We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure

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A FOLIATED HITCHIN-KOBAYASHI CORRESPONDENCE

DAVID BARAGLIA AND PEDRAM HEKMATI

Abstract. We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold.

We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian- Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting.

1. Introduction

In this paper we prove an analogue of the Hitchin-Kobayashi correspondence for compact, taut Riemannian foliated manifolds with transverse Hermitian structure. The two sides of the correspondence, polystable holomorphic bundles and Hermitian-Einstein connections, are replaced by their foliated analogues. Recall that one motivation for the Hitchin-Kobayashi correspondence comes from studying moduli spaces of the anti-self- dual instanton equations on a 4-manifoldX:

(1.1) ∗F_A=−F_A,

where F_A is the curvature of a unitary connection A on a vector bundle E → X. In general it is difficult to understand the topology of the moduli space of instantons on X.

However if X is a Hermitian 4-manifold, one observes that the anti-self-duality equations are equivalent to the equations:

F_A^0,2 = 0 ΛF_A= 0,

where Λ is the adjoint of the Lefschetz operator. These are (a special case of) the Hermitian-Einstein equations for a unitary connection Aon a holomorphic vector bundle.

Note that by the first equation A determines a holomorphic structure on E. Assume that the Hermitian metric on the 4-manifold X is Gauduchon, i.e. ∂∂ω = 0. Then the

Date: August 15, 2022.

This work is supported by the Australian Research Council Discovery Projects DP110103745, DP170101054 and the Royal Society of New Zealand Marsden Fund Grant 17-UOA-061.

1

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Hitchin-Kobayashi correspondence gives necessary and sufficient conditions for a holomorphic vector bundle E to admit a Hermitian-Einstein connection, namely E must be polystable. Consequently, the moduli space of instantons onX of given rank can be identified with the moduli space of polystable holomorphic vector bundles of degree zero and corresponding rank. The advantage of this is that it allows one to study the moduli space of instantons on X using complex analytic tools.

Similarly one can study the Hermitian-Einstein equations in higher dimensions. Suppose Xis a Hermitian manifold of real dimension 2n. We assume the metric onXis Gauduchon, which means that∂∂(ωⁿ⁻¹) = 0. LetE be a unitary vector bundle on Xand Aa unitary connection. The Hermitian-Einstein equations with Einstein factor γ_Aare:

F_A^0,2 = 0 iΛFA=γAIdE.

For 2n >4 these equations have an interpretation as a higher-dimensional analogue of the instanton equation (1.1), at least when E has trivial determinant and γA = 0. Namely, they are solutions of the Ω-instanton equations[6, 8, 23]:

∗F_A=−Ω∧FA, where Ω = ωⁿ⁻² (n−2)!.

Our original motivation for studying a foliated version of the Hitchin-Kobayashi correspondence arose from our study of contact instantons[2, 12], which are an analogue of the anti-self-dual instanton equations for 5-dimensional contact manifolds. Let X be a contact 5-manifold with contact 1-form η. The anti-self-dual contact instanton equations are:

(1.2) ∗F_A=−η∧F_A.

Notice that these are Ω-instantons for Ω =η. In [2], we studied the moduli space of contact instantons for compact K-contact 5-manifolds. Let ξ denote the Reeb vector field of the contact manifoldX. Thenξ generates a 1-dimension foliation ofX. IfXisK-contact, then this is a taut Riemannian foliation and the contact instanton equations (1.2) can be interpreted as saying that Ais a basic connection with respect to the foliation and which is anti-self-dual in the directions transverse to the foliation.

Suppose now that the contact 5-manifold X is Sasakian (the definition is recalled in Section 6.2). Recall that Sasakian geometry is an odd-dimensional analogue of K¨ahler geometry. In particular the geometry of X transverse to the Reeb foliation is K¨ahler. In this case the contact instanton equations admit an interpretation as being connections which are Hermitian-Einstein transverse to the foliation. More generally, if one considers Sasakian manifolds of dimension 2n+ 1 withn≥2 one can consider the following version of the Ω-instanton equations:

∗F_A=−Ω∧F_A, where Ω =η∧ (dη)ⁿ⁻² (n−2)! .

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As in the 5-dimensional case, such connections can be interpreted as basic connections which are Hermitian-Einstein transverse to the Reeb foliation.

From the above considerations we are lead to consider the following very general setup:

let X be a compact oriented manifold with a taut Riemannian foliation which has a transverse Hermitian structure (see Section 2 for definitions). In Theorem 3.10 we prove that after rescaling by a positive smooth basic function, we can assume the transverse metric is transverse Gauduchon (see Definition 3.9). Since X is transverse Hermitian, we may speak of basic differential forms on X of type (p, q) and we may also define the adjoint Λ of the Lefschetz operator, which we regard as acting on basic forms. We may then define transverse Hermitian-Einstein connections as follows:

Definition 1.1.

(i) LetE be a foliated Hermitian vector bundle. A basic unitary connection A on E is called transverse Hermitian-Einsteinif its curvature 2-form F_A is of type (1,1) and satisfies

iΛFA=γAidE,

for some real constant γA, called theEinstein factorof A.

(ii) Let E be a foliated holomorphic vector bundle. A transverse Hermitian metric on E is called a transverse Hermitian-Einstein metric if the associated Chern connection is transverse Hermitian-Einstein.

This is the one side of the transverse Hitchin-Kobayashi correspondence. The other side of the correspondence is the foliated analogue of polystable holomorphic vector bundles.

LetEbe a transverse holomorphic vector bundle onX. The degree deg(E) of a transverse holomorphic vector bundle is defined as follows. Suppose that E admits a transverse Hermitian metric. Note that a transverse Hermitian metric need not exist. Indeed the usual way that one proves the existence of a Hermitian metric on a complex vector bundle is to use a partition of unity. However this fails in the transverse setting since we usually can not find a partition of unity subordinate to a given cover bybasicfunctions. Thus, we will only define deg(E) in the case thatE admits a transverse Hermitian metrich. LetA be the associated Chern connection and F_A its curvature (1,1)-form. We define

(1.3) deg(E) = i

2π Z

X

tr(F_A)∧ωⁿ⁻¹∧χ,

whereχis the leafwise volume form. We show that ifX is transverse Gauduchon and the foliation is taut, then deg(E) does not depend on the choice of transverse Hermitian metric.

