On Penrose integral formula and series expansion of k -regular functions on the quaternionic space Hn

Journal of Geometry and Physics 64 (2013) 192–208

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On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn ✩ Qianqian Kang a , Wei Wang b,∗ a

College of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China

b

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

article

info

Article history: Received 8 April 2012 Received in revised form 1 September 2012 Accepted 5 November 2012 Available online 14 November 2012 MSC: 30G35 32L25 35N05 32L10 Keywords: The k-Cauchy–Fueter operator Penrose integral formula Quaternionic k-regular functions Series expansion The sheaf of holomorphic functions homogeneous of degree m The Čech cohomology group

abstract The k-Cauchy–Fueter operator can be viewed as the restriction to the quaternionic space Hn of the holomorphic k-Cauchy–Fueter operator on C4n . A generalized Penrose integral formula gives the solutions to the holomorphic k-Cauchy–Fueter equations, and conversely, any holomorphic solution to these equations is given by this integral formula. By restriction to the quaternionic space Hn ⊆ C4n , we find all k-regular functions. The integral formula also gives the series expansion of a k-regular function by homogeneous k-regular polynomials. In particular, the result holds for left regular functions, which are exactly 1-regular. It is almost elementary to show the k-regularity of the function given by the integral formula or such series, but the proof of the inverse part that any k-regular function can be provided by the integral formula or such series involves some tools of sheaf theory. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The k-Cauchy–Fueter operators play important roles in quaternionic analysis (cf. [1–9] and references therein). Theorems in [7] proved by the second author imply that there was a 1–1 correspondence between the first cohomology of the sheaf O (−k − 2) over an open subset of the projective space and the solutions to the k-Cauchy–Fueter equations on the quaternionic space Hn . It is quite interesting to find the explicit integral formula realizing this correspondence, since it will allow us to find all solutions to k-Cauchy–Fueter equations, i.e., all k-regular functions. We write a vector in Hn as q = (q0 , . . . , qn−1 ) with ql = x4l + x4l+1 i + x4l+2 j + x4l+3 k ∈ H, l = 0, 1, . . . , n − 1. The usual Cauchy–Fueter operator D : C 1 (Hn , H) → C (Hn , Hn ) is defined as

 ∂ q0 f  .  D f =  ..  , ∂ qn−1 f 

✩ Supported by National Nature Science Foundation in China (No. 11171298).

∗

Corresponding author. E-mail addresses: [email protected] (Q. Kang), [email protected] (W. Wang).

0393-0440/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2012.11.002

(1.1)

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

193

for f ∈ C 1 (Hn , H), where

∂ ql = ∂x4l + i∂x4l+1 + j∂x4l+2 + k∂x4l+3 ,

(1.2)

l = 0, 1, . . . , n − 1. A function f : Hn → H is called (left) regular if D f ≡ 0 on Hn . We want to find all left regular functions on Hn . 1.1. The holomorphic k-Cauchy–Fueter operator We begin with the definition of holomorphic k-Cauchy–Fueter operator on C4n since the k-Cauchy–Fueter operator can ′ be viewed as its restriction to the quaternionic space. We denote a point z ∈ C4n as z = (z AA ), and denote

∇AA′ := ∂z AA′ ,

(1.3)

the holomorphic derivatives on C4n , where A = 0, 1, . . . , 2n − 1, and A′ = 0′ , 1′ . An element of C2 is denoted by (v0′ , v1′ ), while an element of the symmetric power ⊙k C2 is denoted by (vA′ ···A′ ) with vA′ ···A′ ∈ C, A′1 , . . . , A′k = 0′ , 1′ , where vA′ ···A′ 1 k k k 1 1 is invariant under the permutations of subscripts. Therefore,

⊙k C2 ∼ = Ck+1 , and an element v of ⊙k C2 can be written as v = (v0′ ···0′ , v1′ 0′ ···0′ , . . . , v1′ ···1′ )t , where t is the transpose. For k = 1, 2, . . ., define the holomorphic k-Cauchy–Fueter operator D(k) : H (C4n , Ck+1 )

−→ −→

φ

H (C4n , C2n ⊗ Ck ), D(k) φ,

where for φ = (φA′ ···A′ ), 1

D(k) φ





AA′2 ···A′k

k

A′

:= ∇A 1 φA′1 A′2 ···A′k ,

(1.4)

A′1 , . . . , A′k = 0′ , 1′ , A = 0, 1, . . . , 2n − 1, and for some complex vector space V , H (C4n , V ) is the space of all V -valued holomorphic functions on C4n . Here and in the following we use Einstein convention of taking summation over repeated indices. The matrix

ϵ = (ϵA′ B′ ) =





0 −1

1 0

(1.5) A′

is used to raise or lower indices, e.g., ∇A 1 ϵA′ A′ = ∇AA′ . In particular, 1 2

′

∇A0 = ∇A1′ ,

2

′

∇A1 = −∇A0′ .

(1.6)

Then the holomorphic-k-Cauchy–Fueter equations D(k) φ = 0 can be written as

∇A1′ φ0′ A′2 ···A′k − ∇A0′ φ1′ A′2 ···A′k = 0, A = 0, 1, . . . , 2n − 1, A′2 , . . . , A′k = 0′ , 1′ . In particular, the holomorphic 1-Cauchy–Fueter equations D(1) φ = 0 can be written as

 ∇01′    .. (1) φ0′ D = . φ1′ ∇(2n−1)1′  ∂z 01′  .. = . ∂z (2n−1)1′

 −∇00′    φ0′ ..  φ′ . 1 −∇(2n−1)0′  −∂z 00′    φ0′ ..  φ ′ = 0, . 1 −∂z (2n−1)0′

where φ = (φ0′ , φ1′ )t ∈ H (C4n , C2 ).

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

1.2. The k-Cauchy–Fueter operator It is known that the quaternionic algebra H can be embedded in C2×2 by



x0 + ix1 x2 − ix3

x0 + x1 i + x2 j + x3 k −→

 −x2 − ix3 . x0 − ix1

We will use the conjugate embedding

ι : Hn ≃ R4n ↩→ C2n×2 ,

(1.7)

′

(q0 , . . . , qn−1 ) −→ z = (z AA ), A = 0, 1, . . . , 2n − 1, A′ = 0′ , 1′ , defined by



z (2l)0

′

z (2l)1

z (2l+1)0

′

z (2l+1)1

′



 :=

′

x4l − ix4l+1 x4l+2 + ix4l+3

 −x4l+2 + ix4l+3 , x4l + ix4l+1

(1.8)

where l = 0, 1, . . . , n − 1. Pulling back to the quaternionic space Hn ∼ = R4n by the embedding in (1.8), we define differential operators



(2l)0′ ∇

(2l)1′ ∇

(2l+1)0′ ∇

(2l+1)1′ ∇



∂x4l + i∂x4l+1 := ∂x4l+2 − i∂x4l+3 

 −∂x4l+2 − i∂x4l+3 , ∂x4l − i∂x4l+1

(1.9)

on R4n . For k = 1, 2, . . ., we call

 D(k) : C ∞ (R4n , Ck+1 ) φ

−→ −→

C ∞ (R4n , C2n ⊗ Ck ),  D(k) φ,

the k-Cauchy–Fueter operator on R4n , where A′

 (k)   D φ

AA′2 ···A′k

 1 φA′ A′ ···A′ , := ∇ A k 1 2

(1.10)

A′1 , . . . , A′k = 0′ , 1′ , A = 0, 1, . . . , 2n − 1, for φ = (φA′ ···A′ ). A Ck+1 -valued function φ : R4n → Ck+1 is called k-regular if it k 1 satisfies the k-Cauchy–Fueter equations:

 D(k) φ(x) = 0

(1.11)

for any x ∈ R4n . In particular, the 1-Cauchy–Fueter equations  D(1) φ = 0 can be written as



φ0′  D(1) φ1′ 



 01′ 00′ ∇ −∇     φ0′ .. .. =  . . φ1′ (2n−1)1′ −∇ (2n−1)0′ ∇   −∂x2 − i∂x3 −∂x0 − i∂x1 −∂x2 + i∂x3   ∂x0 − i∂x1    .. ..   . .   φ0′ = = 0, −∂x4l − i∂x4l+1  −∂x4l+2 − i∂x4l+3  φ1′  ∂ − i∂  −∂ + i ∂ x4l+1 x4l+2 x4l+3   x4l .. .. . .

