Transversely Holomorphic Cartan Geometries

• Transversely holomorphic Cartan geometries are structures that encode Cartan data on the normal directions to holomorphic foliations, defining a transverse model geometry.
• They employ a holomorphic principal H-bundle with a flat partial connection and a transverse Cartan morphism, with branched and generalized variants enhancing flexibility.
• Rigidity and flatness theorems on complex manifolds, such as rationally connected varieties, Calabi–Yau manifolds, and complex tori, ensure many of these geometries are uniquely determined and flat.

Transversely holomorphic Cartan geometry is the foliated analogue of a holomorphic Cartan geometry: the Cartan data is placed on the normal directions to a holomorphic foliation rather than on the whole tangent bundle. On a complex manifold $(M,\mathcal F)$, it is encoded by a holomorphic principal $H$-bundle with a flat partial connection along the leaf directions and a transverse Cartan morphism modeled on a homogeneous space $G/H$. Branched and generalized variants weaken the transverse isomorphism condition, while flatness yields a developing map that is constant on the leaves. In the compact complex setting, the subject is organized by Atiyah-bundle methods, partial holomorphic connections, and induced holomorphic connections on associated $G$-bundles, with strong rigidity on rationally connected manifolds, Calabi–Yau manifolds, and several classes of foliations on complex tori (Biswas et al., 2018, Biswas et al., 2021, Biswas et al., 2 Oct 2025).

1. Atiyah-bundle formulation and basic definitions

Let $X$ be a connected complex manifold and let

$\mathcal F \subset TX$

be a nonsingular holomorphic foliation, meaning that $\mathcal F$ is a holomorphic subbundle of $TX$ whose sheaf of holomorphic sections is closed under Lie bracket. Its normal bundle is

$N\mathcal F := TX/\mathcal F,$

with quotient map

$q:TX\longrightarrow N\mathcal F.$

There is a canonical flat holomorphic partial connection on $H$0 in directions tangent to $H$1, defined locally by

$H$2

for a local section $H$3 of $H$4, a local section $H$5 of $H$6, and a lift $H$7 of $H$8 to $H$9 (Biswas et al., 2018).

For a complex Lie group $G/H$0 with Lie algebra $G/H$1, and a holomorphic principal $G/H$2-bundle

$G/H$3

the Atiyah bundle is

$G/H$4

and the adjoint bundle is

$G/H$5

They fit into the Atiyah exact sequence

$G/H$6

Relative to the foliation, one defines

$G/H$7

and hence

$G/H$8

A partial holomorphic connection along $G/H$9 is a holomorphic splitting

$G$0

equivalently a holomorphic map

$G$1

splitting the inclusion of $G$2. Its curvature lies in

$G$3

and the partial connection is flat when this curvature vanishes (Biswas et al., 2018, Biswas et al., 2021).

Now let $G$4 be a connected complex Lie group, $G$5 a complex Lie subgroup, and let

$G$6

A transverse holomorphic Cartan geometry of type $G$7 on $G$8 is given by a holomorphic principal $G$9-bundle $X$0 with a flat partial connection $X$1 along $X$2, together with a holomorphic bundle map

$X$3

that is compatible with the natural inclusion

$X$4

and is an isomorphism. Equivalently, the geometry can be expressed by a $X$5-valued holomorphic $X$6-form on $X$7 that is $X$8-equivariant, restricts to the Maurer–Cartan form on fibers, vanishes on the lifted foliation, and induces the transverse isomorphism on the quotient (Biswas et al., 2021).

A persistent point of clarification is that this is not a Cartan geometry on $X$9 itself. The Cartan structure describes the geometry of the normal directions to the leaves. When the foliation is trivial, namely the foliation by points, the transverse theory reduces exactly to the ordinary theory of holomorphic Cartan geometries (Biswas et al., 2021).

