GENERALIZED TRANSVERSELY PROJECTIVE STRUCTURE ON A TRANSVERSELY HOLOMORPHIC FOLIATION

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GENERALIZED TRANSVERSELY PROJECTIVE STRUCTURE ON A TRANSVERSELY HOLOMORPHIC FOLIATION

INDRANIL BISWAS

Received 29 November 2000 and in revised form 4 June 2001

The results of Biswas (2000) are extended to the situation of transversely projective foli- ations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective spaceCPⁿ, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation is also associated with a transversely holomorphic section ofN^⊗−kfor eachk∈[3,n+1], whereNis the normal bundle to the foliation. These transversely holomorphic sections are also flat with respect to the Bott partial connection.

2000 Mathematics Subject Classification: 37F75, 32H02, 51A45.

1. Introduction. A projective structure on a Riemann surfaceXis defined by giving a covering ofXby holomorphic coordinate charts such that all the transition func- tions are restrictions of Möbius transformations. It is well known that the notion of a projective structure can be extended to the situation of foliations (cf. [10]). To define this generalization, letᏲbe a foliation of codimension two on a real manifoldM. Let {Ui}i∈Ibe an open covering ofM, and letφi:Ui→Cbe submersions onto the image such that the fibers ofφiare leaves forᏲ. A transversely projective structure onᏲ is defined by imposing the condition that, for everyi,j∈I, there is a commutative diagram

Ui∩Uj φ_i

Ui∩Uj φ_j

φi

Ui∩Uj

^fi,j

φj

Ui∩Uj

(1.1)

such thatfi,jis a restriction of some Möbius transformation [10].

A holomorphic immersionγ:X→CPⁿof a Riemann surfaceXis called everywhere locally nondegenerate if for everyx∈X, the order of contact of the imageγ(U)at γ(x), where U is a neighborhood ofx in X, with any hyperplane in CPⁿ passing throughγ(x)is at mostn−1 (see [3,9]). Two such immersions are called equivalent if they differ by an automorphism ofCPⁿ. ACPⁿ-structure onX is an equivalence class of an everywhere locally nondegenerate equivariant map of the universal cover ofXintoCPⁿ. ACP¹-structure onXis the same as a projective structure onX.

Iff:X→CPⁿis a holomorphic map such that the image off is not contained in any hyperplane ofCPⁿ, then there is a finite subsetS⊂Xsuch that the restriction of

f to the complementX\Sdefines aCPⁿ-structure onX\S. Any Riemann surface has manyCPⁿ-structures. In [3], it has been shown that the space ofCPⁿ-structures on X, wheren≥2, is canonically identified with the Cartesian product of the space of all projective structures onXwith the direct sum_n+1

i=3H⁰(X,KX^⊗i).

The notion of aCPⁿ-structure can be extended to the situation of foliations which will be called atransversely CPⁿ-structure; seeDefinition 2.3for the definition of a transverselyCPⁿ-structure.

LetᏲbe a transversely holomorphic foliation of complex codimension one. So the normal bundleNis a transversely holomorphic line bundle. The normal bundleNis equipped with the Bott partial connection obtained from the Lie bracket operation of vector fields. The transversely holomorphic structure ofNis compatible with the Bott partial connection.

We prove that, giving a transverselyCPⁿ-structure onᏲ is equivalent to giving a transversely projective structure onᏲtogether with a transversely holomorphic sec- tionωkofN^⊗−k, for eachk∈[3,n+1], such thatωkis flat with respect to the Bott partial connection (seeTheorem 2.4). In particular, setting allωkto be zero we con- clude that, for any transverselyCPⁿ-structure onᏲthere is a canonically associated transversely projective structure onᏲ. When the foliation is trivial, that is,Ᏺ=0, then Theorem 2.4is the main result of [3] (see [3, Theorem 5.5]).

It is not easy to directly construct a transverselyCPⁿ-structure on a holomorphic foliation. In fact, when the foliation is trivial, namely we have a Riemann surfaceX, it is not easy to construct a map of the universal cover ofXtoCPⁿ, which is everywhere locally nondegenerate. However, usingTheorem 2.4we can indirectly construct many examples of transverselyCPⁿ-structures, just as using [3, Theorem 5.5], we can indi- rectly construct examples of everywhere locally nondegenerate maps of the universal cover of a Riemann surface toCPⁿ.

