Real structures on primary Hopf surfaces

Theorem 2.1

Consider the sets I V , I I I , I I a , I I b , I I c ⊂ Aut h ( W ) defined in formula (1).

1. For every primary Hopf surface H, there exists f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c such that H ≃ H f .

2. For f , f ′ ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c , we have H f ≃ H f ′ if and only if either f = f ′ , or f and f ′ belong to I I c , and the corresponding coefficients α , δ , α ′ , δ ′ satisfy α ′ = δ , δ ′ = α .

3. For any f, the group Aut h ( W ) f of holomorphic automorphisms of W commuting with f is given by the table below:

   The class of f	Aut h ( W ) f
   I V	GL ( 2 , C )
   I I I	z w ↦ a z + b w r d w a ∈ C * , d ∈ C * , b ∈ C
   I I a	z w ↦ a r z + b w r a w a ∈ C * , b ∈ C
   I I b	z w ↦ a z + b w a w a ∈ C * , b ∈ C
   I I c	z w ↦ a z d w a ∈ C * , d ∈ C *

4. In each case, the automorphism group Aut h ( H f ) is identified with the quotient group Aut h ( W ) f ⁄ ⟨ f ⟩ .

Proof

We explain below only the part of Theorem 2.1 which is different from Wehler’s original statement.

(1) follows from Kodaira’s [5, Theorem 30, p. 696] quoted in the introduction taking into account that in the case λ ≠ 0 , using a linear rescaling of the first coordinate, we can assume λ = 1 . Note that the argument does not apply if one imposes the strict inequality ∣ α ∣ < ∣ δ ∣ in the definition of the class I I c .

For the statement (2) in the case omitted in Wehler’s [11, Theorem 1], i.e. when f , f ′ belong to the class I I c as defined in formula (1):

If f , f ′ define isomorphic Hopf surfaces, then, as explained in [11, p. 24], there exists a biholomorphism φ : C 2 → C 2 with φ ( 0 ) = ( 0 ) such that f ′ = φ ∘ f ∘ φ − 1 . Taking the Jacobian matrices at 0 of the two sides, we see that the matrices defining f , f ′ must be similar, in particular their spectra coincide. Therefore, we have either α ′ = α , δ ′ = δ , or α ′ = δ , δ ′ = α .

Conversely, if the spectra of the two diagonal matrices coincide, they are obviously conjugate by a linear automorphism of C 2 .

Concerning (3): Note that for the computation of the automorphism groups Wehler quotes Namba’s article [6]. In this article, the class of linear primary Hopf surfaces (i.e. the class of primary Hopf surfaces defined by a linear contraction) is parameterised using the set [6, p. 133]

No inequality between ∣ α ∣ , ∣ β ∣ is imposed (even in the special case t = 0 ). This agrees with the convention we adopted in the definition of the class I I c .□

Remark 2.2

In [11], the case ∣ α ∣ = ∣ δ ∣ is omitted in the definition of I I c , so the exception (2) to the injectivity of the map

is not mentioned either.

Proposition 2.3

Let H = H f = W ⁄ ⟨ f ⟩ be a primary Hopf surface and let π : W → H be the canonical map. Let σ : H → H be a homeomorphism. Then

1. There exists a homeomorphism σ ˆ : W → W such that π ∘ σ ˆ = σ ∘ π .

2. For any such homeomorphism σ ˆ we have σ ˆ ∘ f ∘ σ ˆ − 1 ∈ { f , f − 1 } .

Proof

(4)

1. The composition σ ∘ π remains a covering, and, since W is simply connected, the uniqueness theorem of the universal covering guarantees the existence of a homeomorphism σ ˆ : W → W verifying the equality

   π ∘ σ ˆ = σ ∘ π .