In order to define stability/semistability we need the notion of a transverse coherent subsheaf F ⊂ O(E), whereO(E) is the sheaf of basic holomorphic sections ofE. Trans- verse coherent sheaves are introduced in Definition 3.18. To each transverse coherent sheaf F, one can associate a determinant line bundle det(F) in much the same way as done in the non-foliated setting. We have that det(F) is a transverse holomorphic line bundle. At this point a complication arises compared to the non-foliated setting, namely, it is by no

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means clear whether the line bundle det(F) admits a transverse Hermitian metric and so we can not define the degree of det(F) simply by using (1.3). To get around this problem, we are lead to consider a foliated version of Hironaka’s resolution of singularities, which we carry out in Section 3.2. This is similar to the approach to stability taken in [18]. We use this to define the degree of a transverse coherent subsheaf F ⊂ O(E) of a transverse holomorphic vector bundle E which admits a transverse Hermitian metric, under the further assumption that quotient sheaf O(E)/F is torsion-free. This is sufficient to define stability/semistability:

Definition 1.2. LetE be a transverse holomorphic vector bundle which admits a transverse Hermitian metric. We say that E is stable (resp. semistable) if for each transverse coherent subsheaf F ofE with 0< rk(F)< rk(E) and such that the quotient O(E)/F is torsion-free, we have

deg(F)/rk(F)<deg(E)/rk(E) (resp. deg(F)/rk(F)≤deg(E)/rk(E)).

We also say thatE ispolystableifE is the direct sum of stable bundles of the same slope.

With these definitions at hand, we may now state our main theorem:

Theorem 1.3(The transverse Hitchin-Kobayashi correspondence). LetE be a transverse holomorphic vector bundle which admits transverse Hermitian metrics. Then E admits a transverse Hermitian metric h which is Hermitian-Einstein if and only if E is polystable.

Moreover, if E is simple then h is unique up to constant rescaling.

Our proof is based on the Uhlenbeck-Yau method of continuity proof of the Hitchin- Kobayashi correspondence given in [25] and its exposition in the book [17]. The overall strategy for proving Theorem 1.3 is essentially that of Uhlenbeck-Yau. Therefore most of the work involved in the proof is in adapting each step of the proof to the foliated setting. For example, this requires the introduction of Sobolev spaces of basic sections, establishing embedding, compactness and elliptic regularity results in the basic setting.

Working transverse to a foliation means that we are working with transversally elliptic operators which are not genuinely elliptic and so we also need to make use of the theory of such operators [9].

At this point, the reader may have the impression that essentially any result in gauge theory can be carried over, more or less trivially, to the foliated setting. We wish to em- phasise that this is not the case. In fact, as we will demonstrate in Section 6, the foliated analogue of the weak Uhlenbeck compactness theorem fails. More specifically, we give an example (Example 6.4) of a compact manifold X with a taut Riemannian foliation which is transverse K¨ahler for which one can find sequences of basic connectionsAi on a foliated bundle whose curvatures are uniformly bounded (in our example theA_i are flat) such that there is no weakly convergent subsequence modulo basic gauge transformations (in the L^p,k-norm for anyp, k).

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In Section 6.1, we consider the foliated Hitchin-Kobayashi correspondence in the case where the foliation has complex codimension 1, the foliated analogue of a Riemann sur- face. A number of simplifications occur here, for example, in the definition of stability/semistability, it is enough to consider transverse holomorphic subbundles. So there is no need to consider transverse coherent sheaves or foliated resolutions of singularities. We refer to this special case of Theorem 1.3 as the “transverse Narasimhan-Seshadri theorem”:

Theorem 1.4 (Transverse Narasimhan-Seshadri theorem). Let X be a compact oriented, taut, transverse Hermitian foliated of complex codimension n= 1. Let E be a transverse holomorphic vector bundle which admits transverse Hermitian metrics. Then E admits a transverse Hermitian metric such that the Chern connection A satisfies

F_A=−2πi µ(E)

V ol(X)ω⊗Id_E, if and only if E is polystable.

When E has degree 0, this reduces to the condition that Ais a flat connection. In this case, the transverse Narasimhan-Seshadri correspondence can be neatly summarised as follows:







Isomorphism classes of rankm unitary representations ofπ₁(X)







↔







Isomorphism classes of polystable rank m degree 0 transverse holomorphic vector bundles

admitting transverse Hermitian metrics







In the case of complex codimension 1 foliations we also prove the analogue of the Harder-Narasimhan filtration:

Theorem 1.5 (Transverse Harder-Narasimhan filtration). Let X be a compact oriented, taut, transverse Hermitian foliated of complex codimension n= 1. Let E be a transverse holomorphic vector bundle which admits transverse Hermitian metrics. There exists a uniquely determined filtration of E

0 =E₀ ⊂E₁ ⊂ · · · ⊂E_k=E

by transverse holomorphic subbundles such that the quotients Fi =Ei/Ei−1 are semistable and the slopes are strictly increasing:

µ(F₁)> µ(F₂)>· · ·> µ(F_k).

To prove the Harder-Narasimhan filtration, one has to show that there exists a transverse holomorphic subbundle F ⊆ E which maximises µ(F). In the non-foliated setting this is easy to show using the fact that µ(F) is a rational number with denominator of absolute value at most rk(E). In the foliated setting, the slopes µ(F) can be any real numbers and it is non-trivial to see that the supremum over all slopes of subbundles of E is attained by a subbundle. To prove this, we make use of the notion of (a foliated analogue of) weakly holomorphic subbundles, which were introduced by Uhlenbeck and

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Yau in their proof of the Hitchin-Kobayashi correspondence.

In Section 6.2 we return to our original motivation, the case where X is Sasakian.

We recall the definition of Sasakian manifolds and observe that since they are transverse K¨ahler, Theorem 1.3 applies:

Corollary 1.6. The transverse Hitchin-Kobayashi correspondence holds on any compact Sasakian manifold.

Of course, Theorem 1.3 can be applied to any compact oriented manifold X with taut, transverse K¨ahler foliation. Further examples of such geometries include 3-Sasakian manifolds and co-K¨ahler manifolds.

In Section 6.3, we re-examine the relation between transverse Hermitian-Einstein connections and Ω-instantons. In general, transverse Hermitian-Einstein connections with trivial determinant correspond to solutions of:

(1.4) ∗F_A=−Ω∧F_A, where Ω =χ∧ ωⁿ⁻² (n−2)!,

where we recall that χ is the leafwise volume form. When X is Sasakian of dimension 2n+ 1 (with n ≥ 2), we have χ = η, ω = dη and so (1.4) reduces to the anti-self-dual

“higher contact instanton” equations:

∗F_A=−η∧ (dη)ⁿ⁻² (n−2)! ∧F_A. Combined with Corollary 1.6, we have:

Corollary 1.7. Anti-self-dual SU(r) higher contact instantons on a 2n+ 1-dimensional Sasakian manifold X (with n ≥ 2) correspond to rank r transverse Hermitian-Einstein connections with trivial determinant. If X is compact, then by the transverse Hitchin- Kobayashi correspondence, anti-self-dualSU(r)higher contact instantons onXcorrespond to rank r polystable transverse holomorphic bundles on X with trivial determinant.