2. Transversely projective foliations defined by maps to a projective space. Let Mbe a connected smooth real manifold of dimensiond+2. LetᏲbe aC^∞-subbundle of rankdof the tangent bundleT M.

Definition2.1. A transversely holomorphic structure onᏲis defined by giving the following data (see [5]):

(1) a covering ofM by open subsets Ui, whereiruns over an index setI. So we have

i∈IUi=M;

(2) for eachi∈I, a submersionφiofUito an open subsetDiofC. The restriction Ᏺ|U_i is the kernel of the differential mapdφi:T Ui→φ^∗_iT Di;

(3) for every pairi,j∈I, there is a commutative diagram of maps

Ui∩Uj φ_i

Id Ui∩Uj

φ_j

φi

Ui∩Uj ^fi,j φj

Ui∩Uj ,

(2.1)

wherefi,j is a holomorphic map.

...

Two such data{Ui,φi}i∈Iand{Ui,φi}i∈Jare calledequivalentif their union, namely Ui,φi

i∈I∪J, (2.2)

also satisfies the above conditions. Atransversely holomorphic structure onᏲ will mean an equivalence class of data of the above type satisfying the three conditions.

Next we recall the definition of a transversely projective foliation.

Definition2.2. A transversely projective structure onᏲ is defined by giving a data{Ui,φi}i∈Iexactly as inDefinition 2.1, but satisfying the extra condition (apart from the three conditions) that the holomorphic mapsfi,j in condition (3) are of the formz(az+b)/(cz+d), wherea,b,c,d∈Care constant scalars andad−bc=1, that is, eachfi,jis the restriction of some Möbius transformation; the scalarsa,b,c,d may depend on the indexi. As before, two such data{Ui,φi}i∈I and{Ui,φi}i∈J are calledequivalentif their union{Ui,φi}i∈I∪Jis also a data for a transversely projective structure. Atransversely projectivestructure onᏲwill mean an equivalence class of such data.

Clearly, a transversely projective structure onᏲdefines a transversely holomorphic structure onᏲ. If ¯Ᏺis a transversely holomorphic structure onᏲ, then a transversely projective structure on ¯Ᏺ is a transversely projective structure on Ᏺsuch that, the transversely holomorphic structure defined by it coincides with ¯Ᏺ^.

We now recall the notion of a locally nondegenerate immersion of a Riemann surface into a projective space (see [3,9]).

LetXbe a Riemann surface, that is, a complex manifold of complex dimension one.

LetCPⁿ, n≥1, denote then-dimensional projective space consisting of all lines in Cⁿ⁺¹. A holomorphic immersion

γ:X →CPⁿ (2.3)

is calledeverywhere locally nondegenerateif for everyx∈X, the order of contact of the imageγ(U), whereUis a neighborhood ofxinX, atγ(x)with any hyperplane inCPⁿpassing throughγ(x)is at mostn−1. We need to consider a neighborhood in the definition sinceγmay not be injective.

An alternative description of the above nondegeneracy condition following [9] is given below.

Let

0 →S →V ^q→Q→0 (2.4)

be the universal exact sequence overCPⁿ. The vector bundleV is the trivial vector bundle withCⁿ⁺¹as fiber andSis the tautological line bundleᏻCPⁿ(−1). Consider the differential

dγ:TX →γ^∗TCPⁿ=γ^∗Hom(S,Q) (2.5) of the immersionγ; hereTX is the holomorphic tangent bundle ofX. Sinceγ is an immersion, the homomorphismdγis injective.

Now, the homomorphismdγgives a homomorphism

dγ:TX^∗⊗γ^∗S →γ^∗Q, (2.6)

whereT_X^∗is the holomorphic cotangent bundle ofX. LetS1denote the inverse image q⁻¹(image(dγ)), where the homomorphismqis defined in (2.4). The subbundleS1of γ^∗V defines a map

γ1:X →G(n+1,2) (2.7)

ofXinto the Grassmannian of two planes inCⁿ⁺¹.

Now assume thatγ¹is an immersion. Then repeating the above argument we get a map

γ2:X →G(n+1,3) (2.8)

ofXinto the Grassmannian of three planes inCⁿ⁺¹. More generally, inductively we have a map

γi:X →G(n+1,i+1), (2.9)

wherei∈[1,n−1], by assuming thatγi−1is an immersion. (See also [9, Section 1] for the details of the construction of the mapsγidescribed above.)