2. The group

   Aut H ( W ) = { g : W → W ∣ g homeomorphism , π ∘ g = π }

   of topological automorphisms of the universal covering π (of deck transformations) coincides with the cyclic group ⟨ f ⟩ . On the other hand, the map

   g ↦ σ ˆ ∘ g ∘ σ ˆ − 1

   is a group automorphism of Aut H ( W ) , so it coincides either with id Aut H ( W ) or with the automorphism g ↦ g − 1 . Replacing g by f , we obtain σ ˆ ∘ f ∘ σ ˆ − 1 ∈ { f , f − 1 } as claimed.□

Proposition 2.4

Let σ : H → H be a holomorphic (anti-holomorphic) automorphism of H = H f . Then

1. There exists a a holomorphic (anti-holomorphic) automorphism σ ˆ of W such that π ∘ σ ˆ = σ ∘ π .

2. For any such automorphism σ ˆ , we have σ ˆ ∘ f ∘ σ ˆ − 1 = f .

Proof

Since π is locally biholomorphic, it follows that σ ˆ is holomorphic (respectively anti-holomorphic) if σ is holomorphic (respectively anti-holomorphic). By Hartogs theorem (applied to σ ˆ or to its composition with the conjugation automorphism), it follows that σ ˆ extends to a holomorphic (anti-holomorphic) automorphism σ ˜ of C 2 with σ ˜ ( 0 ) = 0 .

We can suppose that f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c . Any such f is a holomorphic contraction. It follows that for any w 0 ∈ W , we have

in the end compactification W ∪ { 0 , ∞ } of W . Since σ ˆ extends to a homeomorphism σ ˜ of C 2 with σ ˜ ( 0 ) = 0 , it follows that the permutation end ( σ ˆ ) induced by σ ˆ on the set of ends { 0 , ∞ } is id { 0 , ∞ } . Therefore,

whereas lim n → ∞ f − 1 ( w 0 ) = ∞ . Therefore, the case σ ˆ ∘ f ∘ σ ˆ − 1 = f − 1 is ruled out.□

Proposition 2.5

Let f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c . The following conditions are equivalent:

1. The primary Hopf surface H f admits anti-holomorphic automorphisms.

2. Either the coefficients of f are real or f ∈ I I c ′ .

Proof

The H f admits an anti-holomorphic automorphism if and only if H f is biholomorphic to H ¯ f . On the other hand, the conjugation automorphism c : W → W induces an anti-holomorphic isomorphism s : H f → H f , where f ≔ c ∘ f ∘ c − 1 . Recall now that, in general, the data of an anti-holomorphic isomorphism X → Y are equivalent to the data of a holomorphic isomorphism X ¯ → Y .

Therefore, H ¯ f ≃ H f , so H f admits an anti-holomorphic automorphism if and only if H f ≃ H f . Now note that f is obtained from f by conjugating the coefficients of the polynomial expression which defines f . On the other hand, Wehler’s classes are conjugation invariant, in particular f also belongs to I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c . By the classification Theorem 2.1, it follows that H f ≃ H f if and only if either the coefficients of f and f coincide (in other words the coefficients of f are real), or f and f belong to I I c and the coefficients α ¯ , δ ¯ of f are obtained from the coefficients α , δ of f by changing the order. The latter condition is equivalent to f ∈ I I c ′ .□

Remark 2.6

A direct proof of Proposition 2.5 can be obtained using Proposition 2.4 and the Taylor expansion of the anti-holomorphic automorphism σ ˜ of C 2 obtained by applying Hartogs theorem to the lift σ ˆ of an anti-holomorphic automorphism σ .

Proposition 2.7

Let f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c with real coefficients. The set

is given by the table:

In each case, the cyclic group ⟨ f ⟩ acts freely on the set Ah ( W ) f , and Ah ( H f ) is identified with the quotient set Ah ( W ) f ⁄ ⟨ f ⟩ .

Proposition 3.1

Let H = H f be a primary Hopf surface. Let σ : H → H be an anti-holomorphic involution of H and σ ˆ : W → W be a lift of σ (see Proposition 2.4). Then

1. There exists n ∈ Z such that σ ˆ 2 = f n .

2. The parity of n in the previous formula is well defined (depends only on σ , not on the lift σ ˆ ).

Proof

Indeed, since σ 2 = id H f it follows that σ ˆ 2 ∈ Aut H ( W ) = ⟨ f ⟩ . This proves the first claim.

