Affine connections on complex compact surfaces and Riccati distributions
aa r X i v : . [ m a t h . DG ] J a n AFFINE CONNECTIONS ON COMPLEX COMPACTSURFACES AND RICCATI DISTRIBUTIONS
RUBEN LIZARBE
Abstract.
Let M be a complex surface. We show that there is aone-to-one correspondence between torsion-free affine connectionson M and Riccati distributions on P ( T M ). Furthermore, if M is compact, then this correspondence induces a one-to-one corres-pondence between affine structures on M and Riccati foliations on P ( T M ). Introduction
There are numerous papers on compact complex surfaces that admitholomorphic affine connections (see [7], [19], [18], [14], [5], [9] and [3]).In [5], Inoue, Kobayashi and Ochiai give a complete list of all compactcomplex (connected) surfaces admitting affine holomorphic connectionswhich are not necessarily flat. These surfaces are shown to be biholo-morphic (up to a finite covering) to complex tori, primary Kodairasurfaces, affine Hopf surfaces, Inoue surfaces or elliptic surfaces overRiemann surfaces of genus g ≥ C whose changeof coordinate maps are locally constant mappings in the affine groupGL(2 , C ) ⋉ C . These surfaces are said to be affine. It is also knownthat they are quotients of a domain in C by a group consisting of affinetransformations.In [9] Klingler classifies holomorphic affine structures and holomor-phic projective structures on compact complex surfaces. Also, in [3]Dumitrescu classifies torsion-free holomorphic affine connections andshows that any normal holomorphic projective connection on a compact complex surface has zero curvature. Finally, in his PhD dissertation [20]Zhao has studied affine structures and birational structures.The aim of this paper is to study the correspondence between affineconnections on complex surfaces and Riccati distributions. Our resultsare as follows. Theorem 1.1.
There is a one-to-one correspondence between torsion-free affine connections on M and Riccati distributions on P ( T M ) . Fur-thermore, we have a one-to-one correspondence between affine struc-tures on M and parallelizable Riccati foliations on P ( T M ) . Theorem 1.2. If M is compact, then a Riccati distribution on P ( T M ) is parallelizable. In particular, we have a one-to-one correspondencebetween affine structures on M and Riccati foliations on P ( T M ) . In Section 2 we introduce the notion of a Riccati connection on acomplex manifold of dimension n ≥
2. We also define the trace andthe curvature of a Riccati connection. The main result in this sectionguarantees that the existence of a Riccati connection on a compactK¨ahler manifold yields a relationship between its Chern classes (seeTheorem 2.3 and Proposition 2.5). We shall see that in the case ofdimension 2, the K¨ahler hypothesis is not necessary.In Section 3 given a complex surface M we establish a natural equiva-lence between reduced Riccati connetion on M and Riccati distributionson P ( T M ). We also introduce the notion of the curvature of a Riccatidistribution and show that if M is compact, then a Riccati distributionis parallelizable, that is, it has zero curvature. Theorem 1.1 and Theo-rem 1.2 are proved in this section. By using the geometric descriptionof all the affine compact complex surfaces (see the computations in [9,Th´eor`eme 1.2] with a fixed compact complex surface M , where the setof complex affine structures on M compatible with its analytical struc-ture are determined), we were able to calculate the monodromy of allthe Riccati foliations on P ( T M ).In Section 4 we apply the results of Section 3 to get a classificationof regular pencils and k -webs of foliations on compact complex surfaceswhen k ≥ FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 3 Riccati connections
The following definition was motivated by R. Molzon and K. PinneyMortensen [15]. Let M be a complex manifold of dimension n ≥
2. A
Riccati connection on M is a C -bilinear map DR : T M × T M → T M, satisfying(1) DR fX Y = f DR X Y − n X ( f ) Y ,(2) DR X f Y = f DR X Y + X ( f ) Y ,for any local holomorphic functions f and any local vector fields X, Y .In coordinates ( x , . . . , x n ) ∈ U ⊂ C n , a trivialization of T U is givenby the basis ( ∂ x , . . . , ∂ x n ) and the Riccati connection is given by DR X ( Y ) = d ( Y ) + θY − n div( X ) Y, where θ is the matrix of 1-forms associated with the Riccati connection DR and div represents the divergence operator.Letting ( ϕ α , U α ), ( ϕ β , U β ) be two local systems of coordinates on M with U α ∩ U β = ∅ , and denoting the corresponding change of coordinatesby ϕ αβ = ϕ α ◦ ϕ − β , we have θ α = g αβ θ β g − αβ − dg αβ g − αβ + 1 n Tr( dg αβ g − αβ ) I (1)where g αβ represents the Jacobian matrix of ϕ αβ and Tr the trace ope-rator.Note that the trace of θ represents a 1-form on M due to equation(1).Now we will describe two invariants defined from a Riccati connec-tion.2.1. Trace and curvature of a Riccati connection.Definition 2.1.