The condition that the mapγ, together with each mapγi, wherei∈[1,n−1], is an immersion, is equivalent to the condition that the map γ is everywhere locally nondegenerate.

Now, we extend the above notion of everywhere locally nondegenerate map to the context of foliations, which we call transverselyCPⁿ-structure.

Definition 2.3. A transversely CPⁿ-structure onᏲ is defined by giving a data {Ui,φi}i∈Iexactly as inDefinition 2.1satisfying conditions (1) and (2) and the follow- ing stronger version of (3): for everyi∈I, there is an everywhere locally nondegenerate map

γi:Di:=image φi

→CPⁿ (2.10)

such that, for every pairi,j∈I, there is a commutative diagram of maps Ui∩Uj

φ_i

Id Ui∩Uj

φ_j

φi

Ui∩Uj

γ_i

f_i,j

φj

Ui∩Uj

γ_j

CP^{n T} CPⁿ,

(2.11)

whereTis an automorphism ofCPⁿ, that is,T∈GL(n+1,C). As before, two such data {Ui,φi,γi}i∈Iand{Ui,φi,γi}i∈Jare calledequivalentif their union{Ui,φi,γi}i∈I∪Jis

...

also a data for a transverselyCPⁿ-structure. AtransverselyCPⁿ-structureonᏲwill mean an equivalence class of such data.

The above condition forces the mapfi,jto be holomorphic. So, a transverselyCPⁿ- structure onᏲdefines a transversely holomorphic structure onᏲ^{. If ¯}Ᏺis a transversely holomorphic structure onᏲ, then a transverselyCPⁿ-structure on ¯Ᏺis a transversely CPⁿ-structure onᏲsuch that the underlying transversely holomorphic structure co- incides with ¯Ᏺ^.

Note that, a transverselyCP¹-structure onᏲis by definition a transversely projec- tive structure onᏲ.

We fix a transversely holomorphic structure ¯Ᏺ^onᏲ^. The normal bundle

N:=T M

Ᏺ ^(2.12)

is a complex line bundle. Therefore, for every integerk∈Z, we have a complex line bundleN^⊗kobtained by taking thekth tensor power of the complex line bundle N. ByN^⊗−1we mean the dual line bundleN^∗.

Any such line bundleN^⊗khas a natural transversely holomorphic structure. This means that, there is a Dolbeault operator

¯∂N^⊗k:N^⊗k →N^∗⊗N^⊗k=N^⊗k−1 (2.13) satisfying the Leibniz identity. The operator ¯∂_N⊗kis simply the Dolbeault operator on the holomorphic tangent bundleT_C^⊗k of the complex lineCtransported toM using the projectionsφi. It may be noted that, the condition inDefinition 2.1(3) that every fi,jis holomorphic ensures that these locally defined operators patch compatibly to define the global differential operator ¯∂N^⊗k.

Also, the line bundleN, and hence anyN^⊗k, has the Bott partial connection (see [8]).

Recall that, the Lie bracket operation on the sheaf of sections of the tangent bundle T Mdefines the Bott partial connection

N →Ᏺ^∗⊗N (2.14)

along the foliation Ᏺ. The Jacobi identity for Lie bracket ensures that this partial connection is flat.

It is easy to see that both the complex structure ofNand the transversely holomor- phic structure ofNare compatible with respect to the Bott partial connection. In other words, both the complex vector space structure of the fibers ofNand the Dolbeault operator ¯∂Ndefined in (2.13) commute with the differential operator in (2.14) defining the Bott connection. Equivalently, parallel translation (for the Bott connection) along the leaves of the foliation ¯Ᏺof holomorphic sections ofNremain holomorphic. Also, parallel translations for the Bott connection commute with multiplication by√

−1 of the fibers ofN.

The Bott partial connection onNinduces a flat partial connection on anyN^⊗k. All the above compatibility properties of the Bott connection onNevidently remain valid for anyN^⊗k.

Letᐂ_Ᏺ¯(k)denote the space of all globally defined smooth sectionss of the com- plex line bundle N^⊗k such that s is transversely holomorphic for the transversely holomorphic foliation ¯Ᏺand it is flat with respect to the Bott partial connection for ¯Ᏺ^. Soᐂ_Ᏺ¯(k)is a complex vector space; it need not be of finite dimension. However, in the situation whereMis compact, it was proved by Duchamp and Kalka [4, Theorem 1.27, page 323], and also independently by Gómez-Mont [6, Theorem 1, page 169], that the dimension ofᐂ_Ᏺ¯(k)is finite.