Let now σ ˆ , σ ˆ ′ two lifts of σ and n , n ′ ∈ Z be the associated integers. We have σ ˆ ′ ∘ σ ˆ − 1 ∈ Aut H ( W ) , so there exists k ∈ N such that σ ˆ ′ = σ ˆ ∘ f k . Since σ ˆ commutes with f , we obtain σ ˆ ′ 2 = σ ˆ 2 ∘ f 2 k , so n ′ = n + 2 k .□

Definition 3.2

A Real structure σ : H → H on H is said to be:

1. even, if one of the following equivalent conditions is verified:

   1. For any lift σ ˆ : W → W of σ , σ ˆ 2 coincides with an even power of f .

   2. There exists a lift σ ˆ : W → W of σ such that σ ˆ 2 = id W .

2. odd, if one of the following equivalent conditions is verified:

   1. For any lift σ ˆ : W → W of σ , σ ˆ 2 coincides with an odd power of f .

   2. There exists a lift σ ˆ : W → W of σ such that σ ˆ 2 = f .

Example 3.1

Let f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c be with real coefficients. The standard complex conjugation c : W → W induces a Real structure s f on H f , which is obviously even. The Real structure will be called the standard Real structure of H f .

Let now f ∈ I I c ′ be given by f z w = α z α ¯ w , where 0 < ∣ α ∣ < 1 and α ∉ R . The anti-holomorphic automorphism of c ′ : W → W defined by

commutes with f and is involutive, so it defines an even Real structure s f on H f , which will also be called the standard Real structure on H f .

Remark 3.3

Let a : C n → C n be an anti-linear involution on C n . Then there exists a C -linear automorphism l : C n → C n such that a = l ∘ c ∘ l − 1 .

In other words, any anti-linear Real structure on C n is GL ( n , C ) -conjugate to the standard conjugation.

Proposition 3.4

Let f ∈ I V ∪ I I I ∪ I I a ∪ I I b ∪ I I c be with real coefficients and let ϕ be a Real structure on W such that ϕ ∘ f = f ∘ ϕ . Then there exists ψ ∈ Aut h ( W ) f such that ϕ = ψ ∘ c ∘ ψ − 1 with c is the standard conjugation on W.

Proof

1. f ∈ I V . In this case, using Proposition 2.7, we see that ϕ has the form

   ϕ z w = a b c d z ¯ w ¯

   with a b c d ∈ GL ( 2 , C ) . The obvious extension ϕ ˜ : C 2 → C 2 of ϕ to C 2 is an anti-linear Real structure on C 2 , so Remark 3.3 applies and gives a C -linear isomorphism l : C 2 → C 2 such that ϕ ˜ = l ∘ c ∘ l − 1 . Denoting by ψ : W → W the automorphism induced by l (which obviously commutes with f ), we obtain ϕ = ψ ∘ c ∘ ψ − 1 as claimed.

2. f ∈ I I I . In this case, Proposition 2.7 shows that ϕ is given by

   ϕ z w = a z ¯ + b w ¯ r d w ¯ ,

   where a , d ∈ C * , b ∈ C . We have

   ϕ 2 z w = ∣ a ∣ 2 z + ( a b ¯ + b d ¯ r ) w r ∣ d ∣ 2 w .

   The condition ϕ 2 = id becomes

   (5) ∣ a ∣ 2 = 1 a b ¯ + b d ¯ r = 0 ∣ d ∣ 2 = 1 .

   Let now ψ ∈ Aut h ( W ) f . By Wehler’s Theorem 2.1, we have

   ψ z w = A z + B w r D w

   with A , D ∈ C * , B ∈ C . The condition

   ϕ = ψ ∘ c ∘ ψ − 1

   is equivalent to the system

   (6) a = A A ¯ − 1 b = B D ¯ − r − A A ¯ − 1 B ¯ D ¯ − r d = D D ¯ − 1 .

   Since ∣ a ∣ = ∣ d ∣ = 1 we can write a = e i θ , d = e i δ with θ , δ ∈ R . Put A ≔ e i θ 2 , D ≔ e i δ 2 . With this choice, we have A A ¯ − 1 = A 2 = a , D D ¯ − 1 = D 2 = d , so the first and the third equations in (6) are satisfied. We are seeking B ∈ C such that the second equation is also satisfied.