2. Branched and generalized forms, curvature, and developing maps

The branched theory weakens the transverse isomorphism condition. A transversely branched holomorphic Cartan geometry of type $\mathcal F \subset TX$0 on $\mathcal F \subset TX$1 consists of a holomorphic principal $\mathcal F \subset TX$2-bundle $\mathcal F \subset TX$3, a flat partial connection $\mathcal F \subset TX$4 along $\mathcal F \subset TX$5, and a holomorphic morphism

$\mathcal F \subset TX$6

such that $\mathcal F \subset TX$7 is partial-connection preserving, is an isomorphism over a nonempty open set, and fits into the commutative exact-sequence diagram comparing

$\mathcal F \subset TX$8

with

$\mathcal F \subset TX$9

The failure locus is measured by the divisor of

$\mathcal F$0

called the branching divisor. If $\mathcal F$1 is an isomorphism everywhere, one obtains the unbranched transverse Cartan geometry; if it fails only on a divisor, the geometry is branched (Biswas et al., 2018).

A generalized transverse holomorphic Cartan geometry drops the requirement that the transverse dimension of the model equals $\mathcal F$2. In that setting, $\mathcal F$3 need not be generically an isomorphism, but the same formalism still yields a transverse developing map in the flat case (Biswas et al., 2021).

From the transverse Cartan data one constructs a holomorphic connection $\mathcal F$4 on the associated principal $\mathcal F$5-bundle $\mathcal F$6. Its curvature is purely transverse: $\mathcal F$7 The geometry is flat when this curvature vanishes identically. In that case the foliated manifold is locally modeled on the homogeneous space $\mathcal F$8, and on the universal cover there is a developing map

$\mathcal F$9

that is constant on leaves of $TX$0 and equivariant with respect to monodromy. If $TX$1 is simply connected, one obtains a holomorphic map

$TX$2

such that $TX$3 is surjective on a dense open set and

$TX$4

away from the branching locus (Biswas et al., 2018, Biswas et al., 2021).

The same local picture can be expressed as a transverse atlas: for a flat transverse geometry there are local submersions

$TX$5

whose fibers are the leaves of $TX$6, and on overlaps

$TX$7

For branched geometries, the $TX$8 are generically submersive (Biswas et al., 2 Oct 2025).

3. Principal models and constructions from holomorphic maps

Two model geometries recur throughout the theory. For transverse affine geometry one takes

$TX$9

so that $N\mathcal F := TX/\mathcal F,$0. A foliation admits a transversely branched holomorphic affine connection when it carries a branched Cartan geometry of this type; if flat, it is a transversely branched complex affine geometry. For transverse projective geometry one takes

$N\mathcal F := TX/\mathcal F,$1

where $N\mathcal F := TX/\mathcal F,$2 is the maximal parabolic stabilizing a point in $N\mathcal F := TX/\mathcal F,$3, so $N\mathcal F := TX/\mathcal F,$4. The flat case is termed a transversely branched complex projective geometry, equivalently a $N\mathcal F := TX/\mathcal F,$5-geometry (Biswas et al., 2018).

A basic source of examples is pullback from maps to homogeneous spaces. If

$N\mathcal F := TX/\mathcal F,$6

is holomorphic and $N\mathcal F := TX/\mathcal F,$7 is surjective on a dense open set, then $N\mathcal F := TX/\mathcal F,$8 defines a foliation on the open set where $N\mathcal F := TX/\mathcal F,$9 has maximal rank, and pulling back the standard Cartan geometry on $q:TX\longrightarrow N\mathcal F.$0 produces a transversely branched flat Cartan geometry on that foliation. The branching divisor is exactly the locus where $q:TX\longrightarrow N\mathcal F.$1 drops rank. Conversely, on a simply connected manifold, every flat transversely branched Cartan geometry arises from such a developing map (Biswas et al., 2018).

This construction underlies an existence theorem tied to algebraic dimension. If $q:TX\longrightarrow N\mathcal F.$2 is a compact complex manifold of algebraic dimension

$q:TX\longrightarrow N\mathcal F.$3

then, away from a closed analytic subset of positive codimension, $q:TX\longrightarrow N\mathcal F.$4 admits a nonsingular holomorphic foliation of complex codimension $q:TX\longrightarrow N\mathcal F.$5 endowed with a transversely flat branched complex projective geometry. The proof uses algebraic reduction together with a holomorphic map to $q:TX\longrightarrow N\mathcal F.$6 that is generically submersive (Biswas et al., 2018).