(1.12)

where φ = (φ0′ , φ1′ )t ∈ C ∞ (R4n , C2 ). See [7] for the matrix form of the k-Cauchy–Fueter operator. For a function F = F0 + F1 i + F2 j + F3 k on Hn , set φ0′ := F2 − iF3 , φ1′ := −F0 − iF1 . Then we have F = −φ1′ + jφ0′ . The Cauchy–Fueter equations D F = 0 can be written as

AA′ φA′ = 0, ∇

A = 0, 1, . . . , 2n − 1,

which are exactly the 1-Cauchy–Fueter equations. Hence, the C2 -valued function (φ0′ , φ1′ )t is 1-regular if and only if the function F = −φ1′ + jφ0′ is left regular (cf. Section 3.2).

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

195

1.3. The Penrose integral formula Let [z0 , . . . , z2n+1 ] be the homogeneous coordinates for points in the projective space CP2n+1 . Following [7,10], we use notations:

πA′ = (π0′ , π1′ ) := (z0 , z1 ),

(1.13)

ωA = (ω0 , ω1 , . . . , ω2n−1 ) := (z2 , z3 , . . . , z2n+1 ).

Denote PI := CP2n+1 − I, where I := {[πA′ , ωA ] ∈ CP2n+1 ; π0′ = 0, π1′ = 0}. PI can be covered by two coordinate charts

PI0 := {[πA′ , ωA ] ∈ CP2n+1 ; π0′ ̸= 0},

(1.14)

PI1 := {[πA′ , ωA ] ∈ CP2n+1 ; π1′ ̸= 0}. PI0 and PI1 are both biholomorphically equivalent to C2n+1 , and PI0 ∩ PI1 ∼ = C1 \ {0} × C2n .

U := {PI0 , PI1 } is an affine (Stein) covering of PI . We denote by OPI (m) (or briefly O (m)) the sheaf of germs of holomorphic functions on PI , homogeneous of degree m. For any sheaf O over a complex manifold X , we denote by O (U ) the space of sections of the sheaf O over an open subset U ⊆ X . So O (m)(U ) is the space of holomorphic functions on U, homogeneous of degree m, where U is an open subset of PI . By Laurent expansion, we know that any element in O (−k − 2)(PI0 ∩ PI1 ) can be represented as the sum of terms of the following type: hj (ω0 /π0′ , . . . , ω2n−1 /π0′ )

π0k′+2−j π1j ′

,

(1.15)

where hj is holomorphic in ωA /π0′ , A = 0, 1, . . . , 2n − 1, j ∈ Z. The Penrose integral transformation is

P : O (−k − 2)(PI0 ∩ PI1 )

−→ −→

f

C ∞ (C4n , Ck+1 ), Pf,

where

(P f )A′1 ···A′k (z) :=

 Γ

πA′1 · · · πA′k f (πA′ , ωA ) · (π0′ dπ1′ − π1′ dπ0′ ),

′

z = (z AA ) ∈ C4n .

′

(1.16)

′

Here for fixed z = (z AA ), the contour Γ is a smooth closed curve in PI0 ∩ PI1 ∩ {ωA = z AA πA′ } enclosing the singularities of

the integrand, i.e., in the complex curve {(ωA , πA′ ); ωA = z integral is independent of the choice of Γ .

AA′

πA′ }. Since Γ encloses the singularities of the integrand, the

Theorem 1.1. (1) For f ∈ O (−k − 2)(PI0 ∩ PI1 ), the Ck+1 -valued holomorphic function φ with φA′ ···A′ = (P f )A′ ···A′ given 1 k 1 k by (1.16) is symmetric in A′1 , . . . , A′k and satisfies the holomorphic k-Cauchy–Fueter equations: A′

∇A 1 φA′1 A′2 ···A′k = 0,

(1.17)

A = 0, 1, . . . , 2n − 1, A′2 , . . . , A′k = 0′ , 1′ , on C4n . The integral depends only on the equivalent class of f in the first Čech cohomology group H 1 (U, O (−k − 2)). Conversely, for any holomorphic solution φ to Eqs. (1.17), there exists f in O (−k − 2)(PI0 ∩ PI1 ) such that φ = P f . (2) A Ck+1 -valued holomorphic function φ = (φA′ ···A′ ) is a solution to the holomorphic k-Cauchy–Fueter equations if and 1

j

k

only if there exist complex numbers Cm , m ∈ Z2n ≥0 , such that



φA′1 ···A′k (z) =

|m |+a+1

m∈Z2n ≥0

j Cm PAm′ +···+A′ ,j (z), k

1

j =1



z = z AA

′



∈ C4n ,

(1.18)

which is uniformly convergent on any compact subset of C4n , where a = A′1 + · · · + A′k , Pam,j (z) =

1 2π i

2n−1



r a −j |r |=1



′

′

z A0 + z A1 r

mA

dr ,

A=0

is a holomorphic k-regular homogeneous polynomial of degree |m| = m0 + · · · + m2n−1 on C4n .

(1.19)

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208 j

The coefficients Cm in (1.18) are independent of the indices A′1 , . . . , A′k . So the Ck+1 -valued solution φ can be written as

 m  P0,j φ0′ 0′ ···0′ |+a+1 P1m,j   |m φ1′ 0′ ···0′    j  φ= Cm  ..  .  ..  =  . .  j =1 m∈Z2n ≥0 ′ ′ ′ φ1 1 ···1 Pkm,j 

We find the explicit formula of the polynomial Pam,j as follows. Write a vector M ∈ Z4n ≥0 as M = (M0′ , M1′ ) =

(m00′ , . . . , m(2n−1)0′ , m01′ , . . . , m(2n−1)1′ ) with M0′ = (m00′ , . . . , m(2n−1)0′ ),

M1′ = (m01′ , . . . , m(2n−1)1′ ) ∈ Z2n ≥0 .

4n Given m = (m0 , . . . , m2n−1 ) ∈ Z2n ≥0 , we denote by Jm the set of all tuples M ∈ Z≥0 such that

mA0′ + mA1′ = mA ,

A = 0, 1, . . . , 2n − 1.

(1.20)

Then for m = (m0 , . . . , m2n−1 ), we can show that Pam,j (z) =





m



M 1′

M∈Jm , |M1′ |=j−1−a

zM ,

(1.21)

′

where |M1′ | = m01′ + · · · + m(2n−1)1′ , and for z = (z AA ), 2n−1 M

z

:=

1   

z

AA′

mAA′



,

m

:=

M1′

A=0 A′ =0

2n−1



  mA  mA1′

A=0

.

(1.22)

For n = 1, Penrose proved that the Penrose transformation φ = P f satisfies Eqs. (1.17) on C4 , and H 1 (U, O (−k − 2)) is isomorphic to the holomorphic solutions to Eqs. (1.17) by using the method of exact sequence (cf. [10–12] and references therein). We generalize the Penrose integral formula from the case n = 1 to the case n > 1, and show the integral formula (1.16) realizing this isomorphism. Lerner [13] and Penrose [14] sketched the proof of an inverse formula for the case n = 1, i.e., given a solution φ to Eqs. (1.17), they found f such that P f = φ . But their proofs involve too much details of analysis (see [13,15] and pp. 309–316 in [14]). For our purpose, we only need the existence of the inverse. So we give a complete and self-contained proof of the existence of the inverse, by using the diagram chasing of the exact sequence, which seems easier. 1.4. Series expansion of k-regular functions By Theorem 1.1 and the embedding ι in (1.7)–(1.8), we get all k-regular functions on R4n . j

Theorem 1.2. A Ck+1 -valued function φ = (φA′ ···A′ ) is k-regular on R4n if and only if there exist complex numbers Cm , m ∈ Z2n ≥0 , k 1 such that



φA′1 ···A′k (x) =

|m |+a+1

m∈Z2n ≥0

j Cm PAm′ +···+A′ ,j (ι(x)), 1

j =1

x ∈ R4n ,

k

(1.23)

which is uniformly convergent on any compact subset of R4n . Here a = A′1 + · · · + A′k , |m| = m0 + · · · + m2n−1 for a vector m = (m0 , . . . , m2n−1 ), the embedding ι is given by (1.7)–(1.8), and PAm′ +···+A′ ,j are given by (1.19) or (1.21). 1

k

By Theorem 1.2 and the equivalence of 1-regularity and left regularity, we get all left regular functions on Hn . j

Corollary 1.1. A H-valued function on Hn is left regular if and only if there exist complex numbers Cm , m ∈ Z2n ≥0 , such that it equals to

−

|+2   |m m∈Z2n ≥0

j Cm P1m,j (z) + j

j =2

|+1  |m  m∈Z2n ≥0

j Cm P0m,j (z),

(1.24)

j =1 ′

which is uniformly convergent on any compact subset of Hn , (z AA ) are given by z (2l)0 = ′

jql j + kql k

, −2 iql i + ql ′ z (2l+1)0 = j, −2 l = 0, 1, . . . , n − 1.

z (2l)1 = ′

z (2l+1)1

′

jql j − kql k

j,

−2 iql i − ql = , −2

(1.25)