Letᏼ(Ᏺ¯)denote the space of all equivalence classes of transversely projective struc- tures on the transversely holomorphic foliation ¯Ᏺ. Transversely projective structures were defined inDefinition 2.2and transversely projective structures on ¯Ᏺ ^{were de-} fined in the paragraph followingDefinition 2.2. The spaceᏼ(Ᏺ¯)may be empty.

The following theorem is the main result of this section.

Theorem2.4. There is a canonical bijective map from the space of all transversely CPⁿ-structures onᏲ¯and the Cartesian product

ᏼᏲ^¯× _n+1

k=3

ᐂ_Ᏺ(−k) . (2.15)

In particular, a transverselyCPⁿ-structure gives a transversely projective structure on Ᏺ¯by simply taking the zero section inᐂ¯_Ᏺ(−k)for allk∈[3,n+1].

The theorem will be proved after establishing a few lemmas. We start with the definition of jet bundles and differential operators.

LetEbe a holomorphic vector bundle on a Riemann surfaceX, and letnbe a positive integer. Thenth-orderjet bundleofE, denoted byJⁿ(E), is defined to be the following direct image onX:

Jⁿ(E):=p1∗

p2^∗E p^∗2E⊗ᏻX×X

−(n+1)∆

, (2.16)

wherepi:X×X→X,i=1,2, is the projection onto theith factor, and∆is the diagonal divisor onX×X. Therefore, for anyx∈X, the fiberJⁿ(E)xis the space of all sections ofEover thenth-order infinitesimal neighborhood ofx.

Let KX denote the holomorphic cotangent bundle ofX. There is a natural exact sequence

0→KX^⊗n⊗E →Jⁿ(E) →Jⁿ⁻¹(E) →0 (2.17) constructed using the obvious inclusion ofᏻX×X(−(n+1)∆)inᏻX×X(−n∆). The in- clusion mapKX^⊗n

E→Jⁿ(E)is constructed by using the homomorphism

K^⊗nX →Jⁿ ᏻX

, (2.18)

which is defined at anyx∈Xby sending(df )^⊗n, wherefis any holomorphic function withf (x)=0, to the jet of the functionfⁿ/n! atx.

The sheaf ofdifferential operatorsDiffⁿ_X(E,F)is defined to be Hom(Jⁿ(E),F). The homomorphism

σ: Diffⁿ_X(E,F) →Hom

K_X^⊗n⊗E,F

, (2.19)

...

obtained by restricting a homomorphism fromJⁿ(E)toF to the subsheafKX^⊗n

Ein (2.17), is known as thesymbol map.

Let X denote a simply connected open subset ofCP¹. Take a holomorphic map γ:X→CPⁿ. Letζ denote the line bundleγ^∗ᏻCPⁿ(1)overX. In the notation of the exact sequence (2.4), the line bundleᏻCPⁿ(1)isS^∗. Pulling back the universal exact sequence (2.4) toXand then taking the dual, we have

0 →γ^∗Q^∗ →W ^p→ζ →0, (2.20)

whereWis the trivial vector bundle of rankn+1 overXwith fiber(Cⁿ⁺¹)^∗. Of course, (Cⁿ⁺¹)^∗=Cⁿ⁺¹.

The trivialization ofWinduces a homomorphism

p¯:W →Jⁿ(ζ) (2.21)

which can be defined as follows: for any pointx∈Xand vectorw∈Wxin the fiber, let w¯ denote the unique flat section ofWsuch that ¯w(x)=w. Now, ¯p(w)is the restric- tion of the sectionp(w)¯ ofζto thenth-order infinitesimal neighborhood ofx. Recall that, the fiberJⁿ(ζ)xis the space of sections ofζover thenth-order infinitesimal neighborhood ofx.

Lemma2.5. The mapγ is everywhere locally nondegenerate if and only if the ho- momorphismp¯in (2.21) is an isomorphism.

Proof. This is a straightforward consequence of the condition of everywhere lo- cally nondegeneracy. For some pointx∈X, if ¯px:Wx→Jⁿ(ζ)x is not an isomor- phism, then take a nonzero vectorw in the kernel of ¯px, sinceWx=(Cⁿ⁺¹)^∗, the vectorw defines a hyperplaneHinCPⁿ. Clearly,H containsγ(x). The given condi- tion ¯px(w)=0 can be seen to be equivalent to the condition that the order of contact ofHwithγ(X)atγ(x)is at leastn. In other words,γis degenerate atx.