At codimension one there is a particularly explicit model. For

$q:TX\longrightarrow N\mathcal F.$7

a transverse Cartan geometry for a codimension-one foliation is simply a holomorphic $q:TX\longrightarrow N\mathcal F.$8-form $q:TX\longrightarrow N\mathcal F.$9 such that

$H$00

The geometry is flat exactly when

$H$01

the developing map is a primitive of $H$02, and the branching divisor is the zero divisor of $H$03 (Biswas et al., 2021).

4. Rigidity, flatness theorems, and degree obstructions

A central theme is that transverse holomorphic Cartan geometries are often forced to be flat on geometrically constrained compact manifolds. Let $H$04 be a smooth complex projective rationally connected variety, and let $H$05 be a nonsingular foliation on a Zariski open subset $H$06 whose complement has codimension at least two. Then any transversely branched holomorphic Cartan geometry on $H$07 is necessarily flat. Since $H$08 is simply connected, the flat geometry is given by a developing map

$H$09

with

$H$10

The survey formulation strengthens the statement to transverse generalized holomorphic Cartan geometries and adds that if $H$11 is a nontrivial affine algebraic variety, then there is no transverse branched geometry of that type; in particular, there is no transverse branched holomorphic affine connection (Biswas et al., 2018, Biswas et al., 2021).

There is an analogous result on simply connected Calabi–Yau manifolds. If $H$12 is a simply connected compact Kähler manifold with

$H$13

and $H$14 is a nonsingular foliation on a Zariski open subset $H$15, then any transversely branched holomorphic Cartan geometry on $H$16 is flat. In the survey version, for a simply connected Calabi–Yau manifold carrying a transverse generalized holomorphic Cartan geometry, flatness is asserted when $H$17 is simply connected or semisimple; if $H$18 is a nontrivial affine variety, there is no transverse branched geometry (Biswas et al., 2018, Biswas et al., 2021).

Compact Kähler manifolds also impose degree constraints. If $H$19 is compact Kähler and $H$20 admits a transversely branched Cartan geometry of type $H$21 with branching divisor $H$22, then

$H$23

For transversely branched affine geometry, $H$24 is self-dual as an $H$25-module, so

$H$26

and therefore

$H$27

Consequently, if $H$28, no transverse branched holomorphic affine connection exists; if $H$29, any transverse branched affine connection must have trivial branching divisor (Biswas et al., 2018, Biswas et al., 2021).

These results make precise a recurrent phenomenon: flexibility in the branched or generalized definitions does not eliminate global rigidity. Rather, on rationally connected or Calabi–Yau backgrounds, branching and transverse curvature are often forced to collapse to a flat geometry induced by a holomorphic map to a homogeneous model (Biswas et al., 2018).

5. Compact complex tori and smooth turbulent foliations

A recent rigidity theorem concerns transversely holomorphic Cartan geometries on a distinguished class of foliations on compact complex tori. Let

$H$30

be a compact complex torus. Its tangent bundle is trivial, and the global holomorphic vector fields identify with the fiber through the evaluation isomorphism

$H$31

For a holomorphic subbundle

$H$32

the subbundle is called generating if the global holomorphic vector fields that locally lie in $H$33 span all global vector fields: $H$34 A smooth turbulent foliation is a nonsingular holomorphic foliation whose tangent bundle is a generating subbundle of $H$35. When $H$36, this recovers Ghys’s turbulent foliations in codimension one (Biswas et al., 2 Oct 2025).

For such foliations, the main theorem is twofold. First, for a fixed holomorphic principal $H$37-bundle

$H$38

equipped with a fixed flat partial connection $H$39 along $H$40, there is at most one transversely branched holomorphic Cartan geometry of type $H$41 on $H$42 having $H$43 as underlying data. Second, every transversely branched holomorphic Cartan geometry of type $H$44 on $H$45 is flat. The statement applies to branched geometries as well as unbranched ones (Biswas et al., 2 Oct 2025).

