Isoresidual Foliation in Meromorphic Geometry

• Isoresidual Foliation is a residue-fixed decomposition of moduli spaces of meromorphic forms or affine surfaces where invariant residues determine discrete fibers or continuous leaves.
• It uses holomorphic residue maps and combinatorial models, such as decorated trees, to analyze monodromy, resonance, and covering degrees across different settings.
• Applications include computing fiber cardinalities, describing resonance hyperplane effects, and influencing birational geometry in moduli spaces like M₀,n.

Searching arXiv for papers on isoresidual foliation and closely related isoresidual fibrations. Isoresidual foliation denotes a residue-fixed decomposition of a moduli space of meromorphic geometric structures. In the genus-zero theory of meromorphic $1$-forms on $\mathbb{CP}^1$, it arises from the residual map on a stratum $\mathcal{H}(a,-b_1,\dots,-b_p)$, whose fibers consist of differentials with prescribed residues at the labeled poles; in special cases these fibers are finite and form what the source describes as “what one may call an isoresidual foliation” (Gendron et al., 2020). In the theory of meromorphic affine surfaces, the term is used explicitly for a holomorphic foliation whose leaves are the connected components of fibers of a map fixing both holonomy and projective residue data at integral poles (Apisa et al., 31 Jul 2025). Across these settings, the common principle is that residual invariants act as transverse parameters, while the corresponding isoresidual loci are the leaves or fibers.

1. Definition, scope, and dimensional regimes

In the most elementary genus-zero abelian setting, one fixes integers $a\ge 0$, $b_1,\dots,b_p\ge 1$ with

$a+2=\sum_{i=1}^p b_i,$

and considers the stratum

$\mathcal{H}(a,-b_1,\dots,-b_p)$

of meromorphic $1$-forms on $\mathbb{CP}^1$ with a unique zero of order $a$ and poles of orders $\mathbb{CP}^1$0 at labeled points. The residue theorem imposes

$\mathbb{CP}^1$1

so residues live in

$\mathbb{CP}^1$2

and the residual map

$\mathbb{CP}^1$3

is holomorphic and finite because both spaces have complex dimension $\mathbb{CP}^1$4 (Gendron et al., 2020).

This basic picture already displays a recurring dichotomy. In some strata, fixing residues yields zero-dimensional fibers, so the isoresidual foliation degenerates to a finite covering picture. In other strata, especially outside the genus-zero one-zero case, fixing residues leaves positive-dimensional loci. For primitive $\mathbb{CP}^1$5-differentials on $\mathbb{CP}^1$6, the general residue map

$\mathbb{CP}^1$7

has fibers that “can be thought of as leaves of an isoresidual foliation” when $\mathbb{CP}^1$8; in the special family

$\mathbb{CP}^1$9

the domain and target both have dimension $\mathcal{H}(a,-b_1,\dots,-b_p)$0, so the fibers are again zero-dimensional (Chen et al., 2 Oct 2025).

A different, explicitly foliated, framework appears for meromorphic affine surfaces. There the isoresidual foliation is defined on a locus of strata $\mathcal{H}(a,-b_1,\dots,-b_p)$1 with integral poles, and a leaf consists of affine surfaces with fixed holonomy character and fixed projective residue pattern at integral poles. The relevant map is

$\mathcal{H}(a,-b_1,\dots,-b_p)$2

and its fibers form a holomorphic foliation independent of the choice of the tree $\mathcal{H}(a,-b_1,\dots,-b_p)$3 (Apisa et al., 31 Jul 2025).

A common misconception is to treat “isoresidual foliation” as necessarily meaning a foliation by positive-dimensional complex manifolds. The sources show that this is not uniform. In genus-zero abelian and special $\mathcal{H}(a,-b_1,\dots,-b_p)$4-differential cases, the residue-fixed loci are generically finite; in the affine-surface setting, by contrast, the term refers to a genuine holomorphic foliation with tangent space identified cohomologically.

2. Genus-zero abelian case on $\mathcal{H}(a,-b_1,\dots,-b_p)$5

For $\mathcal{H}(a,-b_1,\dots,-b_p)$6, the residual map

$\mathcal{H}(a,-b_1,\dots,-b_p)$7

partitions the stratum into fibers $\mathcal{H}(a,-b_1,\dots,-b_p)$8. Above the complement of a distinguished hyperplane arrangement, the map is an unramified covering of degree

$\mathcal{H}(a,-b_1,\dots,-b_p)$9

so a generic residue vector has exactly $a\ge 0$0 preimages. A striking feature is that this degree is independent of the individual pole orders $a\ge 0$1; it depends only on $a\ge 0$2 and the number of poles $a\ge 0$3, subject to $a\ge 0$4 (Gendron et al., 2020).