   Consider the R -linear map λ : C → C defined by

   λ ( z ) = u z − v z ¯ ,

   where u ≔ D ¯ − r , v ≔ A A ¯ − 1 D ¯ − r = A 2 D ¯ − r . This map is not surjective. By using ∣ u ∣ = ∣ v ∣ = 1 , it follows easily that its image is the real line

   (7) im ( λ ) = { ζ ∈ C ∣ u − 1 ζ + v ζ ¯ = 0 } ⊂ C .

   Now note that b ∈ im ( λ ) , because

   u − 1 b + v b ¯ = D ¯ r b + A 2 D ¯ − r b ¯ = D ¯ − r ( D ¯ 2 r b + A 2 b ¯ ) = D ¯ − r ( d ¯ r b + a b ¯ ) = 0

   by the second equation in (5). Therefore, there exists B ∈ C such that

   b = λ ( B ) = D ¯ − r B − A A ¯ − 1 D ¯ − r B ¯ ,

   which proves that, with this choice, the second equation in (6) is satisfied, too.

3. f ∈ I I a . In this case, Proposition 2.7 shows that ϕ has the form

   ϕ z w = a r z ¯ + b w ¯ r a w ¯

   with a ∈ C * and b ∈ C . The condition ϕ 2 = id W is equivalent to the system

   (8) ∣ a ∣ 2 = 1 a r b ¯ + b a ¯ r = 0 .

   By Wehler’s theorem, an automorphism ψ ∈ Aut h ( W ) f has the form

   ψ z w = A r z + B w r A w

   with A ∈ C * , B ∈ C . The condition

   ϕ = ψ ∘ c ∘ ψ − 1

   is equivalent to the system

   (9) a = A A ¯ − 1 b = B A ¯ − r − A r B ¯ A ¯ − 2 r .

   Since ∣ a ∣ 2 = 1 , we can write a = e i θ with θ ∈ R . Putting A = e i θ 2 , we have A A ¯ − 1 = A 2 = a . As in the previous case, consider the R -linear map λ : C → C

   λ ( z ) = u z − v z ¯ ,

   where this time we choose u ≔ A r and v ≔ A 3 r . We have again ∣ u ∣ = ∣ v ∣ = 1 , so the image of λ is again given by (7). Note that b ∈ im ( λ ) because

   u − 1 b + v b ¯ = A − r b + A 3 r b ¯ = A r ( A 2 r b ¯ + A − 2 r b ) = A r ( a r b ¯ + b a ¯ r ) = 0

   by the second equation in (8). Therefore, there exists B ∈ C such that

   b = λ ( B ) = A r B − A 3 r B ¯ = B A ¯ − r − A r B ¯ A ¯ − 2 r ,

   which shows that ( A , B ) ∈ C * × C is a solution of the system (9).

4. f ∈ I I b . In this case, Proposition 2.7 shows that ϕ has the form

   ϕ z w = a z ¯ + b w ¯ a w ¯ ,

   where a ∈ C * , b ∈ C . The condition ϕ 2 = id W is equivalent to

   (10) ∣ a ∣ 2 = 1 a b ¯ + b a ¯ = 0 .

   By Wehler’s theorem, an automorphism ψ ∈ Aut h ( W ) f has the form

   ψ z w = A z + B w A w ,

   where A ∈ C * , B ∈ C . We use the same arguments as in the case I I a but taking r = 1 in all formulae used in this case.

5. f ∈ I I c . In this case, Proposition 2.7 shows that ϕ has the form

   ϕ z w = a z ¯ b w ¯ ,

   where a , b ∈ C * . The condition ϕ 2 = id W is equivalent to

   (11) ∣ a ∣ 2 = 1 ∣ b ∣ 2 = 1 .

   By Wehler’s theorem, in this case, an automorphism ψ ∈ Aut h ( W ) f has the form

   ψ z w = A z B w ,

   where A , B ∈ C * , and the condition

   ϕ = ψ ∘ c ∘ ψ − 1

   is equivalent to the system

   (12) a = A A ¯ − 1 b = B B ¯ − 1 .

   It suffices to put A ≔ e i θ 2 , B ≔ e i β 2 , where a = e i θ , b = e i β .□