The following is a brief summary of each section of the paper. In Section 2, we cover the background material needed to study gauge theory transverse to a taut Riemannian foliation. In particular, we review the notions of foliated vector bundles, basic connections and basic differential operators. In Section 2.2 we introduce Sobolev spaces of basic sections of a vector bundle and prove basic versions of the Sobolev embedding and compactness theorems. In Section 3, we study the foliated analogues of various notions in complex geometry. In Section 3.1, we prove the existence of Gauduchon metric on compact taut transverse Hermitian manifolds. In Section 3.2, we prove a foliated version of resolution of singularities which we use in Section 3.3 to define stability of transverse holomorphic vector bundles. In Section 4 we introduce the transverse Hermitian-Einstein equations and prove that transverse Hermitian-Einstein implies polystable. In Section 5, we complete the transverse Hitchin-Kobayashi correspondence by proving the converse. Section 6 is

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concerned with some applications and related results. In Section 6.1 we consider the case of foliations of complex codimension 1, which leads to a transverse Narasimhan-Seshadri theorem. We also prove a transverse version of the Harder-Narasimhan filtration and give an example to show the Uhlenbeck compactness fails in general for basic connections on a foliated bundle. In Section 6.2 we recall the definition of Sasakian manifolds and consider the transverse Hitchin-Kobayashi correspondence for them. Finally, in Section 6.3 we relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds.

2. Transverse geometry on Riemannian foliations

2.1. Riemannian foliations. LetX be a smooth oriented manifold with a foliationF of dimension mand codimension 2n. We denote byV =TF the tangent distribution of the foliation andH =T X/V the normal bundle. We assume thatF is aRiemannian foliation [19] which means that X is endowed with a bundle-like metric g, so that the foliation is locally identified with a Riemannian submersion. In other words, g yields an orthogonal splitting T X = V ⊕H and induces a holonomy invariant Riemannian structure on H.

Let volT and χ denote the transverse and leafwise volume forms respectively. These are determined by the metric and the orientations on H and V, which are chosen such that volX =volT ∧χ.

By a foliated charton X, we mean a coordinate chart (U, ϕ), where U ⊆X is an open subset, ϕ:U →V ×W ⊆R²ⁿ×R^ma diffeomorphism, whereV ⊆R²ⁿ,W ⊆R^m are open subsets and ϕ(F |_U) is given by the fibres of the projection V ×W → V. When working with foliated charts we will often drop explicit mention of ϕand simply identify U with V ×W.

A smooth function f on X is called basic if ξ(f) = 0 for all ξ ∈ Γ(X, V). More generally, a differential form α ∈ Ω^k(X) is called basic if i_ξα = 0 and i_ξdα = 0 for all ξ ∈ Γ(X, V). The meaning of this condition is that in a local foliated chart the form depends only on the transverse variables. We let Ω^k_B(X) denote the space of basick-forms and dB: Ω^k_B(X)→Ω^k+1_B (X) the restriction of the exterior derivative dto basic forms.

A foliation is said to betaut ifX admits a metric such that every leaf ofF is a minimal submanifold, or equivalently such that the mean curvature form of the leaves κ is trivial.

Proposition 2.1 (Basic Stokes’ theorem). Let X be a closed oriented manifold with a taut Riemannian foliation of codimension 2n, then

(2.1)

Z

X

dBα∧χ= 0 for all α∈Ω²ⁿ⁻¹_B (X).

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Proof. Rummler’s formula [22] states that

dχ=−κ∧χ+φ

for someφ∈Ω^m+1(X) satisfyingιξ1. . . ιξmφ= 0 for any set{ξ_j}ofmvectors in Γ(X, V).

Thereforeα∧φ= 0 for any basic (2n−1)-formαand by tautness it follows thatd_Bα∧χ= d(α∧χ).

A principal G-bundle π:P → X is said to be a foliated principal bundle if there is a rank m foliation Fe on P with tangent distribution TF ⊆e T P such thatFe is G-invariant and TFe projects isomorphically onto TF under π∗. An isomorphism of foliated principal G-bundles (P,F),e (P⁰,Fe⁰) is a principal bundle isomorphism φ: P → P⁰ such that φ(F) =e Fe⁰. A local section s:U →P is called basic ifs∗ :T U → T P sends TF to TF.e It is known that every foliated principal bundle admits local basic sections [19, Proposi- tion 2.7]. The transition functions {g_αβ} between local basic sections are G-valued basic functions. Conversely, if a principal bundle P is defined by a collection {g_αβ} of basic transition functions, this determines a foliated structure on P. This is because for eachα the obvious lift toU_α×Gof the foliationF |_U_α is preserved by the transition functions. In this way we obtain an equivalence between foliated principal G-bundles and equivalence classes of basic ˇCech cocycles{g_αβ}.

A connectionA onP, thought of as ag-valued 1-form onP is calledadaptedifTFe lies in the kernel of A. It is called basic if as a g-valued 1-form, A is basic with respect to the foliation F. While every foliated principal bundle admits an adapted connection bye extendingTFeto a horizontal distribution, there is an obstruction to the existence of basic connections. This is ultimately due to the lack of a basic partition of unity subordinate to a given cover and the obstruction is a secondary characteristic class of the foliated bundle [13, 19]. However, ifA is a connection onP for which the curvatureFAsatisfiesi_ξFA= 0 for allξ∈TF, then the horizontal lift ofV with respect toAis an integrable distribution and gives P a foliated structure for whichA is basic. We note that if X is a Riemannian foliation, then H admits a basic connection, namely we can take the basic Levi-Civita connection associated to the basic metric on H.

A vector bundle E is foliated if its associated frame bundle is foliated. In this case we will also say that E has a transverse structure or that E is a transverse vector bundle.

We denote by Γ_B(X, E) the space of smooth basic sections of E. In the ˇCech language a basic section is given by a collection {s_α} of basic vector-valued functions such that s_α =g_αβs_β. Atransverse Hermitian metrich on a foliated complex vector bundleE is a Hermitian metric on Ewhich is basic as a section ofE^∗⊗E^∗. Equivalently,hcorresponds to a reduction of structure group of the frame bundle fromGL(n,C) toU(n) as a foliated principal bundle. Such a reduction is however not always possible and therefore not every

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foliated complex vector bundle admits a transverse Hermitian metric.

Let E be a foliated vector bundle. For each d ≥ 0, one can define the d-th basic jet bundle J_B^d(E) whose fibres are the d-jets of local basic sections of E. Clearly J_B^d(E) is a foliated vector bundle in a natural way and there are short exact sequences

(2.2) 0→Sym^d(H^∗)⊗E →J_B^d(E)→J_B^d−1(E)→0.