The singular set in residue space is the resonance arrangement. For every non-empty proper subset $a\ge 0$5, the resonance hyperplane is

$a\ge 0$6

with $a\ge 0$7. Their union

$a\ge 0$8

is a central complex hyperplane arrangement. Geometrically, the condition $a\ge 0$9 is interpreted as a vanishing total residue for the pole set indexed by $b_1,\dots,b_p\ge 1$0, corresponding in flat geometry to degeneracies where certain saddle-connection periods vanish (Gendron et al., 2020).

The resonance arrangement therefore plays the role of singular base locus for the isoresidual decomposition. Over $b_1,\dots,b_p\ge 1$1, the residue-fixed sets are discrete fibers of constant cardinality. Over $b_1,\dots,b_p\ge 1$2, the fiber cardinality drops and the cover ramifies. For a fiber lying only on one resonance hyperplane $b_1,\dots,b_p\ge 1$3, the cardinality is

$b_1,\dots,b_p\ge 1$4

where $b_1,\dots,b_p\ge 1$5 and

$b_1,\dots,b_p\ge 1$6

This number is strictly smaller than the generic degree (Gendron et al., 2020).

An explicit example is $b_1,\dots,b_p\ge 1$7, where $b_1,\dots,b_p\ge 1$8 and $b_1,\dots,b_p\ge 1$9, so the generic degree is

$a+2=\sum_{i=1}^p b_i,$0

The resonance arrangement in $a+2=\sum_{i=1}^p b_i,$1 consists of three lines, and the paper describes how approaching one of them forces a saddle connection to shrink, producing a degenerate object in the WYSIWYG compactification; the fiber over the resonant point has size $a+2=\sum_{i=1}^p b_i,$2 (Gendron et al., 2020).

3. Monodromy, resonance, and combinatorial models

The topology of the genus-zero isoresidual fibration is encoded by monodromy around the resonance arrangement. Writing

$a+2=\sum_{i=1}^p b_i,$3

a generic fiber $a+2=\sum_{i=1}^p b_i,$4 carries a monodromy representation

$a+2=\sum_{i=1}^p b_i,$5

Loops $a+2=\sum_{i=1}^p b_i,$6 around resonance hyperplanes $a+2=\sum_{i=1}^p b_i,$7 generate $a+2=\sum_{i=1}^p b_i,$8, and the corresponding monodromy permutations generate the monodromy group $a+2=\sum_{i=1}^p b_i,$9 (Gendron et al., 2020).

For strata with at most three poles, the monodromy is computed explicitly. If $\mathcal{H}(a,-b_1,\dots,-b_p)$0, the residual map is an isomorphism, so the monodromy is trivial. If $\mathcal{H}(a,-b_1,\dots,-b_p)$1, then $\mathcal{H}(a,-b_1,\dots,-b_p)$2, and the monodromy group satisfies the following case distinction: $\mathcal{H}(a,-b_1,\dots,-b_p)$3 if $\mathcal{H}(a,-b_1,\dots,-b_p)$4; $\mathcal{H}(a,-b_1,\dots,-b_p)$5 embedded exotically into $\mathcal{H}(a,-b_1,\dots,-b_p)$6 if $\mathcal{H}(a,-b_1,\dots,-b_p)$7 and $\mathcal{H}(a,-b_1,\dots,-b_p)$8; $\mathcal{H}(a,-b_1,\dots,-b_p)$9 if $1$0 and $1$1 have the same parity; and $1$2 otherwise (Gendron et al., 2020).

For general strata, the monodromy generators $1$3 admit explicit cycle decompositions in terms of the resonance degree

$1$4

If $1$5 is a singleton, $1$6 decomposes into cycles of length $1$7. If both $1$8 and $1$9 are non-singletons, the cycle lengths are $\mathbb{CP}^1$0, with multiplicities expressed through factorial formulas. Relations among generators are governed by the combinatorics of partitions: commutation for secant partitions, commutators of order $\mathbb{CP}^1$1 in certain parallel cases, and even products of transpositions in the remaining cases (Gendron et al., 2020).