The trace of a Riccati connection DR , denoted byTr( DR ), is the 1-form defined as the trace of a matrix of 1-forms on M associated with DR . We say that DR is a reduced Riccati connection if its trace is zero, i.e., Tr( DR ) = 0. RUBEN LIZARBE
Remark 2.2.
Every Riccati connection DR on M determines a re-duced Riccati connection ˜ DR on M as follows˜ DR X Y = DR X Y − Tr( DR )( X ) n Y. Two Riccati connections DR and ˆ DR on M determine the same re-duced Riccati connection if and only if there is a 1-form γ on M suchthat ˆ DR X Y − DR X Y = γ ( X ) Y .We introduce the notion of curvature of a Riccati connection follo-wing ideas similar to those of Kato [8].Using the same notation as in (1), we define W α := dθ α + θ α ∧ θ α . (2)The matrices of 2-forms W α is called the curvature of the Riccaticonnection DR . Using equation (1) we can verify that W α = g αβ W β g − αβ ,i.e., W ∈ H ( M, Λ T ∗ M ⊗ End(
T M )).Let t be an indeterminate and A an n × n matrix. Define the ele-mentary symmetric polynomials C k bydet( tI + A ) = n X k =0 C n − k ( A ) t k We put R k ( M ) = C k ( i π W α ), for k = 0 , , . . . , n . Theorem 2.3.
Let M be a complex manifold of dimension n ≥ . If M admits a Riccati connection, then c k ( M ) = k X j =0 (cid:18) n − jk − j (cid:19) R j ( M ) (cid:18) c ( M ) n (cid:19) k − j , where c k ( M ) is the k -th Chern form.Proof. We know that there exists a Riccati connection DR , for whichequation (1) holds. Let ∇ be a (smooth) connection on det( T M ).That is, there exist (1,0)-forms η α on U α satisfying η β − η α = Tr( dg αβ g − αβ ) . (3)Note that the connection ∇ has curvature form K α ∇ = dη α and c ( M ) = c (det T M ) = i π dη α . FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 5
Define a matrix-valued smooth (1,0)-form Θ α on U α byΘ α = θ α + 1 n η α , Using equations (1) and (3) we haveΘ α = g αβ Θ β g − αβ − dg αβ g − αβ This shows that { Θ α } α is a (smooth) connection ∇ on T M . Let K α ∇ be the form curvature of ∇ . We can see that K α ∇ = W α + 1 n K α ∇ . Set A = i π K α ∇ , B = i π W α and η = i π K α ∇ . Thendet( tI + A ) = n X j =0 C n − j ( A ) t j = n X j =0 C n − j ( M ) t j , The other side we can rewritedet( tI + A ) = det( tI + B + 1 n ηI ) = det (cid:18) ( t + 1 n η ) I + B (cid:19) , = n X l =0 C n − l ( B )( t + 1 n η ) l = n X l =0 R n − l ( M )( t + 1 n c ( M )) l , = n X l =0 R n − l ( M ) l X s =0 (cid:18) ls (cid:19) t s (cid:18) c ( M ) n (cid:19) l − s , = n X j =0 n X l = j (cid:18) lj (cid:19) R n − l ( M ) (cid:18) c ( M ) n (cid:19) l − j ! t j . Comparing the terms of t j of these two equalities, we get C n − j = n X l = j (cid:18) lj (cid:19) R n − l ( M ) (cid:18) c ( M ) n (cid:19) l − j , j = 0 , . . . , n Exchanging the indices n − j and k we complete the proof of the theo-rem. (cid:3) Using Theorem 2.3 we have R ( M ) = 1, R ( M ) = 0, R ( M ) = c ( M ) − ( n − n c ( M ) , . . . Thus we obtain the following corollary.
RUBEN LIZARBE
Corollary 2.4.
The R k ( M ) forms are d-closed. The de Rham coho-mology classes of the R k ( M ) forms are real cohomology classes andindependent of the choice of Riccati connections. Proposition 2.5.
Let M be a compact complex manifold of dimension n ≥ admitting a Riccati connection. Then R k ( M ) = 0 , for k ≥ n/ .If M is a complex surface, then c ( M ) = c ( M ) . If M is K¨ahler, then the classes R k ( M ) , k ≥ , are zero and c k ( M ) = n − k (cid:18) nk (cid:19) c k ( M ) . Proof.