A basic section ofE determines a basic section ofJ_B^d(E) by prolongation. Suppose thatE admits a basic connection∇. We can use the basic connection to split the sequences (2.2) and thus non-canonically identify J_B^d(E) with Ld

j=0Sym^j(H^∗)⊗E, as foliated vector bundles. Suppose that F is another foliated vector bundle. We define a basic linear differential operator of order d from E toF to be a basic section of the foliated bundle Dif f_B^d(E, F) = Hom(J_B^d(E), F). Clearly a basic linear differential operator defines a linear map D : ΓB(X, E) → ΓB(X, F) which in local foliated coordinates is given by a linear differential operator in the transverse coordinates. From the exact sequence (2.2), there is a natural map σ :Dif f_B^d(E, F) → Hom(Sym^d(H^∗), Hom(E, F)) and we define the symbol of a basic linear differential operatorD of orderdto be the image σ(D) of D under this map. We say that D is transversally elliptic if σ(D)(ξ, . . . , ξ) is invertible for each 06=ξ ∈H^∗.

Remark 2.2. Let E, F be foliated vector bundles and D ∈ Γ_B(X, Dif f_B^d(E, F)) a basic linear differential operator of order d. If E admits a basic connection, then D can be non-canonically extended to a linear differential operator in the ordinary sense. To see this, let J^d(E) be the usual d-th jet bundle of E and Dif f^d(E, F) = Hom(J^d(E), F).

We use the connection to obtain splittings J_B^d(E) ∼= Ld

j=0Sym^j(H^∗) ⊗E, J^d(E) ∼= Ld

j=0Sym^j(T^∗X)⊗E and we use the Riemannian metric g to obtain a non-canonical splitting T^∗X = H^∗⊕V^∗ and thus an inclusion H^∗ → T^∗X. In this way, we obtain a non-canonical homomorphism Dif f_B^d(E, F)→Dif f^d(E, F).

Let E, F be foliated vector bundles and D : Γ_B(X, E) → Γ_B(X, F) a basic linear differential operator. Assume that E, F admit transverse Hermitian structures, in other words they are vector bundles associated to foliated principal bundles with structure group a unitary group. Then we may construct the formal adjoint operator D^∗ : ΓB(X, F) → Γ_B(X, E) [9] which has the property

hDs, ti=hs, D^∗ti

for all s ∈ Γ_B(X, E), t ∈ Γ_B(X, F), where h , i is the usual L²-pairing of sections of a Hermitian vector bundle. Let us give a construction ofD^∗under the assumptions that the foliation is taut and that E admits a basic connection∇. Whenever we take the adjoint of a basic differential operator in this paper, these assumptions will hold. As in Remark 2.2 we can use ∇ to extend D to an ordinary linear differential operator, which has the

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form

D(s) =

d

X

j=0

X

|I|=j

a_(j)(eⁱ¹, eⁱ², . . . , eⁱ^j)∇_e_i

1∇_e_i

2 · · · ∇_e

ijs

where e₁, . . . , e_2n is a local frame of basic sections ofH,e¹, . . . , e²ⁿ the dual coframe and a_(j) is a basic section of Sym^j(H)⊗Hom(E, F). We claim thatD^∗ is given by the usual formula for the adjoint:

D^∗(s) =

d

X

j=0

X

|I|=j

∇^∗_e

ij· · · ∇^∗_e

i2∇^∗_e

i1(a^∗_(j)(eⁱ¹, eⁱ², . . . , eⁱ^j)s), where ∇^∗_e

is = −∇_e_is−div(ei)s and div(ei) is the divergence of ei. To see this, note that by the assumption of tautness, the divergence of Y ∈ Γ(X, H) is determined by div(Y)volT =dB(iYvolT), which shows that ifY is a basic section ofH, thendiv(Y) is a basic function. Therefore the formal adjoint D^∗ is again a basic differential operator.

From the theory of transversally elliptic operators, we have:

Theorem 2.3 ([9]). Let X be a compact Riemannian foliated manifold, E, F foliated complex vector bundles admitting transverse Hermitian metrics and D : Γ_B(X, E) → Γ_B(X, F) a basic linear differential operator of order d. Then D is Fredholm, that is, Ker(D) andKer(D^∗) are finite-dimensional.

Let E be a foliated vector bundle. Denote by Ω^k(X, E) the space of k-form valued sections of E and by Ω^k_B(X, E) the space of basick-form valued basic sections ofE. The latter is given by Ω^k_B(X, E) = Γ_B(X, E⊗ ∧^kH^∗). Suppose that ∇is a basic connection on E. We have that α∈Ω^k(X, E) is a basic section if and only ifiξα= 0 andiξd∇α= 0 for all ξ ∈Γ(X, V). It follows that the restriction of d∇ : Ω^k(X, E) → Ω^k+1(X, E) to basic sections induces a basic first order differential operator d∇: Ω^k_B(X, E)→Ω^k+1_B (X, E).

2.2. Basic Sobolev spaces and elliptic regularity. In this section, we assumeX is a compact, oriented, Riemannian foliation of codimension 2n.

Definition 2.4. Let E be a foliated vector bundle on X equipped with a transverse Hermitian structure and a compatible basic connection ∇. Fork a non-negative integer and p ∈[1,∞), the basic Sobolev space L^p,k_B (E) is defined as the norm closure inL^p,k(E) of the space of smooth basic sections ΓB(X, E) under the Sobolev norm

ksk_p,k=





k

X

j=0

||∇^js||^p_Lp





1 p

, s∈Γ(X, E).

Similarly, for a non-negative integer k we define C_B^k(E) to be the subspace of C^k(E) consisting of basic sections. It is easy to see that the limit of aC^k-convergent sequence of basic sections of E is again basic, henceC_B^k(E) is a closed subspace ofC^k(E).

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Remark 2.5. Note that to defineC_B⁰(E), we need to say what is a continuous basic section of E. We say that a continuous section of E is basic if in each local basic trivialisation of E, the section is given by a basic function, where a continuous function f defined on an open subset A⊆X is called basic if for each x∈A, there exists a local foliated chart U =V ×W ⊆A containing x for which f|_U is the pullback of a continuous function on V under the projection V ×W → V. For differentiable sections, this clearly agrees with our previous notion of basic sections. It is also clear that this notion of basic sections is closed under C⁰-limits.

Theorem 2.6 (Basic Sobolev embedding and compactness).

(i) For all integersk, l such thatk>l>0 and for allp, q∈[1,∞) such thatk−²ⁿ_p >

l−²ⁿ_q , there is a continuous inclusion

L^p,k_B (E),→L^q,l_B(E).

Moreover, if k > l andk−²ⁿ_p > l−²ⁿ_q then the inclusion is compact.

