complex Analysis in Several Variables XVII p. 12/19 Smooth conjugate functions Eikonal equation:( f 2 r ) +( f 2 s ) Plenty of non-analytic solutions: s Γ = 1 f= signed distance to Γ r f(q,r,s) = f(r,s) g(q,r,s) = q } f, g =0 f = g QED
Hayama Symposium on Complex Analysis in Several Variables XVII p. 13/19 Penrose transform τ 1 (U) CP 3 H 1 (τ 1 (U),O( 2)) τ U open S 4 {φ U C ( R/6)φ=0} conformal Laplacian homogeneous vector bundle F 1,2 (C 3 ) L P τ CP 2 L P Fubini-Study metric H 1 (F 1,2 (C 3 ),Θ) Γ(CP 2, 2 Λ 1 ) Γ(CP 2, 2,2 Λ 1 ) Γ(CP 2,Λ 1 ) Γ(CP 2, 2 Λ 1 )
Hayama Symposium on Complex Analysis in Several Variables XVII p. 14/19 Funk-Radon transform on RP 2 R 2 RP 2 γ F 1,2 (R 3 ) Gr 2 (R 3 ) S 2 = RP 2 φ(γ)= γ f projective duality GL(3, R)-invariant Theorem (Funk 1913) C (RP 2 ) C (RP 2) Better Theorem Γ(RP 2,E( 2)) Γ(RP 2,Ẽ( 1))
Hayama Symposium on Complex Analysis in Several Variables XVII p. 15/19 X-ray transform on RP 3 RP 3 F 1,2 (R 4 ) Gr 2 (R 4 ) CP 3 GL(4,R) GL(4,C) Theorem (cf. John 1938) F 1,2 (C 4 ) Gr 2 (C 4 ) S 4 Spin(5, 1)-orbit Γ(RP 3,E( 2)) ker Γ(Gr 2 (R 4 ),Ẽ[ 1]) Γ(Gr 2 (R 4 ),Ẽ[ 3]) X-ray transform ultra-hyperbolic wave operator S 4= R/6
Hayama Symposium on Complex Analysis in Several Variables XVII p. 16/19 Machinery for the X-ray transform Complex analysis comes into play in two ways: constructing a spectral sequence, computing with the spectral sequence. More generally: F 1,2 (C n+1 ) µ ν CP n Gr 2 (C n+1 ) a correspondence dim R = 2n but not a double fibration ν 1 (Gr 2 (R n+1 )) CP n RP n η F 1,2 (R n+1 ) τ Gr 2 (R n+1 ) F Gr 2 (R n+1 )
Hayama Symposium on Complex Analysis in Several Variables XVII p. 17/19 Real blow up F = η CP n (L,P) s.t. L C n+1 is a complex line P R n+1 is a real plane R(L) P (generic equality) = {L s.t. L C n+1 is a complex line} F F 1,2 (R n+1 ) η CP n RP n Real blow up of CP n along RP n! F inherits an involutive structure
Hayama Symposium on Complex Analysis in Several Variables XVII p. 18/19 X-ray transform on CP 2 and CP 3 CP n γ RP n RP n CP n induced by R n+1 C n+1 is totally geodesic. Translates by SU(n+1) too! Model Embeddings µ The X-ray transform on RP n is well-understood. Pullback of tensors under µ is well-understood. Suitable global techniques on CP n are available:- n=2 Penrose transform of H 1 (F 1,2 (C 2 ),Θ)=0 etc. n 3 BGG-like: 0 Λ 1 2 Λ 1 Λ 1 etc.
Hayama Symposium on Complex Analysis in Several Variables XVII p. 19/19 THE END THANK YOU