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

197

In Section 2, we introduce some basic facts about the sheaf O (m) and the Čech cohomology with coefficients in O (m). We prove that P f , as a Ck+1 -valued holomorphic function on C4n , satisfies the holomorphic k-Cauchy–Fueter equations (1.17). The inverse part of Theorem 1.1 will be proved in Section 4. By substituting the cohomology classes in H 1 (U, O (−k − 2)), represented by functions in the form (1.15), into the integral formula, we get all holomorphic solutions to the holomorphic k-Cauchy–Fueter equations, and their series expansions. In Section 3, for any holomorphic solution φ to holomorphic k-Cauchy–Fueter equations, we pull back it to R4n by the embedding ι in (1.7)–(1.8), and show that ι∗ φ(x) = φ(ι(x)) is exactly a k-regular function on R4n . Conversely, any Ck+1 valued k-regular function on R4n can be extended to a Ck+1 -valued holomorphic function on C4n satisfying holomorphic k-Cauchy–Fueter equations. So any k-regular function on R4n has the form of P f (ι(x)) for some f ∈ O (−k − 2)(PI0 ∩ PI1 ) by Theorem 1.1. In particular, we get all left regular functions on Hn . In Section 4, we prove the inverse part of Theorem 1.1, i.e., any holomorphic solution φ to the system of Eqs. (1.17) can be represented as φ = P f for some function f ∈ O (−k − 2)(PI0 ∩ PI1 ) by using the exact sequence constructed in [11,12]. We would like to thank the referees for many useful suggestions. 2. The Penrose integral formula and solutions to the holomorphic k-Cauchy–Fueter equations 2.1. The sheaves O (m) (Cf. Section 2 in [7] and [10]). Let MI := C2n ⊕ C2n ∼ = C4n and FI := MI × CP1 . We will work on the following double fibration: (2.1)

FI @ @@   @@τ  @@    @ η

PI

MI

with

 ′  η(z, πA′ ) = z AA πA′ , πA′ ,

τ (z, πA′ ) = z,

(2.2)

′

for z = (z AA ) ∈ MI . The projection τ is trivial with fiber CP1 , while the fiber η−1 (ωA , πA′ ) of η over the point (ωA , πA′ ) ′ ′ is {(z AA , πA′ ); z AA πA′ = ωA }, isomorphic to C2n . η(τ −1 (z)) is isomorphic to CP1 for fixed z ∈ MI . Let (r , p) and (R, P ) be I coordinates of P0 and PI1 , respectively, where p = (p0 , . . . , p2n−1 ), P = (P 0 , . . . , P 2n−1 ), and

π1′ , π0′ π0′ R := . π1′

ωA , π0′ ωA , P A := π1′

pA :=

r :=

(2.3)

Here [πA , ωA ] are homogeneous coordinates of CP2n+1 in (1.13). For a complex manifold X , we denote by OX the sheaf of germs of holomorphic functions on X . PI is covered by two affine open subsets PI0 and PI1 given by (1.14), where I := {[πA′ , ωA ] ∈ CP2n+1 ; π0′ = 0, π1′ = 0}. The sheaf O (m) is the gluing sheaf of sheaves OPI and OPI by the isomorphism 0

1

ρ10 : OPI |PI ∩PI

−→

OPI |PI ∩PI ,

s(r , p)

−→

S (R, P ) =

0

0

1

1

0

1



π0′ π1′

m

s( r , p ) = R m s



1 P

,

R R



(2.4)

,

π ′

m (cf. pp. 5 and 147 in [16] for gluing sheaf). g10 := ( π0′ )m : PI0 ∩ PI1 → C \ {0} is the transition function of the corresponding 1 line bundle. Under the following isomorphism of sheaves:

O (m)|PI ≃ π0m′ OPI , 0

0

O (m)|PI ≃ π1m′ OPI , 1

(2.5)

1

the sheaf O (m) is isomorphic to the gluing sheaf of sheaves π0m′ OPI and π1m′ OPI by the isomorphism 0

 ρ10 : π

m 0′ OPI0 PI0 ∩PI1

|

s

−→  → −

π

m 1′ OPI1 PI0 ∩PI1

|

1

,

s,

where s ↔ π1m′ S (R, P ) = π0m′ s(r , p) in (2.4). Namely, O (m) is the sheaf of germs of holomorphic functions on PI , homogeneous of degree m.

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

We cover CP1 with V0 := {[π0′ , π1′ ] ∈ CP1 ; π0′ ̸= 0}, V1 := {[π0′ , π1′ ] ∈ CP1 ; π1′ ̸= 0}. Then FI can be trivialized by FI0 := MI × V0 and FI1 := MI × V1 . Similarly, the sheaf OFI (m) is a gluing sheaf of sheaves OFI

0

and OFI by the isomorphism 1

ϱ10 : OFI |FI ∩FI

OFI |FI ∩FI ,

−→

1

1

0

0

s(z, r )

1

0

S (z, R) =

−→



π0′ π1′

m



1

s(z, r ) = R s z, m



R

(2.6)

.

Under the following isomorphism of sheaves:

OFI (m)|FI ≃ π0m′ OFI , 0

OFI (m)|FI ≃ π1m′ OFI ,

0

1

(2.7)

1

the sheaf OFI (m) is isomorphic to the gluing sheaf of sheaves π

m O I 0′ F

0

and π

m O I 1′ F

1

by the isomorphism

 ϱ10 : π0m′ OFI |FI ∩FI −→ π1m′ OFI |FI ∩FI , 1 1 0 1 0 0 s −→ s, where s ↔ π1m′ S (z, R) = π0m′ s(z, r ) in (2.6). Namely, OFI (m) is the sheaf of germs of holomorphic functions on FI , homogeneous of degree m in πA′ . 2.2. The first Čech cohomology group H 1 (U, O (−k − 2)) The set of q-cochains of U with coefficients in the sheaf O (m) will be denoted by C q (U, O (m)). In particular, a 0-cochain is a pair of sections h+ = {h0 , h1 } with hj ∈ O (m)(PIj ), j = 0, 1, while a 1-cochain is a section f ∈ O (m)(PI0 ∩ PI1 ). There is no triple intersection for this cover, so C q (U, O (m)) = 0 for q > 1. The coboundary operator δ is defined as

δ : C 0 (U, O (m))

C 1 (U, O (m)), h0 |PI ∩PI − h1 |PI ∩PI ,

−→ −→

+

h

0

0

1

(2.8)

1

for h+ = {h0 , h1 } ∈ C 0 (U, O (m)). H 1 (U, O (m)) :=

C 1 (U, O (m))

(2.9)

Imδ

is the first Čech cohomology group of the cover U with coefficients in the sheaf O (m) (cf. Appendix A in [17] for Čech cohomology with coefficients in a sheaf). 2.3. The proof of Theorem 1.1 For f ∈ O (−k − 2)(PI0 ∩ PI1 ), we write f = π0−′ k−2 s(r , p) for some s ∈ O (PI0 ∩ PI1 ). Note that π0′ dπ1′ − π1′ dπ0′ = π02′ dr and ′

′

πA′1 · · · πA′k f (πA′ , ωA ) · (π0′ dπ1′ − π1′ dπ0′ ) = r A1 +···+Ak s(r , p)dr .

(2.10)

′ = (z MM ) ∈ MI , we parameterize the complex curve η[τ −1 (z)] ∼ = CP1 by (π0′ , π1′ ): any point 2n−1 [π0′ , π1′ , ω , . . . , ω ] ∈ η[τ −1 (z)] has the form

For fixed z

0

[π0′ , π1′ , z 0M πM ′ , . . . , z (2n−1)M πM ′ ]. ′

′

If π0′ ̸= 0, the point is (r , p) with r = π1′ /π0′ and pM =

ωM ′ ′ = z M0 + z M1 r , π0′

M = 0, . . . , 2n − 1.

(2.11)

It follows from (1.16) and (2.10) that

(P f )A′1 ···A′k (z) =







r A1 +···+Ak s r , z 00 + z 01 r , . . . , z (2n−1)0 + z (2n−1)1 r dr , ′

′

′

′

′

′

(2.12)

|r |=1

where we can take the closed curve Γ = {|r | = 1} in the complex r plane since the only possible singularities of the integrand are the origin and the infinity.

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

199

We can also write f = π1−′ k−2 S (R, P ) for some S ∈ O (PI0 ∩ PI1 ) satisfying

π0−′ k−2 s(r , p) = π1−′ k−2 S (R, P ),

on PI0 ∩ PI1 .