Since R k ( M ) is a holomorphic 2 k -form, if 2 k > n then R k ( M ) =0. Since every c k ( M ) is represented by a real ( k, k )-form, using inductionwe can show that R k ( M ) is represented by a real ( k, k )-form.For k = 2 n , we put γ = R k ( M ) is represented by a real ( k, k )-form η .Since η is real and cohomologous to γ , it is cohomologous to γ . Hence γ and γ are cohomologous to each other. Then Z M γ ∧ γ = Z M γ ∧ γ = 0 . Thus, we get γ = 0.If M is K¨ahler, then R k ( M ) = 0, since R k ( M ) is represented by areal ( k, k )-form. (cid:3) As an immediate consequence of the proposition above and [5, (2.2)Corollary], we have:
Corollary 2.6.
A compact K¨ahler manifold M with c ( M ) = 0 admitsa Riccati connection if and only if it is covered by a complex torus. From now on, unless stated otherwise, M will denote an arbitrarycomplex surface (i.e., not necessarily compact). FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 7 Riccati distributions
Consider the total space S = P ( T M ) of the bundle π : P ( T M ) → M .The three dimensional variety S is called the contact variety .For each point q = ( p, [ v ]) ∈ S , i.e. p ∈ M and v ∈ T p M , one has theplane D q := ( dπ ( q )) − ( C v ). We obtain in this way a two dimensionaldistribution D on S , namely the so called contact distribution .A codimension one holomorphic regular distribution H on S is calleda Riccati distribution if every fibre of π is transverse of H .When S is the total space of the trivial bundle π : P × U → U ((1 : z ) , ( x, y )) ( x, y ) , over the set U with coordinates ( x, y ), we know that D is given by dy − zdx and the contact structure dy = zdx , (we have normalized z ),and hence the Riccati distribution H is given by a 1-form of the type: ω = dz + γ + δz + ηz , (4)with γ , δ , and η dz = − γ − δz − ηz is calledthe Riccati equation .We write θ = − δ − ηγ δ ! . This matrix of 1-form represents a reduced Riccati connection DR . Infact, taking two local systems of coordinates ( ϕ α , U α ) and ( ϕ β , U β ) on M with U α ∩ U β = ∅ and letting ϕ αβ = ϕ α ◦ ϕ − β denote the correspondingchange of coordinates, we have ω α = h αβ ω β , h αβ ∈ O ∗ ( π − U α ∩ π − U β ) , (5)and H | π − Uα = { ω α = 0 } . Using equation (5) we get θ α = g αβ θ β g − αβ − dg αβ g − αβ + 12 Tr( dg αβ g − αβ ) I. (6)Furthermore, the following conditions are equivalent • DR has zero curvature: θ ∧ θ + dθ = 0; • H is integrable (Frobenius): ω ∧ dω = 0. RUBEN LIZARBE
Therefore, we obtain the following.
Proposition 3.1.
We have a one-to-one correspondence between re-duced Riccati connections on M and Riccati distributions on P ( T M ) .Furthermore, this correspondence induces a one-to-one correspon-dence between reduced Riccati connections with zero curvature on M and Riccati foliations on P ( T M ) , i.e. Frobenius integrable Riccati dis-tributions.In particular, finding Riccati foliations on P ( T M ) is equivalent tofinding × holomorphic matrices of 1-forms θ α on U α that satisfy thestructure equations dθ α + θ α ∧ θ α = 0 , (7)Tr( θ α ) = 0 , (8) and whose changes of coordinates satisfy (6). Curvature of a Riccati distribution.
Using the same notationas in (4), we write γδη = γ δ η dx + γ δ η dy, γ i , δ i , η i ∈ O ( U ) . We define the following 1-form κ = ( δ − γ ) dx + ( η − δ dy, which we call the connection form . Now, we will see that κ determinesthe so-called holomorphic connection on det( T M ). Proposition 3.2. If (( x, y ) , U α ) , ((˜ x, ˜ y ) , U β ) are two local systems ofcoordinates on M , with U α ∩ U β = ∅ , then κ α − κ β = 12 Tr( dg αβ g − αβ ) = 12 d (log(det g αβ )) . Proof.
