Turbulent Foliations: Holomorphic and 3-Manifold Models

• Turbulent foliations are a class of structures defined by closed meromorphic forms, featuring compact polar leaves and dense, accumulating leaves on complex tori.
• They are constructed via a combination of translation-invariant holomorphic forms and meromorphic contributions, yielding a rich moduli space and global dynamic implications.
• In 3-manifold topology, turbulization and spiraling introduce Reeb-type models that elucidate torus leaf behavior, holonomy, and criteria for tautness.

Searching arXiv for papers on turbulent foliations and related transverse structures. arxiv_search(query="turbulent holomorphic foliations compact complex tori transversely holomorphic Cartan geometry", max_results=5, sort_by="relevance") Turbulent foliations denote two distinct constructions in foliation theory. In the holomorphic setting of compact complex tori, a nonsingular turbulent foliation is defined by the kernel of globally defined closed meromorphic $1$-forms built from a meromorphic contribution on a quotient torus and a translation-invariant holomorphic contribution on the total space; in codimension one on a product of elliptic curves, such foliations have compact polar leaves, while every other leaf is dense and accumulates on each polar curve (Biswas et al., 2024). In codimension-one foliation theory on $3$-manifolds, turbulization denotes a local model on $T^2\times I$ obtained from Reeb-type holonomy, together with generalizations by spiraling, and it is used to analyze torus leaves, Reeblessness, and tautness (Caillat-Gibert, 2011).

1. Codimension-one holomorphic turbulent foliations on complex tori

Let $T$ be a compact complex torus of dimension two, for instance $T=E_1\times E_2$ with $E_i$ elliptic. A turbulent foliation $\Ff$ on $T$ is, by definition, the kernel of a globally defined, closed, meromorphic $1$-form

$$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$

where $3$0 is a holomorphic elliptic fibration, $3$1 is a meromorphic $3$2-form on $3$3 whose divisor of poles $3$4 is reduced of degree $3$5, and $3$6 is a nowhere-vanishing holomorphic $3$7-form along the fibers of $3$8 (Biswas et al., 2024).

Locally one may trivialize $3$9 by a coordinate $T^2\times I$0 with $T^2\times I$1, trivialize the base $T^2\times I$2 by $T^2\times I$3, and write

$T^2\times I$4

The leaves are then solutions of $T^2\times I$5, so $T^2\times I$6 is constant. Since $T^2\times I$7, $T^2\times I$8 never vanishes identically on a fiber of $T^2\times I$9, and by choosing $T$0 with reduced poles and zeros one checks that $T$1 has no zeros on $T$2. The resulting foliation is therefore nonsingular.

Its dynamics justify the term “turbulent.” The polar curves $T$3 are compact leaves, and every other leaf is dense and accumulates on each polar curve. This sharply contrasts with the linear case in Ghys’ codimension-one dichotomy on compact complex tori, where $T$4 for a global nonzero holomorphic $T$5-form $T$6, so the foliation is translation-invariant (Biswas et al., 2 Oct 2025).

2. Generic families on products of elliptic curves

For the generic construction, fix $T$7, write $T$8 and $T$9, both genus-$T=E_1\times E_2$0 curves, and form

$T=E_1\times E_2$1

Over $T=E_1\times E_2$2 there is a principal $T=E_1\times E_2$3-bundle $T=E_1\times E_2$4 whose fiber over $T=E_1\times E_2$5 is the set of triples $T=E_1\times E_2$6 such that $T=E_1\times E_2$7 is the unique, up to scale, meromorphic form with

$T=E_1\times E_2$8

$T=E_1\times E_2$9, and $E_i$0 rescales $E_i$1 (Biswas et al., 2024).

Pulling back to $E_i$2 via the projections $E_i$3, one sets

$E_i$4

The total parameter space of these nonsingular turbulent foliations is a $E_i$5-dimensional quasi-projective variety

$E_i$6

This parameter count is structurally important. It supplies the ambient moduli against which the dimension of transverse geometric data is compared in the non-existence theorem for nonsingular transversely complex-projective structures. A plausible implication is that the genericity statement is not a local accident of a particular meromorphic form, but a global feature of the family $E_i$7.

3. Nonsingular transversely projective structures and their obstruction

Theorem 5.4 states: if $E_i$8, then there exists a nonempty Zariski-open subset

$E_i$9

such that for every point $\Ff$0 the associated nonsingular turbulent foliation on $\Ff$1 admits no nonsingular transversely complex-projective structure. Equivalently, a generic turbulent foliation of polar degree $\Ff$2 on $\Ff$3 cannot be endowed with any nonsingular projective transverse atlas (Biswas et al., 2024).