Conversely, ifγ is degenerate at a pointx∈X, take a hyperplaneH inCPⁿ con- tainingγ(x)such that the order of contact betweenγ(X)andH atγ(x)is at least n. Letw∈(Cⁿ⁺¹)^∗ be a functional defining the hyperplaneH. It is easy to see that p¯x(w)=0. This completes the proof.

Assume thatγ is everywhere locally nondegenerate. So the homomorphism ¯pin (2.21) gives a trivialization of the jet bundleJⁿ(ζ). Now, from (2.17) it follows that _n+1

Jⁿ(ζ)is canonically isomorphic toKⁿ⁽ⁿ⁺X ^1)/2

ζⁿ⁺¹. The trivialization ofJⁿ(ζ) induces a trivialization ofKX^n(n+1)/2

ζⁿ⁺¹. Fix a square-rootξof the holomorphic tangent bundleTX. In other words,ξis a holomorphic line bundle and an isomorphism betweenTX andξ^⊗2is chosen. The above trivialization ofKX^n(n+1)/2

ζⁿ⁺¹induces an isomorphism

Jⁱ ζ^j

=Jⁱ ξ^nj

⊗ ξ^nj_∗

⊗ζ^j (2.22)

for everyiand j. Indeed, this is an immediate consequence of the fact thatζ and ξⁿdiffer by tensoring with a finite-order line bundle. By a finite-order line bundle we mean a line bundle some tensor power of which has a canonical trivialization.

Consider the homomorphism

pˆ:W →Jⁿ⁺¹(ζ) (2.23)

which sends anyw∈Wxto the restriction of the sectionp(w)¯ ofζto the(n+1)th- order infinitesimal neighborhood ofx. Herepas in (2.20) and ¯was in the definition of the map ¯pin (2.21). From its definition it is immediate that the compositionfn◦p◦ˆ p¯⁻¹ is the identity map ofJⁿ(ζ), wherefnis the projectionJⁿ⁺¹(ζ)→Jⁿ(ζ)defined in (2.17). In other words, ˆp◦p¯⁻¹is a splitting of the jet sequence

0→KXⁿ⁺¹⊗ζ →Jⁿ⁺¹(ζ) →Jⁿ(ζ) →0 (2.24) defined in (2.17).

There is a unique homomorphismJⁿ⁺¹(ζ)→Kⁿ⁺X ¹⊗ζsatisfying the two conditions that its kernel is the image of ˆp◦p¯⁻¹and the composition of the natural inclusion of Kⁿ⁺¹X ⊗ζinJⁿ⁺¹(ζ)(as in (2.17)) with it is the identity map ofKXⁿ⁺¹⊗ζ. By the earlier definition of differential operators given in terms of jet bundles, this homomorphism defines a differential operator

Dγ∈H⁰

X,Diffⁿ⁺¹_X

ζ,KXⁿ⁺¹⊗ζ

. (2.25)

SinceDγ is defined by a splitting of a jet sequence, its symbol is the constant func- tion 1 (the symbol of a differential operator is defined in (2.19)). Now, using (2.22), the differential operatorDγgives a differential operator

D(γ)∈H⁰

X,Diffⁿ⁺_X ¹

ξⁿ,ξ⁻ⁿ⁻²

(2.26) of symbol 1.

It can be deduced from the definition of jet bundles that, for any holomorphic vector bundleE, there is a natural injective homomorphismJ^i+j(E)→Jⁱ(J^j(E))for anyi,j≥0. Therefore, we have a commutative diagram

0 ξ^{⊗(−n−2)} Jⁿ⁺¹

ξ^⊗n

τ

Jⁿ ξ^⊗n

0

0 KX⊗Jⁿ ξ^⊗n

J¹ Jⁿ

ξ^⊗n

Jⁿ ξ^⊗n

0,

(2.27)

where the injective homomorphismτis obtained from the above remark.