(2.13)

Since π0′ dπ1′ − π1′ dπ0′ = −π12′ dR, we have

πA′1 · · · πA′k f · (π0′ dπ1′ − π1′ dπ0′ ) = −Rk−(A1 +···+Ak ) S (R, P )dR. ′

′

On the other hand, the integrand of (2.12) satisfies

−Rk−(A1 +···+Ak ) S (R, P )dR = r A1 +···+Ak s(r , p)dr , ′

′

′

′

on PI0 ∩ PI1 , by (2.13), R = 1/r, dR = −dr /r 2 and r = π1′ /π0′ . Hence, the integrand of (1.16) (or (2.12)) is a well-defined holomorphic 1-form on PI0 ∩ PI1 . Let us verify that P f satisfies the holomorphic k-Cauchy–Fueter equations. Differentiating (2.12) to get

∂(P f )A′1 ···A′k ∂

′ z A0



′

′

r A1 +···+Ak

= |r |=1

 ∂ s  00′ ′ ′ ′ r , z + z 01 r , . . . , z (2n−1)0 + z (2n−1)1 r dr , A ∂p

and

∂(P f )A′1 ···A′k ∂z

A1′



′

′

r 1+A1 +···+Ak

= |r |=1

 ∂ s  00′ 01′ (2n−1)0′ (2n−1)1′ r , z + z r , . . . , z + z r dr , ∂ pA

for any fixed A, A1 , . . . , A′k , by using (2.11). Then, ′

∂(P f )1′ A′2 ···A′k ∂

′ z A0

=

∂(P f )0′ A′2 ···A′k ∂ z A1′

,

for any fixed A, A′2 , . . . , A′k . By (1.6), we find that A′

′

′

∇A 1 (P f )A′1 A′2 ···A′k = ∇A0 (P f )0′ A′2 ···A′k + ∇A1 (P f )1′ A′2 ···A′k = ∇A1′ (P f )0′ A′2 ···A′k − ∇A0′ (P f )1′ A′2 ···A′k = 0, for any fixed A, A′2 , . . . , A′k .

To see that the integral depends only on the Čech cohomology class of f in H 1 (U, O (−k − 2)), let f be an element in the image of δ defined by (2.8). Then we can write f = π0−′ k−2 s + π1−′ k−2 S for some holomorphic functions s and S on PI0 and PI1 , respectively. We find that

 Γ

πA′1 · · · πA′k π0−′ k−2 s · (π0′ dπ1′ − π1′ dπ0′ ) =



′

′



′

′



r A1 +···+Ak s r , z A0 + z A1 r dr = 0, |r |=1

since the integrand is holomorphic on the complex r plane. The integral corresponding to π1−′ k−2 S also vanishes similarly. So we see that P f = 0. The proof of the inverse part of Theorem 1.1 will be given in Section 4. (2) By part (1), given a solution φ to the holomorphic k-Cauchy–Fueter equations, there exists f ∈ H 1 (U, O (−k − 2)) such that φ = P f . Let us calculate the integral (1.16) to get the explicit expression of solutions. Any element f ∈  j m H 1 (U, O (−k − 2)) can be represented by series of (1.15). Let hj (p) = be the Taylor expansion of hj on m∈Z2n Cm p ≥0

j ∈ C, m = (m0 , . . . , m2n−1 ) ∈ Z2n (pA )mA , Cm ≥0 . This power series is uniformly convergent on any ′ compact subset of C . On the complex curve η[τ −1 (z)] for a fixed z = (z AA ) ∈ MI ,   ′ ′ ′ ′ +∞ hj z 00 + z 01 r , . . . , z (2n−1)0 + z (2n−1)1 r  ′ ′ f (πA′ , z 0M πM ′ , . . . , z (2n−1)M πM ′ ) = π0k′+2 r j −∞

C2n , where pm :=

2n−1 A=0 2n

=

+∞ 

1

−∞

π0k′+2 r j

2n−1

 m∈Z2n ≥0

j Cm



′

′

z A0 + z A1 r

mA

.

(2.14)

A=0

This series is obviously uniformly convergent on the circle |r | = 1 for z in a compact subset of C4n by Cauchy’s estimate.

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

Let a = A′1 + · · · + A′k . It follows from (1.16), (2.12) and (2.14) that 1 2π i



1

(P f )A′1 ···A′k (z) =

2π i

Γ

  ′ πA′1 · · · πA′k f πA′ , z AA πA′ · (π0′ dπ1′ − π1′ dπ0′ )

+∞  1 

=



2π i −∞

|r |=1

+∞  

1

=

2π i

j Cm

2n−1



r a −j |r |=1

j=−∞ m∈Z2n ≥0

+∞  

=



′

′

r a−j hj z A0 + z A1 r dr



′

′

z A0 + z A1 r

mA

dr

A=0

j Cm Pam,j (z),

(2.15)

j=−∞ m∈Z2n ≥0 ′

where z = (z AA ), and Pam,j (z) are given by (1.19). The third identity of (2.15) holds since the series (2.14) is uniformly convergent on the circle |r | = 1 for z in a compact subset of C4n . m For m = (m0 , . . . , m2n−1 ) ∈ Z2n ≥0 , we expand the product in the definition of (1.19) of Pa,j to get Pam,j

(z) =

−1  1    m  2n

1

2π i M∈J m





=

M 1′ m





r a+|M1′ |−j dr |r |=1

A=0 A′ =0

M1′

M∈Jm , |M1′ |=j−1−a

′

(z AA )mAA′

zM ,

by

 r

a+|M1′ |−j

 dr =

|r |=1

2π i, 0,

if a + |M1′ | − j = −1, otherwise,

where |M1′ | = m01′ + · · · + m(2n−1)1′ , Jm is given by (1.20),

(2.16)



m M1′



and zM are given by (1.22). It follows from (2.16) that Pam,j

do not vanish only for 1 ≤ j ≤ |m| + a + 1. Conversely, given a Ck+1 -valued series φ = (φA′ ···A′ ) with φA′ ···A′ defined by (1.18), set PAm′ ···A′ ,j (z) := PAm′ +···+A′ ,j (z). k k 1 1 k k 1 1 If we choose f = A′

1 2π i

2n−1

(pA )mA

π ′ 0

π ′ 1

A=0 k+2−j

j

in (2.15), then we have (P f )A′ ···A′ (z) = PAm′ +···+A′ ,j (z) by (2.15), and so automatically, k 1 1 k

∇A 1 PAm′ A′ ···A′ ,j = 0 by (1). By the uniform convergence of the series (1.18) on any compact subset of C4n , we find that 1 2

A′

k

∇A 1 φA′1 ···A′k =

 m∈Z2n ≥0

|m |+a+1

A′

j Cm ∇A 1 PAm′ ···A′ ,j = 0,

j =1

1

k

A′2 , . . . , A′k = 0′ , 1′ , A = 0, . . . , 2n − 1. So the Ck+1 -valued series φ = (φA′ ···A′ ) with φA′ ···A′ given by (1.18) are solutions to 1 k 1 k the holomorphic k-Cauchy–Fueter equations. The theorem is proved.

3. Restriction to the quaternionic space The following theorem is Theorem 5.1 in [7], which will be used in the proof of Theorem 1.2. Theorem 3.1 (The Real Analyticity for k-Regular Functions). For a domain Ω ⊆ Hn , a Ck+1 -valued k-regular function φ on Ω is real analytic. This means that each component of φ is a C-valued real analytic function. For k = 1, each component of a 1-regular function is harmonic, and so it is real analytic.

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

201

3.1. The proof of Theorem 1.2

Lemma 3.1. Suppose that χ is a holomorphic function over C4n and  χ is complex-valued function on R4n such that

 χ (x) = χ (z)|z=ι(x) ,

x ∈ R4n ,

(3.1)

where ι is the embedding in (1.7)–(1.8). Then

AA′  ∇ χ (x) = 2∇AA′ χ (z)|z=ι(x) .