The proof begins with uniqueness. If such a foliation $\Ff$4 admitted a transverse projective structure, then by cohomological obstruction theory it would be unique, because

$\Ff$5

One then uses the fact that $\Ff$6 acts by translations on $\Ff$7 preserving $\Ff$8, hence preserving any hypothetical $\Ff$9-bundle and section $T$0. The problem reduces to data over the base $T$1 together with an auxiliary term in

$T$2

A further constraint comes from the second fundamental form. The section $T$3 must satisfy that its second fundamental form vanishes at exactly $T$4 points of $T$5, namely the points $T$6. Imposing this vanishing-of-order condition cuts down the dimension: the space of all quadruples

$T$7

with the required conditions has dimension at most $T$8. Since the full moduli of turbulent foliations has dimension $T$9, the inequality $1$0, equivalently $1$1, yields the obstruction. The conclusion is a rigid-versus-moving mismatch. No monodromy or explicit residue calculation is needed beyond the cohomological dimension count and the Bott-connection uniqueness on an elliptic base.

Two complementary points delimit the result. First, these examples are the first known nonsingular, codimension-one holomorphic foliations on a projective surface without any nonsingular transversely complex projective structure (Biswas et al., 2024). Second, every turbulent foliation does carry a singular transversely complex projective structure, for instance one coming from the closed-form description $1$2 itself. A common misconception is therefore that the theorem excludes projective transverse structures altogether; it excludes the nonsingular ones in the stated generic range. The bound $1$3 is left open as a possible non-sharp threshold.

4. Higher-codimension turbulent foliations on compact complex tori

The paper “Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry” extends the codimension-one notion to arbitrary codimension by working with generating subbundles (Biswas et al., 2 Oct 2025). Let $1$4 be a compact complex torus of dimension $1$5, and let $1$6 be a holomorphic subbundle of rank $1$7, $1$8. Since $1$9, global holomorphic vector fields $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$0 span each fiber. With the evaluation map

$$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$1

one defines

$$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$2

The foliation is generating if $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$3. An involutive generating subbundle $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$4 is called a smooth turbulent foliation of codimension $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$5.

Equivalently, a codimension-$$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$6 turbulent foliation may be described by a holomorphic surjective submersion

$$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$7

where $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$8 is a compact complex torus of dimension $$\eta \;=\;\pi^*(\omega)\;+\;\beta \;\in\;H^0\bigl(T,\; \Omega^1_{T}(*D)\bigr),$$9, together with meromorphic $3$00-forms $3$01 on $3$02 whose poles lie on a normal-crossing divisor, and linearly independent translation-invariant holomorphic $3$03-forms $3$04 on $3$05 such that generically on a fiber of $3$06 the matrix $3$07 is invertible. One then sets

$3$08

Away from the pole divisor of the $3$09, each $3$10 is a closed holomorphic form whose kernel defines a codimension-one foliation, and their intersection is a codimension-$3$11 foliation.

A corresponding classification statement asserts that if $3$12 is a smooth turbulent foliation of codimension $3$13, then there is a factor torus $3$14 of dimension $3$15 and translation-invariant forms $3$16 on $3$17, together with closed meromorphic forms $3$18 on $3$19, such that

$3$20

where $3$21 is the quotient map. The proof proceeds by showing that the normal bundle $3$22 is generated by translation-invariant sections, hence trivial; dualizing gives a $3$23-dimensional space of closed meromorphic forms whose common kernel is $3$24, and integration produces the quotient torus $3$25.

This higher-codimension theory places Ghys’ codimension-one dichotomy inside a broader framework: every regular integrable generating subbundle is locally of the “kernel of closed forms” type pulled back from a lower-dimensional torus. Example 6.2 makes the construction explicit when

$3$26

with $3$27 elliptic and $3$28 a complementary torus of dimension $3$29.

5. Transversely holomorphic Cartan geometry

For a smooth turbulent foliation $3$30, the relevant transverse geometric framework is that of transversely holomorphic Cartan geometries (Biswas et al., 2 Oct 2025). Given a connected complex Lie group $3$31 and a complex Lie subgroup $3$32, a transversely holomorphic Cartan geometry of type $3$33 consists of a holomorphic principal $3$34-bundle $3$35 equipped with a flat partial connection $3$36 along the leaves of $3$37, the associated principal $3$38-bundle $3$39, and a holomorphic bundle map

$3$40

fitting into the exact diagram

$3$41

and generically an isomorphism. If $3$42 is everywhere invertible, one has a genuine Cartan geometry; otherwise one has a branched Cartan geometry, with branching divisor the zero locus of $3$43.