If

f:Jⁿ ξ^⊗n

→Jⁿ⁺¹ ξ^⊗n

(2.28) is a splitting of the top exact sequence in (2.27), then the compositionτ◦fdefines a splitting of the bottom exact sequence in (2.27). But a splitting of the exact sequence

0 →KX⊗E →J¹(E)→E →0 (2.29)

is a holomorphic connection onE(see [1]). Furthermore, any holomorphic connection on a Riemann surface is flat. Therefore,τ◦fdefines a flat connection onJⁿ(ξ^⊗n). Let

∇^f denote this flat connection onJⁿ(ξ^⊗n)obtained from a splittingf.

...

SinceXis simply connected,∇^f gives a trivialization ofJⁿ(ξ^⊗n). In other words, if we choose a pointz∈X, using parallel translations,Jⁿ(ξ^⊗n)gets identified with the trivial vector bundle overXwithJⁿ(ξ^⊗n)zas the fiber.

Fix an isomorphism of the fiberJⁿ(ξ^⊗n)z withCⁿ⁺¹. As before, letW denote the trivial vector bundle overXwithCⁿ⁺¹as the fiber. So we haveJⁿ(ξ^⊗n)=W.

For any pointy ∈X, consider the one-dimensional subspace(ξ^⊗n⊗K_Xⁿ)y of the fiberJⁿ(ξ^⊗n)ygiven in (2.17). Let

γ:X →CPⁿ (2.30)

denote the map that sends any pointy∈Xto the line inCⁿ⁺¹that corresponds to the line(ξ^⊗n⊗KXⁿ)yby the isomorphism between the fibersJⁿ(ξ^⊗n)yandWy.

If we change the isomorphism between Jⁿ(ξ^⊗n)z and Cⁿ⁺¹by an automorphism A∈GL(n+1,C), then the mapγ is altered by the automorphismAofCPⁿ.

Lemma2.6. Letf:Jⁿ(ξ^⊗n)→Jⁿ⁺¹(ξ^⊗n)be a splitting of the top exact sequence in (2.27). Then the mapγconstructed in (2.30) fromfis everywhere locally nondegenerate.

Proof. The lemma follows fromLemma 2.5and the fact that the connection∇^f, from whichγis constructed, is given by a splittingf(as in (2.28)). In [3], a different but equivalent formulation of the lemma can be found.

Two everywhere locally nondegenerate maps f1 and f2 of X into CPⁿ are called equivalent if there is an automorphismA∈Aut(CPⁿ)=PGL(n+1,C)such thatA◦ f1=f2.

LetᏭdenote the space of all equivalence classes of everywhere locally nondegenerate maps ofXintoCPⁿ.

Take a differential operatorD∈H⁰(X,Diffⁿ⁺¹_X (ξⁿ,ξ⁻ⁿ⁻²))of symbol 1. Since the symbol ofDis 1, it gives a splitting of the top exact sequence in (2.27). Denoting this splitting Jⁿ(ξ^⊗n)→Jⁿ⁺¹(ξ^⊗n)by ¯D, considerτ◦D¯, which, as we already noted, is a flat connection on Jⁿ(ξ^⊗n). It may be noted that sinceξ^⊗2=TX, the line bundle _n+1

Jⁿ(ξ^⊗n)is canonically trivialized.

LetᏮdenote the space of global differential operators D∈H⁰

X,Diffⁿ⁺¹_X

ξⁿ,ξ⁻ⁿ⁻²

(2.31) of symbol1and satisfying the condition that the connection on_n+1

Jⁿ(ξ^⊗n)induced by the connectionτ◦D¯onJⁿ(ξ^⊗n)preserves the trivialization of_n+1

Jⁿ(ξ^⊗n). From the construction of the differential operator D(γ) in (2.26) it follows that D(γ)∈Ꮾ.

Let

F:Ꮽ →Ꮾ (2.32)

be the map that sends any everywhere locally nondegenerate mapγto the differential operatorD(γ)constructed in (2.26).

As above, for a differential operatorD∈Ꮾ, the corresponding splitting is denoted by ¯D. Let

G:Ꮾ →Ꮽ ^(2.33)

be the map that sends any operatorDto the mapγ constructed in (2.30) using the splittingf=D¯as in (2.28).

Lemma2.7. The mapFdefined in (2.32) is one-to-one and onto.

Proof. In fact, unraveling the definitions of the mapsF and G, defined in (2.32) and (2.33), respectively, yields that they are inverses of each other. We omit the details;

it can be found in [3].