The technical model underlying these results is a bijection between real-residue fibers and decorated trees. A decorated tree is an embedded oriented tree in the $\mathbb{CP}^1$2-sphere with labeled vertices $\mathbb{CP}^1$3, together with prescribed half-edge counts and parity constraints. To a meromorphic $\mathbb{CP}^1$4-form with real residues, one associates such a tree by recording poles as vertices, saddle connections between pole domains as oriented edges, and horizontal trajectories from the unique zero to poles as half-edges. Conversely, compatible decorated trees reconstruct unique real-residue differentials in the non-resonant case, and with restrictions in the resonant case (Gendron et al., 2020).

This combinatorial description is not merely auxiliary. It gives the cover degree by counting compatible trees, and it models monodromy by a local surgery in which a short edge separating $\mathbb{CP}^1$5 from $\mathbb{CP}^1$6 is rotated while going around $\mathbb{CP}^1$7. The paper’s “isoresidual foliation” in genus zero is therefore simultaneously geometric, topological, and combinatorial.

4. Higher-order differentials and finite isoresidual covers

For primitive $\mathbb{CP}^1$8-differentials $\mathbb{CP}^1$9, the appropriate invariant is the $a$0-residue. If $a$1 is a pole whose order is a multiple of $a$2, suitable local normal forms define

$a$3

for zeros and poles whose order is not divisible by $a$4, the $a$5-residue is automatically zero. Unlike the abelian case, there is no residue theorem for $a$6: the sum of all $a$7-residues on a compact Riemann surface may be arbitrary (Chen et al., 2 Oct 2025).

In genus $a$8, the paper on finite isoresidual covers studies the special family

$a$9

for which

$\mathbb{CP}^1$00

Here the residue map

$\mathbb{CP}^1$01

is a ramified cover of its image of degree

$\mathbb{CP}^1$02

where

$\mathbb{CP}^1$03

and

$\mathbb{CP}^1$04

Thus the generic isoresidual fibers are again zero-dimensional, but now the degree formula involves the $\mathbb{CP}^1$05-factorial calculus specific to higher-order differentials (Chen et al., 2 Oct 2025).

When all poles have order exactly $\mathbb{CP}^1$06, the degree simplifies to

$\mathbb{CP}^1$07

where

$\mathbb{CP}^1$08

The same paper interprets this cover as a $\mathbb{CP}^1$09-isoresidual cover and shows that ramification is governed by a resonance stratification in residue space. Resonance is no longer linear in the residues themselves: one chooses $\mathbb{CP}^1$10-th roots $\mathbb{CP}^1$11 of the residues $\mathbb{CP}^1$12, lets

$\mathbb{CP}^1$13

and studies hyperplanes

$\mathbb{CP}^1$14

Equivalently, for each subset $\mathbb{CP}^1$15 one considers the homogeneous polynomial

$\mathbb{CP}^1$16

whose vanishing detects resonance (Chen et al., 2 Oct 2025).

The correction to generic fiber cardinality along a single resonance is measured by both $\mathbb{CP}^1$17 and the abelian number

$\mathbb{CP}^1$18

For arbitrary resonance patterns, the fiber cardinality is given by an inclusion–exclusion formula indexed by partitions into resonant subsets. This extends the abelian genus-zero picture to a setting where the resonance equations are polynomial rather than linear, and where the absence of a residue theorem changes the structure of the base space (Chen et al., 2 Oct 2025).

5. Meromorphic affine surfaces

The affine-surface setting gives the term “isoresidual foliation” its most literal meaning. A complex affine surface is described as a triple $\mathbb{CP}^1$19, where $\mathbb{CP}^1$20 is a compact Riemann surface, $\mathbb{CP}^1$21 is a finite set of cone points, and

$\mathbb{CP}^1$22

is a holomorphic connection on the cotangent bundle with at worst simple poles along $\mathbb{CP}^1$23. At a cone point $\mathbb{CP}^1$24, in a local coordinate $\mathbb{CP}^1$25,

$\mathbb{CP}^1$26

and $\mathbb{CP}^1$27. The cone order at $\mathbb{CP}^1$28 is $\mathbb{CP}^1$29, and the local holonomy is $\mathbb{CP}^1$30 (Apisa et al., 31 Jul 2025).

The isoholonomic foliation fixes the holonomy character

$\mathbb{CP}^1$31

The isoresidual foliation refines this by fixing projective residue data at integral poles. If

$\mathbb{CP}^1$32

is the set of integral poles, one chooses a tree of arcs

$\mathbb{CP}^1$33

from $\mathbb{CP}^1$34 to $\mathbb{CP}^1$35. There exists a nonzero flat meromorphic $\mathbb{CP}^1$36-form $\mathbb{CP}^1$37 on a neighborhood of $\mathbb{CP}^1$38, unique up to scalar, and the residues $\mathbb{CP}^1$39 define a projective residue vector

$\mathbb{CP}^1$40

This produces a holomorphic map

$\mathbb{CP}^1$41

On the locus of non-translation surfaces with some integral pole of nonzero residue, the fibers of

$\mathbb{CP}^1$42

define the isoresidual foliation (Apisa et al., 31 Jul 2025).