(3.2)

Proof. Note that

∂χ ∂χ ∂χ ∂ χ ∂χ ∂ χ = (2l)0′ + (2l+1)1′ , = −i (2l)0′ + i (2l+1)1′ , ∂ x4l ∂ x4l+1 ∂z ∂z ∂z ∂z ∂χ ∂χ ∂ χ ∂χ ∂χ ∂ χ = − (2l)1′ + (2l+1)0′ , = i (2l)1′ + i (2l+1)0′ , ∂ x4l+2 ∂ x4l+3 ∂z ∂z ∂z ∂z by using (1.8). Consequently,

∂ χ ∂ χ ∂χ ∂ χ ∂χ ∂ χ +i = 2 (2l)0′ , −i = 2 (2l)1′ , − ∂ x4l ∂ x4l+1 ∂ x4l+2 ∂ x4l+3 ∂z ∂z ∂ χ ∂ χ ∂ χ ∂χ ∂ χ ∂χ −i = 2 (2l+1)0′ , −i = 2 (2l+1)1′ . ∂ x4l+2 ∂ x4l+3 ∂ x4l ∂ x4l+1 ∂z ∂z AA′ in (1.9). Now the result follows by definitions of ∇AA′ in (1.3) and ∇



For a Ck+1 -valued holomorphic function φ = (φA′ ···A′ ) on C4n , we define a Ck+1 -valued function on R4n : 1

k

     φA′1 ···A′k (x) := φA′1 ···A′k (ι(x)) , φ(x) = 

(3.3)

for x ∈ R4n . It follows from (3.2) that  D(k) φ = 0, i.e.,  φ is k-regular, if D(k) φ = 0. k+1 Conversely, for any C -valued k-regular function  φ = ( φA′ ···A′ ) on R4n , we want to extend it to a Ck+1 -valued 1

k

holomorphic function φ satisfying both the relation (3.3) and the holomorphic k-Cauchy–Fueter equations D(k) φ = 0. Denote

 φa :=  φA′1 ···A′k ,

(3.4)

φ can be written as  φ = ( φ0 ,  φ1 , . . . ,  φk )t . The real where a = A′1 + · · · + A′k . Then the Ck+1 -valued k-regular function  4n part and the imaginary part of each component  φa must be real analytic on R by Theorem 3.1. So we can write  φa as a convergent power series on R4n with complex coefficients: 

 φa (x) =

Ca;m xm ,

(3.5)

m∈Z4n ≥0 m

xi i for x = (x0 , . . . , x4n−1 ) ∈ R4n and m = (m0 , . . . , m4n−1 ) ∈ Z4n ≥0 . 4n  First, we extend φa to a holomorphic function Φa on C by the above power series expansion of  φa , i.e., set

where Ca;m ∈ C, xm =

Φa (w) :=



4n−1 i =0

Ca;m wm

m∈Z4n ≥0

for w = (w0 , . . . , w4n−1 ) ∈ C4n . Let wl = xl + iyl , l = 0, . . . , 4n − 1. Then

 φa (x) = Φa (x + i0),

for x ∈ R4n .

(3.6)

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

We claim that the Ck+1 -valued holomorphic function φ = (φA′ ···A′ ) on C4n , defined as follows, is the extension satisfying k 1 the requirements. Set φA′ ···A′ := φa with a = A′1 + · · · + A′k and k

1

φa (z) := Φa (ι−1 (z)),

(3.7)

where the map ι is extended to a linear transformation ι : C4n −→ C4n , w → z, defined by z (2l)0

 z

z (2l)1

′

′

(2l+1)0′

z



(2l+1)1′

 =

w4l − iw4l+1 w4l+2 + iw4l+3

 −w4l+2 + iw4l+3 , w4l + iw4l+1

and so the inverse ι−1 is defined as z (2l)0 + z (2l+1)1 ′

w4l =

2

w4l+2 =

z

(2l)1′

′

z (2l)0 − z (2l+1)1 ′

,

w4l+1 =

(2l+1)0′

−z −2

,

w4l+3 =

′

, −2i ′ ′ z (2l)1 + z (2l+1)0 2i

,

l = 0, 1, . . . , n − 1. By (3.6) and (3.7), we get

φa (z)|z=ι(x) = Φa (ι−1 (z))|z=ι(x) = Φa (x) =  φa (x),

(3.8)

for each a. Namely, φ satisfies the relation (3.3). Then by (3.4) and (3.8), we can apply Lemma 3.1 to χ = φ and  χ = φ to get

AA′  ∇ φa (x) = 2∇AA′ φa (z)|z=ι(x) , for each A, A′ . Since the Ck+1 -valued k-regular function  φ satisfies k-Cauchy–Fueter equations, we have ′

′

A  A1  ′ ′ ′ 2∇A 1 φA′ A′ ···A′ (z)|z=ι(x) = ∇ A φA A ···A (x) = 0, 1 2

k

k

1 2

x ∈ R4n ,

(3.9) (k)

(k)

for A = 0, 1, . . . , 2n − 1, and A2 , . . . , Ak = 0 , 1 , by using (1.6). Consequently, D φ ≡ 0 on C since D φ , as a C ⊗ Ck valued holomorphic function, vanishes on the totally real subspace ι(R4n ) ⊂ C4n by (3.9). This construction of Ck+1 -valued φ is the same as that in Section 3 of [7]. Now we have shown that a Ck+1 -valued  φ on R4n is k-regular if and only if  φ(x) = φ(ι(x)) for some Ck+1 -valued holomorphic function φ on C4n satisfying D(k) φ = 0. By Theorem 1.1, any solution to the holomorphic k-Cauchy–Fueter equations D(k) φ = 0 is given by formula (1.18). Consequently, we know that any k-regular functions is given by formula (1.18) when restricted to z = ι(x). ′

′

′

′

4n

2n

3.2. Series expansion of left regular functions We have the equivalence of 1-regularity and left regularity (cf. Section 1 in [7]). Proposition 3.1. For an H-valued function F = F0 + F1 i + F2 j + F3 k on Hn , set φ0′ = F2 − iF3 , φ1′ = −F0 − iF1 . Then F = −φ1′ + jφ0′ , and it is left regular if and only if the C2 -valued function (φ0′ , φ1′ ) is 1-regular. Proof. The Cauchy–Fueter equations for F can be written as

∂ ql F = (∂x4l + i∂x4l+1 + j∂x4l+2 + k∂x4l+3 )(−φ1′ + jφ0′ ) = −∂x4l φ1′ − i∂x4l+1 φ1′ − ∂x4l+2 φ0′ − i∂x4l+3 φ0′   + j −∂x4l+2 φ1′ + i∂x4l+3 φ1′ + ∂x4l φ0′ − i∂x4l+1 φ0′ = 0,

(3.10)

which is equivalent to

−∂x4l φ1′ − i∂x4l+1 φ1′ − ∂x4l+2 φ0′ − i∂x4l+3 φ0′ = 0, −∂x4l+2 φ1′ + i∂x4l+3 φ1′ + ∂x4l φ0′ − i∂x4l+1 φ0′ = 0, i.e., by (1.9)

(2l)0′ φ1′ + ∇ (2l)1′ φ0′ = 0, −∇  (2l+1)1′ φ0′ = 0, −∇(2l+1)0′ φ1′ + ∇

(3.11) ′

A φA′ = 0, A = 0, 1, . . . , 2n − 1, by using (1.6). l = 0, 1, . . . , n − 1. (3.11) are exactly ∇ A 2 Conversely, if (3.11) holds for a C -valued function (φ0′ , φ1′ ), then (3.10) holds for F = −φ1′ + jφ0′ . The result follows.



Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

203

4. The proof of the inverse part Given a Ck+1 -valued holomorphic function φ satisfying D(k) φ = 0 on C4n , we will find f ∈ O (−k − 2)(PI0 ∩ PI1 ) such that P f = φ along the line of the proof in [12] for the case n = 1, although in [12] the author proved the isomorphism between H 1 (U, O (−k − 2)) and ker D(k) . We show that the holomorphic section f obtained by diagram chasing is exactly what we want, i.e., it satisfies P f = φ . 4.1. Sheaves T (−k − 2) and Zk (m) on FI ′ Let f be a holomorphic function on FI homogeneous in πA′ . If it is constant on the fiber η−1 η(z, πA′ ) = η−1 (z AA πA′ , πA′ ),



then f (z, πA′ ) = g (z

AA′

′

π A ∇AA′ f = π A

′

πA′ , πA′ ) for some function g on PI , where z = (z

AA′



). So we have

∂g πA′ = 0, ∂ωA

by ′

′

′

π A πA′ = π 0 π0′ + π 1 π1′ = 0,

(4.1)

which follows from ′

′

π 0 = π1′ ,

π 1 = −π0′ ,

(4.2)

by (1.5). ′ ∂f π ′ ∂f Conversely, without loss of generality, we assume π1′ ̸= 0. Suppose π A ∇AA′ f = 0. We have A0′ − A1′ π0′ = 0. If we ′

′

′

′

∂z

∂z

1

f . Then f (z A0 , ωA , πA′ ) for some  use coordinates {z A0 , ωA , πA′ } for FI1 (z A1 = (ωA − z A0 π0′ )/π1′ ), we can write f (z, πA′ ) = 

∂ f ∂

=

′ z A0

′

∂f ∂z

A0′

+

∂ f ∂ z A1 ∂f ∂ f π0′ = 0. ′ ′ = ′ − A1 A0 A0 ∂z ∂z ∂z ∂ z A1′ π1′

Hence,  f is constant for fixed ωA , πA′ , and so f is constant on the fiber η−1 η(z, πA′ ) = η−1 (ωA , πA′ ).





I From the above, we know that a holomorphic  function f (z, πA′ ) on F homogeneous in πA′ pushes down to a function on PI if it is constant on each fiber η−1 η(z, πA′ ) or equivalently if ′

′

πA′ ∇AA f = π A ∇AA′ f = 0.