Letᏼ(X)denote the space of all projective structures on the Riemann surfaceX. It is known thatᏼ(X)is an affine space for the space of quadratic differentials, namely, H⁰(X,K²X)(see [7]).

Lemma2.8. There is a natural bijective map betweenᏮand the Cartesian product ᏼ(X)×

_n+1

i=3

H⁰ X,KX^⊗i

(2.34)

ifn≥2. Ifn=1, thenᏮis in bijective correspondence withᏼ(X).

Proof. The key input in the proof is [2, Theorem 6.3, page 19]. Now we recall its statement.

LetY be a Riemann surface equipped with a projective structure. Letk,l∈Zand letn∈Nbe such thatk∉[−n+1,0]andl−k−j∉{0,1}for any integerj∈[1,n]. Then,

H⁰

Y ,Diffⁿ_Y

ᏸ^k,ᏸ^l= n

i=0

H⁰

Y ,ᏸ^{l−k−2n+2i}, (2.35) whereᏸis the square-root of the canonical bundle defined by the projective structure.

A clarification of the above statement is needed. In [2], a projective structure means an SL(2,C)structure. But here projective structure means a PGL(2,C)structure. But we know that a PGL(2,C)structure on a Riemann surface always lifts to an SL(2,C)struc- ture [7]. Furthermore, the space of such lifts is in bijective correspondence with the space of theta-characteristics (square-root of the holomorphic cotangent bundle) ofY. Therefore, given a PGL(2,C)structurePonX, the pair(P ,ξ)determines a unique SL(2,C)structure.

Now, setk= −nandl=n+2 in (2.35). This yields an isomorphism F:H⁰

X,Diffⁿ⁺¹_X

ξⁿ,ξ⁻ⁿ⁻²

→ n

i=0

H⁰ X,K_X^⊗i

. (2.36)

For anyD∈H⁰(X,Diffⁿ⁺¹_X (ξⁿ,ξ⁻ⁿ⁻²)), the component ofF(D)in H⁰

X,KX^⊗0

=H⁰ X,ᏻX

(2.37)

is the symbol ofD. Furthermore, the condition in the definition ofᏮthat, the con- nection on_n+1

Jⁿ(ξ^⊗n)induced by the connectionτ◦D¯ onJⁿ(ξ^⊗n)preserves the trivialization of_n+1

Jⁿ(ξ^⊗n), is actually equivalent to the condition that the com- ponent ofF(D)inH⁰(X,KX)vanishes (see [3]). Therefore, usingF, the spaceᏮgets

...

identified with the direct sum

n+1

i=2

H⁰ X,KX^⊗i

, (2.38)

ifXis equipped with a projective structure.

Using the fact that the space of projective structures onX, namelyᏼ(X), is an affine space forH⁰(X,KX²), it is easy to deduce that given any

D∈H⁰

X,Diffⁿ⁺¹_X

ξⁿ,ξ⁻ⁿ⁻²

, (2.39)

there is a unique projective structureP ∈ᏼ(X)such that, for the map F in (2.36) corresponding toP, the component of F(D)inH⁰(X,K_X^⊗2)vanishes identically. Let F(D)denote the projection ofF(D)in_n+1

i=3H⁰(X,KX^⊗i);Fcorresponds to this unique projective structure. Now, we have a bijective map

F¯:Ꮾ →ᏼ(X)×

_n+1

i=3

H⁰ X,K^⊗iX

, (2.40)

that sends anyDto the pair(P ,F(D))constructed above. (See [3, Section 4] for the details.)

Ifn=1, then using [2, Theorem 6.3] and the fact thatᏼ(X)is an affine space for H⁰(X,K^⊗X²), it follows immediately thatᏮ=ᏼ(X). This completes the proof of the lemma.

For the first part of the proof ofLemma 2.8, we should have directly used [2, Corol- lary 6.6] instead of deriving it using [2, Theorem 6.3]. Unfortunately, in the statement of [2, Corollary 6.6], the word “compact” is used which technically makes it useless for our purpose. But, of course, compactness is not used in the proof of [2, Corollary 6.6]. When [2,3] were written, we had primarily compact Riemann surfaces in mind.

Combining Lemmas2.7and2.8, we have the following corollary.

Corollary2.9. There is a natural bijective map Γ:Ꮽ →ᏼ(X)×

_n+1

i=3

H⁰ X,K^⊗iX

(2.41)

forn≥2. Ifn=1thenᏭis in bijective correspondence withᏼ(X).