The central structural theorem states that

$\mathbb{CP}^1$43

is a holomorphic submersion on that locus, its fibers form a well-defined holomorphic foliation, this foliation does not depend on $\mathbb{CP}^1$44, and it descends to moduli space. Here the leaves are genuine positive-dimensional holomorphic submanifolds, not discrete fibers (Apisa et al., 31 Jul 2025).

Infinitesimally, the deformation theory is expressed by the two-term complex

$\mathbb{CP}^1$45

with

$\mathbb{CP}^1$46

The tangent space to an isoholonomic leaf is $\mathbb{CP}^1$47, where $\mathbb{CP}^1$48 is the derivative of the framing map. For the isoresidual foliation, one introduces the sheaf of translation vector fields

$\mathbb{CP}^1$49

and the tangent space to an isoresidual leaf is

$\mathbb{CP}^1$50

On the $\mathbb{CP}^1$51-holonomy locus, the paper further equips these leaves with a nondegenerate leafwise indefinite Hermitian metric arising from the cup product on

$\mathbb{CP}^1$52

This extends Veech’s metric from the isoholonomic setting to the residue-refined foliation (Apisa et al., 31 Jul 2025).

6. Relations to isoperiodic theory, quadratic residues, and birational geometry

Isoresidual and isoperiodic structures coincide in certain genus-zero situations and diverge sharply in higher genus. For the stratum $\mathbb{CP}^1$53 on elliptic curves, the natural foliation is isoperiodic rather than isoresidual: the unique double pole has residue $\mathbb{CP}^1$54, so residues do not parametrize the leaves. The paper explicitly states that for strata of meromorphic $\mathbb{CP}^1$55-forms on the Riemann sphere, isoperiodic foliation coincides with the isoresidual fibration defined by the residue vector, whereas in $\mathbb{CP}^1$56 the leaf geometry is governed by absolute periods and yields Loch Ness Monster leaves in the marked stratum and complex disks in the unmarked stratum (Faraco et al., 2023).

A complementary residue-based perspective appears for meromorphic quadratic differentials with poles of order exactly two. In that setting, a measured foliation with centers determines the real parts of the complex residues through

$\mathbb{CP}^1$57

where $\mathbb{CP}^1$58 is the transverse measure of a small loop around the pole and $\mathbb{CP}^1$59 is the residue in the normalized local form. Gupta and Wolf prove that, for fixed compatible complex residues, there exists a unique meromorphic quadratic differential realizing a given measured foliation with centers. For fixed residues, this yields a bijection between the corresponding differentials and the measured foliations with compatible loop measures. This suggests an isoresidual rigidity statement rather than a foliation theorem in the strict affine-surface sense (Gupta et al., 2016).

The genus-zero residue formalism also has birational consequences. For

$\mathbb{CP}^1$60

on $\mathbb{CP}^1$61, one has $\mathbb{CP}^1$62, and the residue map induces a rational map

$\mathbb{CP}^1$63

Its fibers are isoresidual subvarieties, and when $\mathbb{CP}^1$64 the general fibers are curves. After resolution by the multi-scale compactification, the class of a general fiber spans an extremal ray of the moving cone. In the specific case

$\mathbb{CP}^1$65

on $\mathbb{CP}^1$66, the resulting extremal moving curve has an orthogonal pseudo-effective face of rank $\mathbb{CP}^1$67, from which it follows that the pseudo-effective cone of $\mathbb{CP}^1$68 is not polyhedral for $\mathbb{CP}^1$69, and hence that $\mathbb{CP}^1$70 is not a Mori Dream Space for $\mathbb{CP}^1$71 (Mullane, 29 Oct 2025).

Taken together, these results show that “isoresidual foliation” is best understood as a family of residue-fixed structures rather than a single uniform object. In genus-zero abelian and special $\mathbb{CP}^1$72-differential strata, it is a finite-cover phenomenon with resonance, ramification, and monodromy. In meromorphic affine geometry, it is a holomorphic foliation with cohomological tangent model and a leafwise indefinite Hermitian metric. In higher-order and quadratic settings, fixed residues organize existence, uniqueness, and degeneration. The recurring invariant is the pole residue, but the geometric realization of the isoresidual condition depends decisively on the ambient moduli problem.