(ii) For all integers k, l with k > l > 0 and for all p ∈ [1,∞) such that k−²ⁿ_p > l, there is a continuous inclusion

L^p,k_B (E),→ C_B^l(E).

Moreover, the inclusion is compact.

Proof. The argument is essentially that of [15, Theorem 9, 10] which we now sketch. Con- sider for instance the case L^p,k_B (E),→L^q,l_B(E). Letf ∈L^p,k_B (E). Then f is the L^p,k-limit of a sequencef_i of basic smooth functions. Consider a local foliated chartU =V×W over which E is trivialised as a foliated bundle. We assume that the foliated chart is chosen so that the closure U is contained in a slightly larger foliated chart ˜U = ˜V ×W˜ and that the local trivialisation of E extends to ˜U. We also assume that ˜V ⊂ R²ⁿ, ˜W ⊂ R^m are bounded. Then f_i|_U is given by a smooth vector-valued function on V. Taking the L^p,k- limit, we see that f|_U is given by an L^p,k function on V. Under the stated assumptions, the Sobolev embedding theorem forV gives a continuous injectionL^p,k(V)→L^q,l(V) (see for instance [1]). By considering a collection of such foliated charts coveringX, we see that the sequence {f_i} converges in the L^q,l-norm and therefore f ∈L^q,l_B(E) by the definition of L^q,l_B(E). The Sobolev embedding theorem also implies an estimate ||f_i||_q,l ≤C||f_i||_p,k, for a constant C which does not depend on f or thef_i. Hence||f||_q,l ≤C||f||_p,k, so that the inclusion L^p,k_B (E),→L^q,l_B(E) is continuous.

We now argue that if k > l and k− ²ⁿ_p > l − ²ⁿ_q , the inclusion is compact. Let fi be a bounded sequence in L^p,k_B (E). Then in each foliated chart U = V ×W of the type previously described, we can apply Sobolev compactness of the inclusion L^p,k(V)→ L^q,l(V) (see [1]) to deduce that there is a subsequence of {f_i}which converges inL^q,l(V × W). Since we can cover X by finitely many such charts, we can find a subsequence of {f_i_j}which converges inL^q,l(E). But since eachfibelongs to the closed subspaceL^q,l_B(E),

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it follows that the convegent subsequence {f_i_j}converges to an element ofL^q,l_B(E), which proves compactness. The case of L^p,k_B (E),→ C_B^l (E) is proved similarly.

Theorem 2.7(Basic Sobolev multiplication). LetE, F be foliated vector bundles equipped with transverse Hermitian metrics and basic unitary connections. Let k, k⁰, l be integers andp, p⁰, q∈[1,∞)be such that

k− ²ⁿ_p +

k⁰−²ⁿ_p0

>

l−²ⁿ_q

,k, k⁰ ≥l, (k−l)p <2n and (k⁰−l)p⁰ <2n. Then multiplication of smooth basic sections extends to a continuous map

L^p,k_B (E)×L^p_B⁰^,k⁰(F)→L^q,l_B(E⊗F).

Proof. First, let us reduce this result to the case where k = k⁰ = l. If k > l, then the assumption (k−l)p <2nensures that there exists ˜p∈[1,∞) such that k−²ⁿ_p =l−²ⁿ_p_˜. Then we can use the basic Sobolev embedding theorem to replace L^p,k_B (E) with L^p,l_B^˜ (E).

A similar argument applies if k⁰ > l.

We now assumek=k⁰ =l. In this case,k, p, p⁰, qsatisfy _2n^k +¹_q < ¹_p+_p¹0. Ifk= 0, then the fact that multiplication extends to a continuous map L^p_B(E)×L^p_B⁰(F)→ L^q_B(E⊗F) follows from the H¨older inequality. Now we proceed by induction on k. The rest of the proof is exactly the same as the proof of [26, Lemma B.3], except using the basic Sobolev

embedding theorem in place of the ordinary one.

We also need to use a transverse analogue of elliptic regularity. The following version suffices for our purposes:

Lemma 2.8 (Basic elliptic regularity). Let E be a transverse vector bundle with admits a transverse Hermitian metric and a basic connection ∇. Let L: Γ_B(X, E)→Γ_B(X, E) be a second order transverse elliptic differential operator from E to itself. Suppose that k≥2 andp∈[1,∞). Let s∈L^p,k(E)∩L^p,1_B (E) be such that L(s)∈L^p,k−1(E). Then we have that s∈L^p,k+1(E)∩L^p,1_B (E).

Proof. By Remark 2.2, we can (non-canonically) extend L to a second order differential operator Le : Γ(X, E)→ Γ(X, E). Let ∇_V : Γ(X, E)→ Γ(X, E ⊗V^∗) denote the composition of ∇: Γ(X, E)→ Γ(X, E⊗T^∗X) with the orthogonal projectionT^∗X → V^∗. Let

∇^∗_V : Γ(X, E⊗V^∗)→Γ(X, E) be the formal adjoint of∇_V. DefineD: Γ(X, E)→Γ(X, E) to be the second order differential operator

D(a) =∇^∗_V∇_Va+L(a).e

We have that Dis elliptic because Lis transverse elliptic. Now let s∈L^p,k(E)∩L^p,1_B (E) and suppose that L(s) ∈ L^p,k−1(E). Note that s ∈ L^p,1_B (E) implies that ∇_Vs = 0.

This is because s is the L^p,1-limit of a sequence s_i of basic smooth sections. Since the si are basic and smooth, they satisfy ∇_Vsi = 0, hence also ∇_Vs = 0. It follows that D(s) = L(s) ∈ L^p,k−1(E). By the usual elliptic regularity of linear elliptic differential

operators, we have s∈L^p,k+1(E).

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Remark 2.9. Note that for k≥1, we haveL^p,k_B (E)⊆L^p,k(E)∩L^p,1_B (E). We do not know whether this inclusion is an equality. Fortunately, the above lemma suffices for the results of this paper.

3. Transverse complex geometry

3.1. Transverse Hermitian structures and transverse Gauduchon metrics.

Definition 3.1. Let X be a foliated manifold such that the foliation has codimension 2n. Let V =TF be the tangent distribution to the foliation and H =T X/V the normal bundle. A transverse almost complex structure is an endomorphismI :H→H such that I² = −Id. We say that the almost complex structure is integrable and that X has a transverse complex structure ifX can be covered by foliated chartsU_α =V_α×W_α, such that eachVα is an open subset of Cⁿand such that I|_U_α agrees with the natural complex structure on V_α obtained from the inclusion V_α ⊆Cⁿ.