(4.3)

Let T (−k − 2) be the sheaf of germs of holomorphic functions f (z, πA′ ) on FI , homogeneous of degree −k − 2 in πA′ , ′ satisfying Eqs. (4.3). Obviously, η−1 O (−k − 2) ⊆ T (−k − 2). For a germ f in T (−k − 2), since πA′ ∇AA f = 0, we know that ′

f is constant on the fiber η−1 (ωA , πA′ ), where ωA = z AA πA′ . Hence, f (z, πA′ ) = g (ωA , πA′ ) for some germ g in O (−k − 2). Then we have

η−1 O (−k − 2) = T (−k − 2).

(4.4)

Denote by Zk (m) the sheaf of germs of Ck+1 -valued holomorphic functions (φA′ ···A′ (z, πA′ )) on FI , homogeneous of degree k m in πA′ , satisfying the holomorphic k-Cauchy–Fueter equations: A′

∇A 1 φA′1 A′2 ···A′k = 0.

(4.5)

Suppose that A, B and C are sheaves over a complex manifold X , a sequence of sheaves g

h

A −→ B −→ C is called exact if the induced sequence on stalks gx

hx

Ax −→ Bx −→ Cx ,

(4.6)

satisfies Im(gx ) = Ker(hx ) for each x ∈ X . Recall that an element of stalk Ax is a germ of A at point x, represented by a local section of A on a neighborhood of x. The exactness of the following sequence (4.7) for n = 1 is proved in Chapter 10 of [11] and Section 4 of [12]. Since it is important for Theorem 1.1, we will give a complete proof.

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Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

Lemma 4.1. The sequence of sheaves over FI : ν1

ν2

0 −→ T (−k − 2) −→ Zk+1 (−1) −→ Zk (0) −→ 0,

(4.7)

is exact, where for any point ζ ∈ F , f ∈ T (−k − 2)ζ and ψ ∈ Zk+1 (−1)ζ , I

(ν1 f )A′1 ···A′k+1 = πA′1 · · · πA′k+1 f , ′

(ν2 ψ)A′1 ···A′k = π Ak+1 ψA′1 ···A′k A′k+1 . Before proving Lemma 4.1, we give a standard lemma which is a consequence of Taylor’s formula. Let D(ζ ; ε) be the polydisc D(ζ1 ; ε) × · · · × D(ζm+1 ; ε) for some point ζ = (ζ1 , . . . , ζm+1 ) ∈ Cm+1 . Lemma 4.2. Suppose that f is a holomorphic function on a polydisc D((ζ1 , . . . , ζm , 0); ε) in Cm+1 and f (z , 0) = 0 for any (z , 0) ∈ D((ζ1 , . . . , ζm , 0); ε). Then there exists a holomorphic function g such that f (z1 , . . . , zm+1 ) = zm+1 g (z1 , . . . , zm+1 ) on this polydisc. Proof. Since f is holomorphic, we can write its Taylor’s expansion in variable zm+1 as f (z1 , . . . , zm+1 ) = f (z1 , . . . , zm , 0) +

∞  1 ∂ lf (z1 , . . . , zm , 0)zml +1 = zm+1 g (z1 , . . . , zm+1 ), l l ! ∂ zm +1 l =1

where g (z1 , . . . , zm+1 ) is a holomorphic function on this polydisc.



Proof of Lemma 4.1. For a fixed point ζ in FI , let us show ν1

ν2

0 −→ T (−k − 2)ζ −→ Zk+1 (−1)ζ −→ Zk (0)ζ −→ 0

(4.8)

is exact. Obviously, the map ν1 is injective. First, let us show Imν1 = ker ν2 . For f ∈ T (−k − 2)ζ , we have ν2 (ν1 f ) = ′

′

π Ak+1 πA′1 · · · πA′k πA′k+1 f = π Ak+1 πA′k+1 πA′1 · · · πA′k f = 0 by (4.1). Thus, Imν1 ⊆ ker ν2 . Now let us show ker ν2 ⊆ Imν1 . Our sheaves can be viewed as subsheaves of OF , the sheaf of germs of holomorphic functions on the space F = C4n × C2 \ {0} with variables (z, π0′ , π1′ ). According to the position of ζ in F , there are three cases: ζ = (z, 0, π1′ ); ζ = (z, π0′ , 0); ζ = (z, π0′ , π1′ ) with π0′ π1′ ̸= 0. We only show the first case. The other two cases are similar. Now we can assume ζ = (z, 0, 1). A′ Let ψ ∈ ker ν2 . So π k+1 ψA′ ···A′ A′ = 0, i.e., by (4.2), k k+1 1 π1′ ψA′1 ···A′k 0′ (z, π0′ , π1′ ) = π0′ ψA′1 ···A′k 1′ (z, π0′ , π1′ ),

(4.9)

for (z, π0′ , π1′ ) in a small polydisc D(ζ ; ε) ⊂ F . If we set π0′ = 0 in (4.9), we get ψA′ ···A′ 0′ = 0 on D(z; ε) × {0} × D(1; ε), k 1 since we can choose the polydisc D(ζ ; ε) so small that π1′ ̸= 0 on it. Hence, there exist holomorphic functions gA′ ···A′ such 1 k that

ψA′1 ···A′k 0′ = π0′ gA′1 ···A′k , on this polydisc D(ζ ; ε), by Lemma 4.2. On the other hand, define GA′ ···A′ = ψA′ ···A′ 1′ /π1′ since π1′ ̸= 0 on this polydisc. k k 1 1 Then we have

ψA′1 ···A′k 1′ = π1′ GA′1 ···A′k . Now by (4.9), we get π1′ π0′ gA′ ···A′ = π0′ π1′ GA′ ···A′ , which implies gA′ ···A′ = GA′ ···A′ when π1′ π0′ ̸= 0. Thus, 1

k

ψA′1 ···A′k A′k+1 = πA′k+1 gA′1 ···A′k ,

1

k

1

k

1

k

(4.10)

on the whole polydisc D(ζ ; ε). Since ψA′ ···A′ is invariant under the permutations of subscripts by the definition of 1 k+1 Zk+1 (−1), we have

πA′k+1 gA′1 ···A′k−1 A′k = πA′k gA′1 ···A′k−1 A′k+1 . In particular, for A′k = 0′ , A′k+1 = 1′ , we get

π1′ gA′1 ···A′k−1 0′ = π0′ gA′1 ···A′k−1 1′ .

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

As above there exist holomorphic functions hA′ ···A′ 1

k−1

such that gA′ ···A′

a function f holomorphic on the polydisc D(ζ ; ε) such that

1

A′

k−1 k

205

= πA′k hA′1 ···A′k−1 . Repeating this procedure, we get

ψA′1 ···A′k+1 (z, πA′ ) = πA′1 · · · πA′k+1 f (z, πA′ ).

(4.11) A′

Since ψ is homogeneous of degree −1 in πA′ , we know that f is homogeneous of degree −k − 2 in πA′ . ∇A 1 ψA′ A′ ···A′ = 0 k+1 1 2 implies ′

πA′ ∇AA f = 0. Then f ∈ T (−k − 2)ζ . Second, we show that the map ν2 is onto, i.e., given φ ∈ Zk (0)ζ , we can find ψ ∈ Zk+1 (−1)ζ such that ′

π Ak+1 ψA′1 ···A′k A′k+1 = φA′1 ···A′k

(4.12)

in a neighborhood of ζ . ′ ′ Fix π 0 = 1, π 1 = 0. (4.12) becomes

ψA′1 ···A′k 0′ = φA′1 ···A′k .