We denote by κ = κ α = κ dx + κ dy , ˜ κ = κ β = ˜ κ dx + ˜ κ dy and g = g αβ = g g g g ! . Let DR be the Riccati connection inducedby H , and we define T ( X, Y ) = DR X Y − DR Y X − [ X, Y ]. We see that T ( ∂ x , ∂ y ) = − κ ∂ x + κ ∂ y . FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 9
We can verify the following properties: T is C -bilinear and T ( f X, gY ) = f gT ( X, Y ) + 12 ( f Y ( g ) X − gX ( f ) Y ) , where f, g ∈ O M , X, Y ∈ T M . Using these properties and the factthat ∂ ˜ x = g ∂ x + g ∂ y and ∂ ˜ y = g ∂ x + g ∂ y , we get T ( ∂ ˜ x , ∂ ˜ y ) = ( − (det g ) κ + 12 h ) ∂ x + ((det g ) κ + 12 h ) ∂ y , (9)where, h = g ( ∂ x ( g ) + ∂ y ( g )) − g ( ∂ x ( g ) + ∂ y ( g )) ,h = g ( ∂ x ( g ) + ∂ y ( g )) − g ( ∂ x ( g ) + ∂ y ( g )) . On the other hand, we have T ( ∂ ˜ x , ∂ ˜ y ) = ( − ˜ κ g + ˜ κ g ) ∂ x + ( − ˜ κ g + ˜ κ g ) ∂ y . (10)Comparing (9) and (10), we get(det g ) κ = (det g )˜ κ + 12 (( h g − h g ) d ˜ x + ( h g − h g ) d ˜ y ) . It is not difficult to verify that(det g ) Tr( dg.g − ) = ( h g − h g ) d ˜ x + ( h g − h g ) d ˜ y. This completes the proof of the proposition. (cid:3)
We define the curvature of H as K ( H ) = dκ ∈ Ω ( M ) , We say that H is parallelizable if K ( H ) = 0. Theorem 3.3. If M is compact and H is a Riccati distribution on P ( T M ) , then H is parallelizable.Proof. By Proposition 3.2, the 1-forms { κ α } α define a holomorphicconnection on det( T M ). The curvature of this connection is 2 dκ , whichrepresents c (det T M ) = c ( M ). We conclude by following the sameideas as Proposition 2.5. (cid:3) Riccati distributions and affine connections.
An affine con-nection on M is a (linear) holomorphic connection on the tangent bun-dle T M , i.e., a C -bilinear map ∇ : T M × T M → T M satisfyingthe Leibnitz rule ∇ X ( f · Z ) = f · ∇ X ( Z ) + df ( X ) Z and ∇ fX ( Z ) = f ∇ X ( Z ), for any holomorphic function f and any vector fields X, Z .The connection ∇ is torsion free when the torsion vanishes, that is ∇ X Z − ∇ X Z − [ X, Z ] = 0, for all vector fields
X, Z , and the curvatureof ∇ is denoted by K ∇ = ∇ · ∇ . The connection ∇ is flat when thecurvature vanishes, that is K ∇ = ∇ · ∇ = 0.We describe a map from the set of connections to into the set ofdistributions as follows: In coordinates ( x, y ) ∈ U ⊂ C , a trivializationof T U is given by the basis ( ∂ x , ∂ y ) and the affine connection is givenby ∇ ( Z ) = d ( Z ) + θZ, θ = θ θ θ θ ! , where Z = z ∂ x + z ∂ y and θ ij ∈ Ω ( U ). On the projectivized bundle P ( T U ), with trivializing coordinate z = z /z , the equation ∇ = 0induces a Riccati distribution H ∇ that is locally given by ω = dz + θ + ( θ − θ ) z − θ z . (11) Theorem 3.4.
This map induces a one-to-one correspondence betweentorsion free affine connections on M and Riccati distributions on P ( T M ) .Furthermore, there are one-to-one correspondences between:(1) Torsion free affine connections ∇ on M with zero curvature ofthe trace of connection Tr( ∇ ) , i.e., K Tr( ∇ ) = 0 , and paralleliza-ble distributions on P ( T M ) .(2) Torsion free affine connections ∇ on M with K ∇ = K Tr( ∇ ) I and Riccati foliations on P ( T M ) .(3) Affine structures on M and parallelizable Riccati foliations on P ( T M ) .Proof. To verify that we have a bijection we will describe the inversemapping as follows: Let H be a Riccati distribution on P ( T M ). ByProposition 3.1 we have a reduced Riccati connection DR induced by H that verifies θ α = g αβ θ β g − αβ − dg αβ g − αβ + Tr( dg αβ g − αβ ) I , where θ α FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 11 is the matrix of 1-forms of DR in U α . By Proposition 3.2 we have Tr( dg αβ g − αβ ) = κ α − κ β , where κ α ∈ Ω ( U α ) is the connection form.Define ˜ θ α = θ α − κ α I , so that ˜ θ α = g αβ ˜ θ β g − αβ − dg αβ g − αβ . Then the1-forms { ˜ θ α } α define an affine connection ∇ H on M .To verify items 1., 2. and 3. it is sufficient to see that K Tr( ∇ ) = − K ( H ∇ ) and K ∇ = W − K ( H ∇ ) I , where W is the curvature of theRiccati connection induced by H ∇ . The theorem is proved. (cid:3) Corollary 3.5.