Definition 3.2. A transverse Hermitian structureon a Riemannian foliated manifold X is a pair (g, I) consisting of a bundle-like Riemannian metricgand a transverse integrable complex structure I such that g and I are compatible in the usual sense: g(IX, IY) = g(X, Y) for all X, Y ∈ H. In this case we define the associated Hermitian 2-form ω ∈ Γ(X,∧²H^∗) byω(X, Y) =g(IX, Y). Pulling back by the natural projection T X →H, we identify ω with a 2-form on X. Using that fact that g is bundle-like and I is integrable, one sees that ω is a real basic 2-form, ω∈Ω²_B(X).

Throughout this section, unless stated otherwise we assume X is a compact oriented, taut, transverse Hermitian foliated manifold of complex codimensionn. We may introduce transverse analogues of all the usual notions in Hermitian geometry. For instance we have the Lefschetz operator:

L: Ω^j_B(X)→Ω^j+2_B (X), L(α) =ω∧α and the contraction operator, the adjoint of the Lefschetz operator:

Λ : Ω^j_B(X)→Ω^j−2_B (X), Λ(α) =L^∗(α).

The transverse complex structure I allows us to speak of basic differential forms of type (p, q). We denote the space of such forms as Ω^p,q_B (X,C). The exterior derivative d, restricted to basic differential forms can be decomposed as d = ∂ +∂, where

∂ : Ω^p,q_B (X,C) → Ω^p+1,q_B (X,C) and ∂ : Ω^p,q_B (X,C) → Ω^p,q+1_B (X,C). Integrability of I ensures that ∂² = ∂² = 0. If E is a foliated complex vector bundle we can also define Ω^p,q_B (X, E), the space of basic (p, q)-form valued sections of E.

Definition 3.3. LetE be a foliated complex vector bundle. A basic∂-connection onE is a first order basic differential operator ∂_E : Ω⁰_B(X, E)→ Ω^0,1_B (X, E) satisfying∂_E(f s) =

∂(f)s+f ∂_E(s) for all local basic sections s and local basic functions f. We extend ∂_E to a basic differential operator ∂E : Ω^p,q_B (X, E) →Ω^p,q+1_B (X, E) in the usual way. We say that ∂E isintegrable if∂²_E = 0. A transverse holomorphic structure onE is by definition

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an integrable transverse ∂-connection. A basic section of E is said to be holomorphic if

∂Es= 0.

Let (E, ∂E) be a transverse holomorphic vector bundle. The integrability condition

∂²_E = 0 implies that locallyEadmits a local frame of basic holomorphic sections. With respect to such frames, the transition functions{g_αβ}are basic holomorphicGL(r,C)-valued functions (wherer is the rank ofE). Conversely, given a cocycle{g_αβ}ofGL(r,C)-valued basic holomorphic functions, the associated vector bundleEhas a natural transverse holomorphic structure.

Many of the constructions one can do with holomorphic vector bundles have counter- parts in the transverse setting. For instance, suppose thatE is a transverse vector bundle with transverse Hermitian metric and basic unitary connection∇_E. Then we can decom- pose∇_E into its (1,0) and (0,1)-parts∇_E =∂E+∂E, when acting on basic sections. Then

∂_E is a basic ∂-operator. Moreover, ∂_E is integrable if and only if the curvature of ∇_E has type (1,1). In the other direction, if (E, ∂_E) is a transverse holomorphic vector bundle which admits a transverse Hermitian metric, then we can complete ∂E to a uniquely determined basic unitary connection ∇_E =∂_E+∂_E, which we call theChern connection associated to ∂E. Lastly, if E is a transverse holomorphic vector bundle which admits a transverse Hermitian metric, then we can take adjoints of the operators d_E, ∂_E, ∂_E and form their associated Laplacians ∆_d_E =d^∗_Ed_E +d_Ed^∗_E, etc. Here d_E : Ω^p(E)→ Ω^p+1(E) is the composition of ∇_E : Ω^p(E) → Γ(X, T^∗X⊗ ∧^pT^∗X⊗E) with the wedge product T^∗X⊗ ∧^pT^∗X⊗E → ∧^p+1T^∗X⊗E.

Definition 3.4. LetEbe a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h. TheP-operator associated to (E, h), denotedPE is given by

PE : Ω⁰_B(E)→Ω⁰_B(E), PE =iΛ∂E∂E.

When E is the trivial line bundle equipped with the standard Hermitian metric we write P instead ofP_E.

Lemma 3.5. Let E be a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h. Then P_E, P_E^∗ are transverse elliptic operators of index zero, i.e.

dim(Ker(P_E)) =dim(Ker(P_E^∗)).

Proof. A direct computation shows that the symbol of PE coincides with the symbol of

∆_∂_E. It follows thatP_E is transverse elliptic and that P_E and ∆_∂_E differ by a first order operator. Now considerPE,∆_∂_E as Fredholm operatorsL^2,2_B (E)→L²_B(E). The difference is a first order differential operator, which by basic Sobolev compactness is compact as an operator from L^2,2_B (E) to L²_B(E). The Fredholm operators PE and ∆∂E differ by a compact operator, so they have the same index. But ∆_∂_E is self-adjoint, so it has index

zero and hence P_E has index zero as well.

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Lemma 3.6([17] Lemma 7.2.4). LetE be a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h. The adjointP_E^∗ of theP-operator PE is given by

P_E^∗ = i

(n−1)!∗_B∂E∂ELⁿ⁻¹, where ∗_B denotes the basic Hodge star∗_B :∧^jH^∗ → ∧^2n−jH^∗.

Proof. Bearing in mind the discussion in Section 2.1 on computing the adjoint of a basic differential operator in the case of a taut Riemannian foliation, the computation of the adjoint P_E^∗ proceeds exactly as in the non-foliated setting [17, Lemma 7.2.4].

Lemma 3.7. For the trivial line bundle, we have Ker(P) =C and Im(P|_C^∞

B(X,R)) con- tains no basic functions of constant sign other than the zero function.

Proof. Both statements follow easily from the maximum principle, as shown in [17, Lemma

7.2.7].

Corollary 3.8.

(i) We havedim(Ker(P^∗)) = 1and every functionf ∈Ker(P^∗|_C^∞

B(X,R))has constant sign.

(ii) We have a direct sum decomposition C_B^∞(X,R) = Im(P|_C^∞

B(X,R))⊕R, where the second summand denotes constant real valued functions.