(4.13)

Note that the Ck+2 -valued function ψ will be in Zk+1 (−1)ζ , i.e., ψ satisfies the holomorphic k-Cauchy–Fueter equations (4.5), if ψ satisfies

∂ψ1′ ···1′ 1′ ∂ψ1′ ···1′ 0′ = ′ A0 ∂z ∂ z A1′



∂φ1′ ···1′ = ∂ z A1′



,

for each A, since other equations for ψ in the holomorphic k-Cauchy–Fueter equations (4.5) follows from the equations satisfied by φ . For fixed φ ,

∂φ1′ ···1′ ∂ψ1′ ···1′ 1′ = , ∂ z A0′ ∂ z A1′

A = 0, . . . , 2n − 1,

(4.14)

is a system of linear partial differential equations of first order. Note that

∂ 2 ψ1′ ···1′ 1′ ∂ 2 φ1′ ···1′ ∂ 2 φ1′ ···1′ 0′ , ′ ′ = ′ ′ = A0 B0 A1 B0 ∂z ∂z ∂z ∂z ∂ z A1′ ∂ z B1′ where the second identity holds since φ ∈ Zk (0)ζ also satisfies Eqs. (4.5). Thus,

∂ 2 ψ1′ ···1′ 1′ ∂ 2 ψ1′ ···1′ 1′ = , ′ ′ ∂ z A0 ∂ z B0 ∂ z B0′ ∂ z A0′ i.e., Eqs. (4.14) are compatible. Hence, we can integrate (4.14) to get a unique solution locally in FI by choosing the initial ′ value ψ1′ ···1′ 1′ = 0 on the hyperplane defined by z A0 = constant, A = 0, . . . , 2n − 1. Hence, ν2 is surjective when ′ ′ π 0 = 1, π 1 = 0. ′ ′ ′ ′ ′ Now we allow πA′ to vary. Set w = zN, i.e., (w A0 , w A1 ) := (z A0 , z A1 )N, A = 0, . . . , 2n − 1, where N = (NAB′ ) is a 2 × 2 ′

′

′

′

non-degenerate matrix. Let M = (MAB′ ) be the inverse matrix of N. Then z = wM, i.e., z AA = w AB MBA′ . In the following, the matrices M and N may depend on πA′ , but are independent of z. For the given Ck+1 -valued function φ = (φA′ ···A′ ) ∈ Zk (0)ζ , define a new Ck+1 -valued holomorphic function 1

B′

ΦA′1 ···A′k (w, π0′ , π1′ ) := MA′1 · · · 1

B′ MA′k k

k

φB′1 ···B′k (z, π0′ , π1′ )|z=wM

(4.15)

on a neighborhood of ζ . We claim that Φ = (ΦA′ ···A′ ) also satisfies the holomorphic k-Cauchy–Fueter equations (4.5). k 1 Namely, these equations are invariant under the transformation (4.15) (cf. [18] for a similar transformation for n = 1). B′ ···B′

B′

B′

2

k

Denote MA′2 ···Ak′ := MA′2 · · · MA′k . We have 2

k

∂ Φ1′ A′2 ···A′k ∂wA0′

=

B′ ···B′ B′ MA′2 ···Ak′ M1′1 2

k



∂φB′1 ···B′k ∂ z A0′ ∂φB′1 ···B′k ∂ z A1′ + ∂ z A0′ ∂wA0′ ∂ z A1′ ∂wA0′



  ∂φ1′ B′2 ···B′k ∂φ0′ B′2 ···B′k ∂φ1′ B′2 ···B′k ′ ′ ∂φ0′ B′2 ···B′k B′ ···B′ 1′ 0′ 0′ 1′ 1′ 1′ = MA′2 ···Ak′ M10′ M00′ + M M + M M + M M , ′ ′ ′ ′ ′ ′ 1 0 1 0 0 1 2 k ∂ z A0′ ∂ z A0′ ∂ z A1′ ∂ z A1′

206

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

and

∂ Φ0′ A′2 ···A′k

=

∂wA1′

∂φB′1 ···B′k ∂ z A0′ ∂φB′1 ···B′k ∂ z A1′ + ∂ z A0′ ∂wA1′ ∂ z A1′ ∂wA1′



B′ ···B′ B′ MA′2 ···Ak′ M0′1 k

2



  ∂φ1′ B′2 ···B′k ∂φ0′ B′2 ···B′k ∂φ1′ B′2 ···B′k ′ ′ ∂φ0′ B′2 ···B′k B′ ···B′ 1′ 0′ 0′ 1′ 1′ 1′ = MA′2 ···Ak′ M00′ M10′ + M M + M M + M M . 0′ 1′ 0′ 1′ 1′ 0′ k 2 ∂ z A0′ ∂ z A0′ ∂ z A1′ ∂ z A1′ It follows from

∂φ1′ B′ ···B′ 2

∂ z A0

∂ Φ1′ A′2 ···A′k ∂w

=

∂φ0′ B′ ···B′ 2

k

′

∂ z A1

∂ Φ0′ A′2 ···A′k

=

A0′

k

′

∂wA1′

that

.

(4.16)

The claim is proved. Now we define ΨA′ ···A′ 0′ on a neighborhood of ζ as 1

k

ΨA′1 ···A′k 0′ = ΦA′1 ···A′k ,

(4.17)

and Ψ1′ ···1′ 1′ is determined by the following equations:

∂ Φ1′ ···1′ ∂ Ψ1′ ···1′ 1′ = , ′ A0 ∂w ∂wA1′

A = 0, . . . , 2n − 1,

(4.18)

similar to (4.14). By above, Eqs. (4.18) are compatible and the unique holomorphic solution Ψ1′ ···1′ 1′ exists locally on a ′ neighborhood of ζ with initial condition Ψ1′ ···1′ 1′ = 0 on the hyperplane defined by w A0 = constant, A = 0, . . . , 2n − 1. Then (4.16)–(4.18) imply that Ψ = (ΨA′ ···A′ A′ ) satisfies holomorphic k-Cauchy–Fueter equations: k k+1

1

∂ Ψ1′ A′2 ···A′k+1 ∂w

A0′

∂ Ψ0′ A′2 ···A′k+1

=

′

.

∂wA1′

(4.19)

′

For given (π 0 , π 1 ), we can choose matrix N such that ′

′

′

π A NA0′ = 1,

′

π A NA1′ = 0

(4.20)

i.e., N =



1

(π 0′ )2 + (π 1′ )2

′

π0 ′ π1

−π 1 ′ π0

′



,

(4.21)

and take M to be the inverse matrix of N. Set A′

A′

ψC1′ ···Ck′ +1 (z, π0′ , π1′ ) := NC ′1 · · · NC ′k+1 ΨA′1 ···A′k+1 (w, π0′ , π1′ )|w=zN .

(4.22)

k+1

1

Then (4.15), (4.17) and (4.20) implies that ′

A′

′

A′

A′

1

k

π Ck+1 ψC1′ ···Ck′ Ck′ +1 = π Ck+1 NC ′k+1 NC ′1 · · · NC ′k ΨA′1 ···A′k+1 k+1

=

A′ NC ′1 1

···

A′ NC ′k ΦA′ ···A′ k 1 k

= φC1′ ···Ck′ ,

(4.23)

where we have used identity B′

B′

A′

A′

B′

B′

1

k

1

k

1

k

MA′1 · · · MA′k NC ′1 · · · NC ′k = δC 1′ · · · δC k′ . The fact that Ψ satisfies the holomorphic k-Cauchy–Fueter equations (4.19) implies that ψ also satisfies these equations by the invariance of these equations under the transformation (4.22). Note that φ , M and N are holomorphic on a neighborhood of ζ , homogeneous of degree 0, 1 and −1 in πA′ , respectively. We know that ψ is holomorphic on a neighborhood of ζ , homogeneous of degree −1 in πA′ by (4.15) and (4.22). Hence, ψ ∈ Zk+1 (−1)ζ and ν2 ψ = φ by (4.23). The lemma is proved. 

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

207

4.2. Proof of the inverse part of Theorem 1.1 Let V := {FI0 , FI1 } be an open cover of FI . For an open subset X of FI , the short exact sequence of sheaves on FI in Lemma 4.1 induces the following long exact sequence of Čech cohomology groups: 0 −→ H 0 (X , T (−k − 2)) −→ H 0 (X , Zk+1 (−1)) −→ H 0 (X , Zk (0)) −→ H 1 (X , T (−k − 2))

−→ H 1 (X , Zk+1 (−1)) −→ H 1 (X , Zk (0)) −→ · · · .

(4.24)

Note that for any sheaf O , H 0 (X , O ) = O (X ) and C 0 (V , O ) = H 0 (FI0 , O ) ⊕ H 0 (FI1 , O ),

C 1 (V , O ) = H 0 (FI0 ∩ FI1 , O ).

(4.25)

Also H 1 (FI0 , O ) = H 1 (FI1 , O ) = 0 for a coherent sheaf O , since FI0 and FI1 are both affine. But H 1 (FI0 ∩ FI1 , O ) may be ′

non-vanishing (FI0 ∩ FI1 is not affine). Note that T (−k − 2) is coherent since it is the kernel of the morphism πA′ ∇AA : OFI (−k − 2) → OFI (−k − 1) between coherent sheaves. Applying (4.24) to X = FI0 , FI1 and FI0 ∩ FI1 , we get the following commutative diagram of exact sequences of Čech cochains: ν1

0−→

C 0 (V , T (−k − 2)) −→

0−→

C (V , T (−k − 2)) −→

↓δ

ν2

C 0 (V , Zk+1 (−1)) −→

↓δ

ν1

1

ν2

C (V , Zk+1 (−1)) −→ 1

C 0 (V , Zk (0)) −→ 0,

↓δ

(4.26)

C (V , Zk (0)). 1

A solution φ to the holomorphic k-Cauchy–Fueter equations (1.17) on MI defines a 0-cochain φ + ∈ C 0 (V , Zk (0)) by setting φ + := {φ, Φ } with Φ (z, πA′ ) := φ(z). Obviously, δ(φ + ) = 0. Now let us find a section f ∈ O (−k − 2)(PI0 ∩ PI1 ), such that P f = φ , by diagram chasing. By the surjectivity of ν2 in the first line in (4.26), there exists a 0-cochain ψ + = {ψ, Ψ } ∈ C 0 (V , Zk+1 (−1)), with ψ ∈ Zk+1 (−1)(FI0 ) and Ψ ∈ Zk+1 (−1)(FI1 ), such that ν2 ψ + = φ + , i.e., ′

π Ak+1 ψA+′ ···A′ A′ 1

k k+1

= φA+′ ···A′ . 1

(4.27)

k

Here (4.27) means ′

on FI0 ,

′

on FI1 .