The following assertions are equivalent. • M admits an affine connection, • M admits a Riccati connection, • P ( T M ) admits a Riccati distribution. It is worth mentioning that a Riccati connection on a surface inducesa (normal) projective connection. Using this fact together with theclassification of normal projective connections [10, 11] and Proposition2.5 we can obtain another proof of Corollary 3.5 in the case of a compactcomplex surface.
Corollary 3.6. If M is compact, then we have a one-to-one corres-pondence between Riccati foliations on P ( T M ) and affine structures on M .Proof. This follows from Theorem 3.3 and Theorem 3.4. (cid:3)
Monodromy of a Riccati foliation.
The monodromy represen-tation of the Riccati foliation H on S = P ( T M ) is the representation ρ H : π ( M ) → Aut( P )defined by lifting paths on M to the leaves of H . The image of ρ H in Aut( P ) is, by definition, the monodromy group of H , denoted byMon( H ). Let’s see some particular cases.3.3.1. Pencils of foliations and Riccati foliations.
A regular pencil offoliations P on U is a one-parameter family of foliations {F t } t ∈ P definedby F t = [ ω t = 0] for a pencil of 1-forms { ω t = ω + tω ∞ } t ∈ P with ω , ω ∞ ∈ Ω ( U ) and ω ∧ ω ∞ = 0 on U . The pencil of 1-forms defining {F t } t ∈ P is unique up to multiplication by a non vanishing function:˜ ω t = f ω t for all t ∈ P and f ∈ O ( U ). In fact, the parametrization by t ∈ P is not intrinsic; we will say that {F t } t ∈ P and { ˜ F t } t ∈ P definethe same pencil on U if there is a M¨obius transformation ϕ ∈ Aut( P )such that ˜ F t = F ϕ ( t ) for all t ∈ P .The graphs S t of the foliations F t are disjoint sections (since foliationsare pairwise transversal) and form a codimension one foliation H on P ( T U ) transversal to the projection π : P ( T U ) → U . The foliation H is a Riccati foliation, i.e. a Frobenius integrable Riccati distribution: H : [ ω = 0] , ω = dz + γ + δz + ηz , ω ∧ dω = 0 . In local coordinates ( x, y ) such that F and F ∞ are defined by dx = 0and dy = 0 respectively, we can assume the pencil is generated by ω = dx and ω ∞ = u ( x, y ) dy (we have normalized ω ) with u (0 , = 0.Then the graph of each foliation F t is given by the section S t = { z = − tu ( x,y ) } ⊂ P ( T U ). These sections are the leaves of the Riccati foliation H : [ dz + duu z = 0].We know that the curvature K ( P ) of a regular pencil P = {F t } t ∈ P is a 2-form (see [13]). For instance, if P is in normal form P = dx + tu ( x, y ) dy , then the curvature is given by K ( P ) = − (ln u ) xy dx ∧ dy. We recall that a pencil P is flat when its curvature is zero, that is K ( P ) = 0.On the other hand, P induces a Riccati foliation H : [ dz + duu z =0] on P ( T U ). So we have κ = u x u dx − u y u dy . Thus the curvature of K ( H ) = K ( P ) = dκ , this means that the definition of the curvature ofa Riccati distribution extends the definition of the curvature of a pencilof foliations. So, we have the following proposition. Proposition 3.7.
The following data are equivalent: • a regular pencil of foliations {F t } t ∈ P on M , • a Riccati foliation H on P ( T M ) with trivial monodromy, i.e., Mon( H ) = { id } .Furthermore, if M is compact and P is a regular pencil on M , then P is flat. The following lemma exhibits the normal form of a flat pencil thatrepresents an affine structure.
FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 13
Lemma 3.8. ( [13, Lemma 2.1.4] ). Let P be a flat regular pencil definedin a neighborhood of the origin ∈ C . Then, there is a change of localcoordinates ϕ ( x, y ) sending P to the pencil defined by dx + tdy , t ∈ P . Webs and pencils of foliations.
Let W = F ⊠ F ⊠ F ⊠ F bea regular 4-web on ( C , F i = [ X i = ∂x + e i ( x, y ) ∂y ] = [ η i = e i dx − dy ] , i = 1 , , , . The cross-ratio ( F , F , F , F ) := ( e − e )( e − e )( e − e )( e − e )is a holomorphic function on ( C ,
0) intrinsically defined by W . Then,we have: Proposition 3.9. If W = F ⊠ F ⊠ F ∞ is a regular 3-web on ( C , ,then there is a unique pencil {F t } t ∈ P that contains F , F and F ∞ asits elements. More precisely, F t is the only foliation such that ( F t , F , F , F ∞ ) = t. Conversely, any Riccati foliation comes from a 3-web: it suffices tochoose 3 elements of a pencil. In particular, any 4 elements of a pencil {F t } t ∈ P have constant cross-ratio.In general, we can define a cross-ratio for a k -web, k >
3, on acompact complex surface M . Indeed, in a finite cover of M any fourfoliations define a constant cross-ratio, and therefore they define a cross-ratio on M , thus giving a Riccati foliation on M .The proposition below follows from the previous considerations. Proposition 3.10. If M is compact and W is a regular k -web on M , k ≥ , then there is a Riccati foliation H on P ( T M ) induced by W suchthat Mon( H ) ⊂ Sym(1 , . . . , k ) . See [16] and [6] for more details.