Proof. (i) From Lemmas 3.5 and 3.7, we have dim(Ker(P^∗)) =dim(Ker(P)) = 1. Now suppose that f ∈Ker(P^∗|_C^∞

B(X,R)) is positive at some points and negative at some other points. So there exists an interval [a₊, b₊]⊂R withb₊> a₊ >0 for which f⁻¹([a₊, b₊]) has positive measure. Let ϕ+ :R →R be a smooth non-negative function with support on the positive real axis and with ϕ|_[a₊_,b₊_]= 1. Let g₊=ϕ₊◦f :X →R. Then g₊ is a basic smooth function onX,g+ ≥0 everywhere and I+ =R

Xf·g+dvol_X >0. Similarly, since f is negative somewhere, we can find a smooth non-negative function ϕ− :R→ R with support on the negative real axis such that g−=ϕ−◦f :X →R is a basic smooth function on X, g− ≥ 0 and I− = R

Xf ·g−dvol_X < 0. Now setting g = I₊g−−I−g₊ we have that g is a basic smooth function, g ≥ 0 everywhere and R

Xf ·gdvolX = 0.

Since Ker(P^∗|_C^∞

B(X,R)) is 1-dimensional, this shows that g∈Ker(P^∗)^⊥=Im(P|_C^∞

B(X,R)).

But g has constant sign, so this contradicts Lemma 3.7. Lemma 3.7 also implies that Im(P|_C^∞

B(X,R))∩R={0} and this implies (ii), sincedim(Coker(P)) = 1.

Definition 3.9. LetX be compact oriented and transverse Hermitian foliated. Letg be the transverse Hermitian metric onX. We saygis(transverse) Gauduchonif∂∂(ωⁿ⁻¹) = 0.

Theorem 3.10. LetXbe a compact oriented, taut, transverse Hermitian foliated manifold of complex codimension n and let g be the transverse Hermitian metric on X. Then g can be conformally rescaled by a basic positive real valued smooth function such that the rescaled transverse metric g₀ is Gauduchon. If X is connected and n≥2, then g₀ is the unique transverse Gauduchon metric within its conformal class up to constant rescaling.

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Proof. This proof adapts [17, Theorem 1.2.4] to the foliated setting. If n = 1 there is nothing to show, so assume n≥2. Consider the following second order transverse elliptic differential operator

Q:C_B^∞(X)→ C_B^∞(X), Q(ϕ) =i∗_B∂∂(ωⁿ⁻¹ϕ).

If we can find a smooth basic function ϕ satisfying Q(ϕ) = 0 and which is everywhere positive, then g₀ = ϕⁿ⁻¹¹ g will be a transverse Gauduchon metric. By Lemma 3.6, we have Q = −(n−1)!P^∗. By Corollary 3.8, Ker(Q) is 1-dimensional, so if g0 exists then it is unique up to scale. Let ϕ₀ spanKer(Q). By Corollary 3.8 we may assume ϕ₀ ≥0.

It remains only to show that ϕ₀ is non-vanishing. This follows by applying the maximum principle in exactly the same manner as in the proof of [17, Theorem 1.2.4].

Definition 3.11. Let E be a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h. Let ∇_E be the associated Chern connection. The mean curvatureof E, denoted byKE ∈Ω⁰_B(End(E)) is defined asKE =iΛFE, whereFE is the curvature of ∇_E.

From the above definition it follows that

inF_E∧ωⁿ⁻¹ =K_Eωⁿ.

Lemma 3.12. Suppose that Xis a compact oriented, taut, transverse Hermitian foliation of complex codimension n with transverse Gauduchon metric. LetE be a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h. Then onΩ⁰(E)we have

∆_E(a) =P_E(a) +P_E^∗(a)−K_E(a).

Proof. This is a local computation, so it is essentially the same as the non-foliated setting,

see [17, Lemma 7.2.5].

Lemma 3.13. Suppose that Xis a compact oriented, taut, transverse Hermitian foliation of complex codimension n with transverse Gauduchon metric. LetE be a transverse holomorphic vector bundle equipped with a transverse Hermitian metric h and suppose that K_E = 0. Then (as operators on Ω⁰_B(E)) we have:

Ker(P_E) =Ker(P_E^∗) =Ker(∆_E) =Ker(d_E).

Proof. Clearly Ker(d_E) =Ker(∆_E). Ifd_E(a) = 0, then∂_E(a) = 0 and henceP_E(a) = 0, soKer(dE)⊆Ker(PE). On the other hand, if KE = 0, then by Lemma 3.12, we have

h∆_Ea, ai=hP_E(a), ai+ha, P_E(a)i=hP_E^∗(a), ai+ha, P_E^∗(a)i,

from which it follows that Ker(P_E)⊆Ker(∆_E) =Ker(d_E) andKer(P_E^∗)⊆Ker(∆_E) = Ker(d_E). It remains only to show thatKer(d_E)⊆Ker(P_E^∗). But this follows easily from

Lemma 3.6 and the fact that X is Gauduchon.

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3.2. Transverse resolution of singularities.

Definition 3.14. Let X be a foliated manifold with transverse complex structure. A subsetS⊆X is called atransverse analytic subvariety ofXif for everys∈S, there exists a foliated chart U =V ×W containing sfor which S∩U is the common zero locus of a finite number of basic holomorphic functions on U. SinceS is locally given in a foliated coordinate chartU =V×W as the pre-image underV×W →V of an analytic subvariety S_V ⊆V in the ordinary sense, we can for eachs∈S define thecodimension ofS atsto be the codimension of the corresponding point in SV (with codimension taken with respect to V). Clearly this does not depend on the choice of foliated chart. We then define the codimension of S to be the infimum over alls∈S of the codimension of S ats.

The collection of all transverse analytic subsets of X gives a topology on X (which is certainly not Hausdorff as it does not separate points which lie in the same leaf of the foliation). Using this topology, we may speak of irreducible transverse analytic subsets.

Note that our transverse analytic varieties are always taken to be reduced. That is to say that whenever we consider the zero locus of basic holomorphic functions, we are considering the underlying reduced analytic space structure on the zero locus.

Any local properties in complex analytic geometry can easily be extended to the setting of transverse analytic subvarieties. Thus for example we may speak of singular or non-singular transverse analytic subvarieties of X and if S ⊆ X is a transverse analytic subvariety we may speak of the singular locus Ssing ⊂ S, which is again a transverse analytic subvariety of X.

Theorem 3.15. Let X be a foliated manifold with transverse complex structure and let ι : Y → X be a transverse analytic subvariety. There exists a foliated manifold Ye with transverse complex structure and a proper map q:Ye →Y such that

(i) The compositionι◦q:Ye →X is a smooth map.

(ii) ι◦q is transverse holomorphic in the sense that there exists covers of Y˜ and X by foliated coordinate charts of the form U1 =V1×W1 ⊆Ye and U2 =V2×W2 ⊆X on whichι◦q is given by

V1×W1 3(v, w)7→(q1(v), q2(v, w))∈V2×W2, where q1 is holomorphic.

(iii) q : Ye → Y is an isomorphism of foliated manifolds with transverse holomorphic structure over the non-singular part of Y.