The main technical input is the vanishing statement: if $3$44 is an involutive generating subbundle with normal bundle $3$45, and $3$46 is any vector bundle on $3$47 of nonpositive maximal slope, then

$3$48

In particular,

$3$49

whenever $3$50 admits a holomorphic connection.

From this, the flatness and uniqueness theorem follows. For any transversely branched Cartan geometry of type $3$51 on a smooth turbulent foliation, the induced holomorphic connection on $3$52 has curvature valued in $3$53; by the vanishing lemma, that curvature must vanish, so the geometry is branchwise flat. Moreover, if $3$54 are two Cartan forms on the same underlying $3$55, then their difference lies in $3$56 and must vanish. Thus the Cartan form is unique.

The projective case is a special instance: when there is a flat transversely branched Cartan geometry of model $3$57, with $3$58 a Borel subgroup, the foliation acquires a possibly branched transversely projective structure. The theorem shows that on a turbulent foliation this structure, if it exists, is automatically flat and unique. The paper leaves open the problem of characterizing exactly which higher-codimension turbulent foliations admit a given flat Cartan structure of type $3$59.

6. Turbulization and spiraling in codimension-one $3$60-manifold foliations

In $3$61-manifold topology, “turbulization” refers to a different construction, centered on a foliation of $3$62 obtained from a Reeb-type model (Caillat-Gibert, 2011). In $3$63 with coordinates $3$64, the submersion

$3$65

has level sets defining a foliation by cylinders and paraboloids limiting on the vertical cylinder $3$66. Restricting to the solid torus $3$67 with $3$68, and deleting its core $3$69, yields a foliation on

$3$70

for which $3$71 is a leaf and $3$72 is everywhere transverse, carrying a circle-foliation. This foliation on $3$73 is the turbulization component, denoted $3$74; with a chosen transverse orientation one writes $3$75 or $3$76.

An equivalent suspension description uses

$3$77

with coordinates $3$78, and a strictly increasing diffeomorphism $3$79 satisfying $3$80 and $3$81. For each $3$82, one defines the suspension annulus

$3$83

Quotienting $3$84 yields a foliation isotopic to $3$85. The holonomy map along the transverse annulus $3$86 is exactly $3$87, so one also writes $3$88.

This local model is inserted into ambient foliated manifolds along a torus $3$89 carrying a circle-foliation. Removing a collar $3$90 and gluing in $3$91 turns $3$92 into a compact torus leaf bounding a genuine Reeb component. Generalized turbulization $3$93 replaces the circle-foliated annulus by an arbitrary foliation $3$94 of $3$95 whose boundary leaves are circles; if $3$96 is an irrational rotation, one obtains a Reebless, dense torus-foliation on the top boundary and all non-compact leaves of type $3$97. Spiraling $3$98 extends the construction to $3$99, where $T^2\times I$00 is a closed orientable surface of genus $T^2\times I$01, by inserting suspension along one annulus and infinitely many gluings along another so as to produce spiral-annuli winding into the top leaf.

These constructions organize several criteria and examples. Proposition 2.6 states that if a foliation carries a compact separating leaf, or if all boundary leaves carry the same transverse orientation, then the foliation cannot be taut. Proposition 3.2 states that any foliation admitting a torus leaf must contain either a $T^2\times I$02-component or a $T^2\times I$03-component in a neighborhood of that torus, unless the manifold is itself a product by $T^2\times I$04. Theorem 1.2 (4.19) states that on a manifold whose boundary is a union of torus leaves and has no interior Reeb annuli, the foliation is taut if and only if at least two boundary tori carry opposite transverse orientations; a bad orientation forces non-tautness. The examples include the classical Reeb foliation of $T^2\times I$05, a non-taut Reebless foliation $T^2\times I$06 on $T^2\times I$07, foliations on a mapping torus $T^2\times I$08 with good and bad orientations, and the Waldhausen manifold $T^2\times I$09, which admits both taut and non-taut Reebless foliations with a single torus boundary leaf.

The holomorphic and $T^2\times I$10-manifold usages of “turbulent foliation” are therefore distinct. In the former, turbulence is encoded by closed meromorphic forms on complex tori and by dense leaves accumulating on polar curves; in the latter, turbulization is a local Reeb-type or spiraling model governing torus-leaf neighborhoods, holonomy, and tautness. The shared terminology reflects a common emphasis on nontrivial transverse behavior, but the ambient categories, local models, and transverse geometric questions are different.