When X is a compact Riemann surface, the above corollary is [3, Theorem 5.5].

Again since “compactness” condition is thrown in [3] indiscriminately, a vast part of it is technically useless for our present purpose. Nevertheless, the ideas of [3] have been borrowed here.

LetY⊂Xbe a simply connected open subset. LetᏭYdenote the space of all equiva- lence classes of everywhere locally nondegenerate maps ofYintoCPⁿ. In other words, ᏭY is obtained by substitutingY in place ofXin the definition ofᏭ. The space of all projective structures onY is denoted byᏼ(Y ).

The restriction ofξtoY defines a square-root of the tangent bundleTY. There is a natural restriction mapᏼ(X)→ᏼ(Y )and also there are homomorphisms

H⁰ X,KX^⊗i

→H⁰ Y ,KY^⊗i

(2.42)

for everyi∈Zdefined by restriction of sections. Similarly, we have a mapᏭ →ᏭY, which sends a mapγofXtoCPⁿto the restriction ofγ toY.

Let

ΓY:ᏭY →ᏼ(Y )× _n+1

i=3

H⁰ Y ,KY^⊗i

(2.43)

be the isomorphism forY obtained inCorollary 2.9. The mapΓ inCorollary 2.9has the property that the following diagram commutes:

Ꮽ ^Γ ᏼ(X)×_n+1 i=3H⁰

X,KX^⊗i

ᏭY ΓY

ᏼ(Y )×_n+1 i=3H⁰

Y ,KY^⊗i

.

(2.44)

The vertical maps are defined by restriction. The commutativity of this diagram is indeed easy to see from the construction ofΓ.

Now that we haveCorollary 2.9and (2.44), we are ready to proveTheorem 2.4.

Proof ofTheorem 2.4. Assume thatn≥2, since the theorem is obvious in the case ofn=1.

Suppose we are given a transversely CPⁿ-structure, as defined in Definition 2.3.

We assume that all the subsets Di:=image(φi) of Cin Definition 2.1 are simply connected. Clearly, this is a harmless assumption.

Consider a triplet(Ui,φi,γi)as inDefinition 2.3. Now, using the mapΓ inCorollary 2.9, from the everywhere locally nondegenerate mapγiwe have a projective structure onDi=image(φi)together with a holomorphic section ofTD^⊗−l_i for alll∈[3,n+1]. This projective structure onDiis denoted byᏼi, and the holomorphic section ofT_D^⊗−l_i obtained above is denoted byω^l_i. The projective structureᏼiinduces a transversely projective structure on the open subsetUi of M. We denote this transversely pro- jective structure onUi by ¯ᏼi. The pullback, using the map φi, of the holomorphic sectionω^l_i ofT_D^⊗−l_i defines a section of N^⊗−lover Ui. This section of N^⊗−l overUi

is denoted by ¯ω^l_i. Since ω^l_i is holomorphic, we have the section ¯ω^l_i over Ui to be transversely holomorphic. Furthermore, ¯ω^l_iis obviously flat with respect to the Bott partial connection. The proof of the theorem is completed by showing that all these locally defined transversely projective structures ¯ᏼi(resp., transversely holomorphic flat sections ¯ω^l_i) patch compatibly to define globally onM a transversely projective structure (resp., transversely holomorphic flat section ofN^⊗−l).

If we take another triplet(Uj,φj,γj),j∈I, as inDefinition 2.3, then the two pro- jective structures onDi∩Dj, namelyᏼiandᏼj, coincide. This is an immediate con- sequence of the commutativity of the diagram (2.44). Therefore, we have a projective

...

structure on the unionDi∪Dj, and hence the two transversely projective structures, namely ¯ᏼi and ¯ᏼj, coincide overUi∩Uj. Consequently, the transversely projective structures{ᏼ¯i}i∈Ipatch together compatibly to define a transversely projective struc- ture on ¯Ᏺ. Similarly, from the commutativity of the diagram (2.44), it follows that the two sections ¯ω^l_iand ¯ω^l_jcoincide overUi∩Uj. In other words, these local sections ¯ω^l_i ofN^⊗−lpatch together to give an element ofᐂ¯_Ᏺ(−l). This completes the proof of the theorem.

Theorem 2.4can be considered as a generalization of [10, Theorem 6.1].

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Indranil Biswas: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay400005, India

E-mail address:[email protected]