Calculations of the monodromy of a Riccati foliation.
Whatwe will do is the following: calculate the Riccati connections on M ofzero curvature and zero trace, and hence the space of Riccati foliationson P ( T M ). If M has at least one affine structure, the Riccati con-nections are the elements of H ( M, V ), where V = End ( T M ) ⊗ T ∗ M (vector bundle, Higgs field)=:Higgs bundle. This follows from (6) since g αβ , can be chosen constant so that dg αβ = 0. Now pull back V viathe universal cover π : ˜ M → M of M , and compute the holomorphicsections of π ∗ ( V ); those invariant by π ( M ) will be the holomorphicsections of V .The universal cover ˜ M of each of these surfaces is a subdomain of C and the covering transformations are affine so that the standardflat holomorphic linear connection on C restricted to ˜ M ⊂ C canbe ”pulled down” to M . In this case, M has a natural or usual affineconnection whose corresponding affine coordinates are those of ˜ M ⊂ C defined locally on M . Since π ∗ ( V ) = ˜ M × C , the Riccati connectionson ˜ M are of the form θ = A ( x, y ) dx + A ( x, y ) dy where A k ( x, y ) isa holomorphic 2 × x, y ) are the global coordinates on˜ M ⊂ C . The Riccati connections on M are the 2-forms θ that areinvariant by π ( M ). Using the structure equations (7) and (8), thezero curvature condition becomes0 = ∂A ∂x − ∂A ∂y + [ A , A ] , and the relation Tr( θ ) = 0 turns into Tr( A ) = Tr( A ) = 0.3.4.1. Complex torus surfaces.
In this case M is the quotient of C bysome lattice Γ = ⊕ i =1 ( k i , l i ) Z . Then θ = A dx + A dy , where A , A are complex 2 × A A = A A and Tr( A ) = Tr( A ) = 0. (12)The pairs of matrices ( A , A ) are divided into conjugacy classes:( A , A ) is said to be conjugate to ( B , B ) iff there exists C ∈ GL(2 , C )such that CB C − = A and CB C − = A . Clearly, if we can findone member ( B , B ) of a conjugacy class satisfying (12) then the othermembers ( A , A ) are easily obtained. So we may assume that B , B FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 15 are in Jordan normal form, and in this case what we obtain is given inthe listed below:Type B = B = Mon = hi − a a ! − b b ! z exp( ak i + bl i )2 ! c ! z − ( k i + cl i )3 ! ! z − l i a, b, c ∈ C , i = 1 , , , z ) is also included.We write C = e fg h ! . Then, in general, for A and A we have: Type Riccati foliation induced by θ Mon = hi dz + C ( g − ez )( − h + f z )( adx + bdy ) z exp( ak i + bl i )2 dz + C ( g − ez )( − g + ez )( dx + cdy ) ( z − gf ( k i + cl i ) , if e = 0 ,z − e ( k i + cl i ) , if e = 0 . dz + C ( g − ez )( − g + ez ) dy ( z − gf l i , if e = 0 ,z − el i , if e = 0 . where a, b, c ∈ C and i = 1 , , , Lemma 3.11.
Let H be a Riccati foliation on P ( T ( C / Γ)) .(1)
Mon( H ) is trivial if and only if θ = 0 , i.e., a = b = 0 .(2) Mon( H ) is not trivial if and only if there exist non periodicelements in Mon( H ) . Lemma 3.12.
Let M be a compact complex surface and H be a Riccatifoliation on P ( T M ) . If M is covered by a complex torus surface and Mon( H ) is finite, then M is a complex torus surface and Mon( H ) istrivial. Primary Kodaira surfaces.
A primary Kodaira surface K is thequotient of C by some group G generated by g ( x, y ) = ( x, y + 1) , g ( x, y ) = ( x + 1 , ax + y ) ,g ( x, y ) = ( x, y + τ ) , g ( x, y ) = ( x + τ , bx + y ) , where τ , τ are generators of the fundamental domain of the modulargroup and a, b ∈ C satisfy aτ − b = mτ for some positive integer m .See more details in [19].A Riccati connection on K has the form θ = e c h ! dx + e − h ! dy, for some constants e , c , h . Therefore the Riccati foliations on P ( T K )are of the form dz + cdx , c ∈ C . So, the monodromy group Mon isgenerated by f ( z ) = z − a − c , and f ( z ) = z − b − cτ . From this wecan conclude that there are no regular pencils P and regular k -webs W on a primary Kodaira surface, for k ≥ Hopf surfaces.