π Ak+1 ψA′1 ···A′k A′k+1 = φA′1 ···A′k , π Ak+1 ΨA′1 ···A′k A′k+1 = ΦA′1 ···A′k ,

(4.28)

By commutativity of (4.26), we have ν2 (δψ + ) = δ(ν2 ψ + ) = δ(φ + ) = 0. Hence, δψ + is in the kernel of the map ν2 in the second line.By the exactness of the second line in (4.26), we get δψ + = ν1 g for some g ∈ C 1 (V , T (−k − 2)) = T (−k − 2)(FI0 FI1 ), i.e.,

(δψ + )A′1 ···A′k+1 = πA′1 · · · πA′k+1 g .

(4.29)

Automatically, δ g = 0 since C q (V , T (−k − 2)) = 0 for q ≥ 2. By (4.4), we can write ′

g (z, πA′ ) = f (z AA πA′ , πA′ ),

(4.30) AA′

for some f ∈ O (−k − 2)(PI0 ∩ PI1 ), where z = (z ). Now let us show that P f gives φ . By (4.29) and the definition of δ , we have

πA′1 · · · πA′k+1 f = (δψ + )A′1 ···A′k+1 = ψA′1 ···A′k+1 − ΨA′1 ···A′k+1 . Since ψA′ ···A′ 1

k+1

(4.31)

∈ OFI (−1)(FI0 ), we can write it as

ψA′1 ···A′k+1 (z, πA′ ) = hA′1 ···A′k+1 (z, r )

1

π0′

,

(4.32)

for some holomorphic function hA′ ···A′ (z, r ) on FI0 = MI × V0 , by using (2.7), where r = π1′ /π0′ is the coordinate on V0 . 1 k+1 Similarly,

ΨA′1 ···A′k+1 (z, πA′ ) = HA′1 ···A′k+1 (z, R)

1

π1′

,

(4.33)

208

Q. Kang, W. Wang / Journal of Geometry and Physics 64 (2013) 192–208

for some holomorphic function HA′ ···A′ (z, R) on FI1 = MI × V1 , where R = π0′ /π1′ is the coordinate on V1 . Substitute (4.33) k+1 1 into the second equation of (4.28) to get

φA′1 ···A′k (z) = π1′ ΨA′1 ···A′k 0′ (z, πA′ ) − π0′ ΨA′1 ···A′k 1′ (z, πA′ ) = HA′1 ···A′k 0′ (z, R) − HA′1 ···A′k 1′ (z, R)R, on FI1 by (4.2). Since φA′ ···A′ is independent of πA′ , we have k

1

φA′1 ···A′k (z) = HA′1 ···A′k 0′ (z, 0).

(4.34)

Substitute (4.31) into the Penrose integral formula (1.16) and use (4.32)–(4.34) to get

(P f )A′1 ···A′k (z) =

 Γ

πA′1 · · · πA′k f · (π0′ dπ1′ − π1′ dπ0′ )

π0′ dπ1′ − π1′ dπ0′ π0′ π1′   Γ dr dr ΨA′1 ···A′k 0′ π1′ ψA′1 ···A′k 0′ π1′ − = r r |r |=1 |r |=1   dr = hA′ ···A′ 0′ (z, r )dr − HA′ ···A′ 0′ (z, 1/r ) k k 1 1 

πA′1 · · · πA′k π0′ f π1′

=

|r |=1

|r |=1

r

= −2π iHA′1 ···A′k 0′ (z, 0) = −2π iφA′1 ···A′k (z).

(4.35)

Here the integral of hA′ ···A′ 0′ (z, r ) vanishes since it is holomorphic in the complex variable r. 1

k

Now for a holomorphic solution φ on MI to the holomorphic k-Cauchy–Fueter equations (1.17), we have found f ∈ O (−k − 2)(PI0 ∩ PI1 ) such that −21π i P f = φ . Remark 4.1. (1) In [19], the authors proved that given a ∂ -closed (0, 1)-form f with coefficients in the (−k − 2)-th power of the hyperplane section bundle H −k−2 , there is a Radon–Penrose type integral representation P f such that ι∗ (P f ) is a solution to the k-Cauchy–Fueter equations, where ι is an embedding of the quaternionic space Hn into C4n . (2) Theorems in [8] proved by the second author imply that there is a 1–1 correspondence between some cohomology and the solutions to the tangential k-Cauchy–Fueter equations on the quaternionic Heisenberg group. It is interesting to find explicit integral formula realizing this correspondence. References [1] W. Adams, C. Berenstein, P. Loustaunau, I. Sabadini, D. Struppa, Regular functions of several quaternionic variables and the Cauchy–Fueter complex, J. Geom. Anal. 9 (1999) 1–15. [2] W. Adams, P. Loustaunau, Analysis of the module determining the properties of regular functions of several quaternionic variables, Pacific J. Math. 196 (2001) 1–15. [3] W. Adams, P. Loustaunau, V. Palamodov, D. Struppa, Hartogs’ phenomenon for polyregular functions and projective dimension of releted modules over a polynomial ring, Ann. Inst. Fourier 47 (1997) 623–640. [4] J. Bureš, A. Damiano, I. Sabadini, Explicit resolutions for several Fueter operators, J. Geom. Phys. 57 (2007) 765–775. [5] F. Colombo, I. Sabadini, F. Sommen, D. Struppa, Analysis of Dirac Systems and Computational Algebra, in: Progress in Mathematical Physics, vol. 39, Birkhäuser, Boston, 2004. [6] F. Colombo, V. Souček, D.C. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys. 56 (2006) 1175–1191. [7] W. Wang, The k-Cauchy–Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic k-regular functions, J. Geom. Phys. 60 (2010) 513–530. [8] W. Wang, The tangential Cauchy–Fueter complex on the quaternionic Heisenberg group, J. Geom. Phys. 61 (2011) 363–380. [9] W. Wang, On the optimal control method in quaternionic analysis, Bull. Sci. Math. 135 (2011) 988–1010. [10] M. Eastwood, R. Penrose, R. Wells, Cohomology and massless fields, Comm. Math. Phys. 78 (3) (1980) 305–351. [11] S.A. Huggett, K.P. Tod, An Introduction to Twistor Theory, in: London Math. Society Student Texts, vol. 4, Cambridge University Press, 1994. [12] R. Penrose, On the twistor descriptions of massless fields, in: Complex Manifold Techniques in Theoretical Physics, in: Res. Notes. Math., vol. 32, 1979, pp. 55–91. [13] D.E. Lerner, The inverse twistor function for positive frequency fields, in: Advance in Twistor Theory, in: Res. Notes. Math., vol. 37, 1979, pp. 65–67. [14] R. Penrose, Twistor theory, its aims and achievements, in: C.J. Isham, R. Penrose, D.W. Sciama) (Eds.), Quantum Gravity, An Oxford Symposium, Clarendon Press, Oxford, 1975, pp. 267–407. [15] N.P. Buchdahl, The inverse twistor function revisited, in: Further Advance in Twistor Theory, Volume I: The Penrose Transform and its Applications, in: Res. Notes. Math., vol. 231, 1990, pp. 88–94. [16] H. Grauert, R. Remmert, Theory of Stein Space, in: Grundlehren der Mathematischen Wissenschaften, vol. 236, Springer-Verlag, Berlin, New York, 1979, (translated from the German by A. Huckleberry). [17] R. Wells, Differential Analysis on Complex Manifolds, in: Graduate Texts in Mathematics, vol. 65, Springer-Verlag, New York, Berlin, 1980. [18] S.G. Gindikin, G.M. Henkin, Penrose transformation and complex integral geometry, J. Soviet Math. 21 (4) (1983) 508–551. [19] Q. Kang, W. Wang, On Radon–Penrose transformation and k-Cauchy–Fueter operator, Sci. China Math. 55 (9) (2012) 1921–1936.