Proof. This is basically a consequence of the existence of a functorial resolution of singularities for analytic varieties [27]. Choose an open cover of X by foliated chartsU_α = Vα×Wα

ϕα

−→ X. The charts can be chosen so that the overlaps U_αβ = ϕ⁻¹_α (U_β) ⊆ Uα

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have the form U_αβ =V_αβ×W_αβ with V_αβ ⊆V_α,W_αβ ⊆W_α and the transition maps V_βα×W_βα=U_βα=ϕ⁻¹_β (U_α) ^ϕ^αβ^=ϕ

−1 α ◦ϕ_β

//ϕ⁻¹_α (U_β) =U_αβ =V_αβ×W_αβ

have the form

ϕ_αβ(u, v) = (f_αβ(u), g_αβ(u, v))

for some f_αβ :V_βα→V_αβ and someg_αβ :V_βα×W_βα→W_αβ, where thef_αβ are holomorphic.

Since Y is a transverse analytic subvariety, the image Y_α = ϕ⁻¹_α (Y ∩U_α) of Y in the chart Uα has the form Yα =π_α⁻¹(Zα), where Zα⊆Vα is an analytic subvariety of Vα and πα :Uα =Vα×Wα →Vα is the projection. By Hironaka’s resolution of singularities [11]

(see, eg [27, Theorem 2.0.1] for the case of analytic varieties), there exists a canonical desingularisation p_α: ˜Z_α→Z_α, where ˜Z_α is smooth,p_α is proper, bimeromorphic and an isomorphism over the non-singular locus of Zα. Set ˜Yα = ˜Zα×Wα and let qα : ˜Yα →Yα

be given by qα(˜z, w) = (pα(˜z), w), where ˜z∈Z˜α,w∈Wα. Observe that the composition ϕ⁻¹_α ◦ι◦q_α : ˜Z_α×W_α = ˜Y_α →U_α=V_α×W_α has the form

(˜z, w)7→(p_α(˜z), w).

In particular, ι◦qα satisfies (i), (ii) and (iii) (restricted to Uα). If we can show that the local desingularisations{q_α : ˜Y_α →Y_α}glue together on the overlaps of coordinate charts, we will have obtained our desired desingularisation q : ˜Y →Y. In fact, this is easily seen to follow from the functoriality property of the desingularisations ˜Zα →Z [27, Theorem 2.0.1 (3)], which shows that the ˜Z_α agree on overlaps. In more detail, this means that the change of coordinate maps f_αβ :Z_β∩V_βα →Z_α∩V_αβ lift to ˜f_αβ :p⁻¹_β (V_βα)→p⁻¹_α (V_αβ) satisfying an associativity condition on triple overlaps (by functoriality of the ˜fαβ). Let Y˜_αβ =p⁻¹_α (V_αβ)×W_αβ ⊆Z˜_α×W_α= ˜Y_α. We define transition maps ψ_αβ : ˜Y_βα→Y˜_αβ by

ψαβ(˜z, w) = ( ˜fαβ(˜z), gαβ(pβ(˜z, w)).

Then it is easy to check that the ψ_αβ satisfy the appropriate associativity condition on triple overlaps (because the ˜fαβ and the ϕαβ = (fαβ, gαβ) satisfy such conditions), hence allow us to glue the{q_α: ˜Y_α→Y_α}together to obtain the desired q: ˜Y →Y. 3.3. Transverse coherent sheaves and stability. LetX be a compact oriented, taut, transverse Hermitian foliation of complex codimensionnwith transverse Gauduchon metric g.

Definition 3.16. LetEbe a transverse holomorphic vector bundle of rankrwhich admits a transverse Hermitian metric h. We define the degree of E, denoted deg(E) to be the real number

deg(E) = i 2π

Z

X

tr(F_E)∧ωⁿ⁻¹∧χ,

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whereF_E is the curvature of the Chern connection associated toh. This is independent of the choice of transverse Hermitian metric, because tr(FE) is independent ofhup to a∂∂- exact term, which by the basic Stokes’ theorem and the Gauduchon property∂∂ωⁿ⁻¹= 0 does not alter the degree. The slope ofE, denoted µ(E) is defined by µ(E) = deg(E)/r.

Remark 3.17. Note that deg(E) depends on the choice of leafwise volume formχ. More- over, our argument that deg(E) is independent of the choice of transverse Hermitian metric only holds in the case that the foliation is taut, since otherwise when applying Stokes’ theorem there would be an additional term which in general can change the degree. Let us also point out that the degree of a transverse holomorphic vector bundleE is only defined when E admits a transverse Hermitian metric.

Definition 3.18. LetX be a foliated manifold with transverse complex structure and let Odenote the sheaf of basic holomorphic functions onX. A sheaf ofO-modulesF is called a transverse coherent sheaf if locally, F is given as the cokernel of a sheaf map O^p → O^q for somep, q.

By this definition, in a local foliated chartU =V×W ⊂X, a transverse coherent sheaf is the same thing as a coherent sheaf on V. In particular it follows that to any local property of coherent sheaves, there is a corresponding local property for transverse coherent sheaves. In particular, we may speak of torsion free, reflexive and locally free transverse coherent sheaves. It is easy to see that locally free transverse coherent sheaves correspond to transverse holomorphic vector bundles by taking the sheaf of basic holomorphic sections. To any transverse coherent sheaf F, we may associate a determinant det(F) which is a transverse holomorphic line bundle. The determinant det(F) is constructed exactly as in the non-foliated setting [14, Chapter V, §6]. Adapting the proofs in [14, Chapter V,

§5], we also find that a torsion-free (resp. reflexive) transverse coherent sheaf is locally free outside a transverse analytic subvariety of codimension at least 2 (resp. 3).

In order to define stability of transverse holomorphic vector bundles, we need to define the degree of transverse coherent subsheaves. However, there is a complication due to the fact that if F is a coherent subsheaf of a transverse holomorphic vector bundle E, then even if E admits a transverse Hermitian metric it is not at all clear whether the determinant line bundle det(F) associated to F admits a transverse Hermitian metric. Thus we can not simply define deg(F) to be deg(det(F)). We get around this problem using a foliated resolution of singularities.

Let E be a transverse holomorphic vector bundle of rankr which admits a transverse Hermitian metric. Let s be a positive integer less than r and let q : Gr(s, E) → X be the associated Grassmannian bundle of E whose fibre over x ∈ X is the Grassmannian Gr(s, r) of s-dimensional complex subspaces of Ex. We note that Gr(s, E) has a natural taut, Riemannian foliation. To see this, note that the vector bundle E is constructed by patching together local trivialisations over foliated charts of X such that the transition functions g_αβ are basic holomorphic. Then Gr(s, E) is just the associated Grassmannian