A compact complex surface M whose universalcovering space is biholomorphic to C − { } is called a Hopf surface.
1. Primary Hopf surfaces . We consider first the primary Hopf sur-faces, which are quotients of C − { } by the infinite cyclic group gen-erated by an automorphism g of C − { } . According to [12, Part II], § g has the form: g ( x, y ) = ( ax + λy m , by ) , for some positive integer m and some complex numbers a , b , λ with0 < | a | ≤ | b | < a − b m ) λ = 0. FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 17
By using [5, (7.5) Theorem] we have:Condition Riccati foliation induced by θ Mon = hi λ = 0, m = 1 dz zb + λa λ = 0, a = b dz zba λ = 0, a = b dz − cz dy zb where c ∈ C .From the above list we deduce directly the following. Lemma 3.13.
Let M be a primary Hopf surface and H be a Riccatifoliation on P ( T M ) .(1) Mon( H ) is trivial if and only if θ = 0 ,i.e., λ = 0 and a = b .(2) H is induced by a regular k -web, k ≥ , if and only if λ = 0 and a = ξb , where ξ is a primitive j -th root of 1, with ≤ j ≤ k .2. Secondary Hopf surfaces . A secondary Hopf surface M is thequotient of C −{ } by the free action of a group Γ containing a central,finite index subgroup generated by an automorphism g of the abovetype. The primary Hopf surface N = C − { } /g is a finite ´etale coverof M and the corresponding finite subgroup is generated by e ( x, y ) =( ǫ x, ǫ y ), where ǫ and ǫ are primitive l -th roots of unity and ( ǫ − ǫ m ) λ = 0. Then we have:Condition Riccati foliation induced by θ Mon = hi λ = 0, m = 1 dz zǫ ǫ λ = 0, ǫ = ǫ dz zǫ ǫ λ = 0, ǫ = ǫ dz − cz dy zǫ where c ∈ C .Hopf surfaces with fundamental group isomorphic to Z ⊕ Z /l Z , where l ≥ g ( x, y ) = ( ax, ay ) with 0 < | a | <
1, and e ( x, y ) = ( ǫx, ǫy ) where ǫ isa primitive l -th root of 1. We denote them by H a,l . From these twoprevious lists we conclude the following. Lemma 3.14.
Let M be a Hopf surface and H be a Riccati foliationon P ( T M ) .(1) Mon( H ) is trivial if and only if M = H a,l .(2) H is induced by a regular k -web, k ≥ , if and only if H a, is afinite cover of M .(3) If Mon( H ) is finite, then Mon( H ) is cyclic. Inoue surfaces.
Inoue surfaces are compact complex surfaces oftype
V II , see [1] for more details. In [4] Inoue shows that these surfacesare obtained as quotients of H × C by a group Γ of affine transformationsof C preserving the open set H ( H being the Poincar´e upper half-plane). In particular, each Inoue surface inherits an affine structureinduced by the canonical affine structure of C , which is unique by[9, Lemma 4.3]. Up to a double unramified cover, Inoue surfaces areobtained by one of the following two procedures [4].1. Surfaces S M . Consider a matrix M ∈ SL(3 , Z ) with eigenvalues α, β, β such that α > β = β . Choose a real eigenvector ( a , a , a )associated with the eigenvalue α and an eigenvector ( b , b , b ) associ-ated with the eigenvalue β . Consider also the group Γ of (affines)transformations of C generated by: γ ( x, y ) = ( αx, βy ) ,γ i ( x, y ) = ( x + a i , y + b i ) , with i = 1 , , C preserves H × C and the quotient is the compactcomplex surface S M .In this case, there is only one Riccati foliation on P ( T S M ) and it hasthe form dz . Thus, the monodromy group is generated by f ( z ) = βα z .2. Surfaces S + N,p,q,r,t . Let N = ( n ij ) ∈ SL(2 , Z ) be a diagonalizablematrix on R with eigenvalues α > α − and eigenvectors ( a , a )and ( b , b ) respectively. Choose r ∈ Z ∗ , p , q ∈ Z , t ∈ C and realsolutions c , c of the equation( c , c ) = ( c , c ) N t + ( e , e ) + 1 r ( b a − b a )( p, q ) , FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 19 where e i = n i ( n i − a b + n i ( n i − a b + n i n i b a and N t denotes the transpose of N .In this case Γ is generated by the transformations γ ( x, y ) = ( αx, y + t ) ,γ i ( x, y ) = ( x + a i , y + b i x + c i ) , ( i = 1 , ,γ ( x, y ) = ( x, y + r − ( b a − b a )) . This group is discrete and acts properly and discontinuously on H × C and the quotient we obtain is the compact complex surface S + N,p,q,r,t .Also in this case, there is only one Riccati foliation on P ( T S + N,p,q,r,t )which has the form dz . Moreover, the monodromy group is generatedby f ( z ) = zα and f i ( z ) = z b i z , i = 1 , P nor regular k -webs W on a Inoue surface, for k ≥ Elliptic surfaces over a Riemann surface of genus g ≥ , withodd first Betti number. The existence of a holomorphic affine structureon a elliptic surface over a Riemann surface of genus g ≥