The mechanism is a vanishing theorem controlled by the generating property. If $H$46 and $H$47 is a holomorphic vector bundle on $H$48 with

$H$49

then

$H$50

The proof shows that

$H$51

hence

$H$52

At the same time, the curvature of the induced holomorphic connection $H$53 on $H$54 satisfies

$H$55

so the curvature has nowhere to live and must vanish. Uniqueness follows because the difference of two transverse Cartan geometries with the same underlying $H$56 is a section of

$H$57

which also vanishes (Biswas et al., 2 Oct 2025).

In codimension one, flatness is automatic for formal reasons because

$H$58

when $H$59. The higher-codimension theorem is therefore the substantive extension. It generalizes the earlier codimension-one result of Biswas–Dumitrescu on uniqueness of transversely complex projective structures and places smooth turbulent foliations among the most rigid known carriers of transversely holomorphic Cartan geometry (Biswas et al., 2 Oct 2025).

6. Relation to ordinary holomorphic Cartan geometries

The transverse theory is best understood as a foliated extension of the ordinary theory. In the ordinary setting, a holomorphic Cartan geometry of type $H$60 on a complex manifold $H$61 consists of a holomorphic principal $H$62-bundle

$H$63

and a $H$64-valued holomorphic $H$65-form

$H$66

that is $H$67-equivariant, restricts on each fiber to the Maurer–Cartan form, and is a vector bundle isomorphism at every point. Its curvature is

$H$68

and flatness means local isomorphism with the homogeneous model $H$69 (Biswas et al., 2021).

Several rigidity results for ordinary holomorphic Cartan geometries provide context for the transverse theory. On compact Kähler manifolds bearing a holomorphic Cartan geometry, rational curves force a canonical reduction: the geometry drops to a lower-dimensional base, while the rationally connected fibers carry model geometries. This is described in terms of a holomorphic foliation by directions along which curvature vanishes, and every rational curve lies in a leaf (Biswas et al., 2010). On compact connected Kähler manifolds with nef tangent bundle, the existence of a holomorphic Cartan geometry forces the Demailly–Peternell–Schneider fibration, after a finite unramified Galois covering, to have rational homogeneous fibers $H$70, and the associated principal $H$71-bundle over the torus admits a flat holomorphic connection (Biswas et al., 2011). These are not transverse theorems, but they show that holomorphic Cartan geometry often organizes compact complex manifolds into homogeneous directions and flat torus directions.

A complementary torus rigidity result concerns genuine holomorphic Cartan geometries rather than foliated transverse ones. If $H$72 is a complex affine Lie group and $H$73 is a complex Lie subgroup, then every holomorphic Cartan geometry of type $H$74 on any complex torus is translation invariant. Here the key mechanism is the canonical flat connection on the associated principal $H$75-bundle, which lifts translations of the torus and preserves the full Cartan geometry (Biswas et al., 2017). The paper explicitly notes that it does not primarily discuss transversely holomorphic Cartan geometries; its significance here is comparative, since it exhibits a torus-based rigidity phenomenon parallel in spirit to the flatness theorem for smooth turbulent foliations (Biswas et al., 2017, Biswas et al., 2 Oct 2025).

The deformation theory of ordinary holomorphic Cartan geometries also runs through Atiyah-type complexes. For a holomorphic Cartan geometry $H$76, the infinitesimal deformations are parametrized by the hypercohomology of a two-term complex

$H$77

and, in the flat case, the natural forgetful map from infinitesimal deformations of the flat holomorphic Cartan geometry to infinitesimal deformations of the associated flat principal $H$78-bundle is an isomorphism (Biswas et al., 2022). The paper does not explicitly study transversely holomorphic Cartan geometries, but its formalism is very much in the same spirit: in both settings, the Atiyah sequence, induced connections, and hypercohomological control of infinitesimal structure are central (Biswas et al., 2022).

Transversely holomorphic Cartan geometry therefore sits between foliation theory and holomorphic Cartan geometry proper. Its core distinction is that the Cartan structure lives on $H$79 rather than on $H$80, yet many of the decisive tools—Atiyah exact sequences, adjoint bundles, curvature localization, developing maps, and bundle-theoretic flatness—are direct analogues of the ordinary theory (Biswas et al., 2018, Biswas et al., 2021).