2, of odd firstBetti number, is a result due to Maehara [14]. The global geometry ofaffine holomorphic structures and holomorphic affine connections wi-thout torsion on these surfaces are studied in [9] and [3] respectively.Up to a finite covering and a finite quotient, this surface M is con-structed as follows. Let Γ be a discrete torsion-free subgroup of PSL(2 , R )such that Σ = Γ \ H , where H denotes the Poincar´e half-plane. If wethink of Γ as a subgroup of SL(2 , R ), then the action of an element γ = a bc d ! ∈ SL(2 , R ) of Γ on C × H is given by the followingformula: γ ( x, y ) = ( x + log ( cy + d ) , γy ), for all ( x, y ) ∈ C × H , wherelog denotes a branch of the logarithm function and the action of γ on H comes from the canonical action of SL(2 , R ) on H . See [9] for moredetails.The quotient of C × H by the action of Γ is the complex surfacecompact M . A torsion-free flat connection on M has the form θ = dx dy dx ! + f ( y ) dy h ( y ) dy ! , where h ( γy ) = h ( y )( cy + d ) and f ( γy ) = f ( y )( cy + d ) + ch ( y )( cy + d ) ,for all γ = a bc d ! ∈ Γ. Therefore the Riccati foliations on P ( T M )are of the form ω = dz + dy + h ( y ) yz − f ( y ) dyz . In this case it wasnot possible to calculate the monodromy.It follows from a result by Puchuri [17] that there are no regularpencils, and therefore there are no regular k -webs on M .4. Applications
The following theorem was proved in [17] by using foliation tech-niques.
Theorem 4.1. If M is compact and P is a regular pencil on M , then M is either a complex torus M = C / Γ or a Hopf surface M = H a,l .Moreover, P is unique pencil in both cases, and it is generated by { dx + tdy } t ∈ P on C (in the first case) or on C \ { (0 , } (in thesecond case). When M is not elliptic surface over a Riemann surface of genus g ≥ Proof.
Let H be the Riccati foliation on P ( T M ) induced by P . ByProposition 3.7 we have that M on ( H ) is trivial. Using the calculationsof the monodromy from Subsection 3.4, we conclude the proof. (cid:3) Theorem 4.2. If M is compact and W is a regular k -web on M , k ≥ ,then (up to a finite unramified cover) M is either a complex torus C / Γ or a Hopf surface H a, as above, and W ⊂ P , where P is the uniquepencil in both cases.Proof. By Proposition 3.10, there is a Riccati foliation H on P ( T M )induced by W with finite monodromy. Up to a finite unramified coverwe can suppose that the monodromy is trivial and W = F ⊠ · · · ⊠ F k is a (completely decomposable) regular k -web on M . Now we applyTheorem 4.1 to complete the proof. (cid:3) FFINE CONNECTIONS AND RICCATI DISTRIBUTIONS 21
Theorem 4.3. If M is compact and H is a Riccati foliation on P ( T M ) with finite monodromy, then M is either a complex torus C / Γ and Mon( H ) is trivial or a Hopf surface and Mon( H ) is cyclic.Proof. This follows readily from the calculations of the monodromyfrom the previous subsection. (cid:3)
Finally, we point out that the classification of regular 2-webs followsdirectly from [2, Theorem C].
Acknowledgements
The author wishes to express his deepest gratitude to Frank Lorayfor lots of fruitful discussions about the content of this paper, and alsofor reading the drafts and suggesting improvements. The author alsowishes to thank Cesar Hilario for the suggestions and comments on themanuscript. The author acknowledge support from CAPES/COFECUB(Ma 932/19 ”Feuilletages holomorphes et int´eration avec la g´eom´etrie”/ process number 88887.356980/2019-00). The author is grateful tothe Institut de Recherche en Math´ematique de Rennes, IRMAR andthe Universit´e de Rennes 1 for their hospitality and support.
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