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arxiv: 2604.08464 · v1 · submitted 2026-04-09 · 🧮 math.AG · math.CV

Formulae for indices of holomorphic foliations via reduction of singularities

Maycol Falla Luza , Percy Fern\'andez S\'anchez , David marin This is my paper

Pith reviewed 2026-05-10 17:06 UTC · model grok-4.3

classification 🧮 math.AG math.CV
keywords holomorphic foliationsreduction of singularitiesMilnor numberCamacho-Sad indexBaum-Bott indexsaddle-nodesdiscrepancy vectorseparatrices
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The pith

General formulas for Milnor numbers and classical indices of arbitrary holomorphic foliations are obtained by summing the contributions of saddle-nodes that appear during desingularization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit expressions for the discrepancy vector, Milnor and intrinsic Milnor numbers, and several indices along separatrices, including Camacho-Sad, Variation, Gómez-Mont-Seade-Verjovsky, and Baum-Bott. These formulas extend earlier results that were restricted to generalized curve foliations by isolating and adding the numerical effects of saddle-nodes in the resolved configuration. A sympathetic reader cares because the invariants classify foliation germs and recover known theorems of Brunella and others inside one numerical setting. The work also supplies characterizations of generalized curve foliations by index values and of second-type foliations by the discrepancy vector.

Core claim

We provide general expressions for the discrepancy vector, the Milnor and intrinsic Milnor numbers, and classical indices along a separatrix as Camacho-Sad, Variation, Gómez-Mont-Seade-Verjovsky and also the Baum-Bott index. These extensions require a careful analysis of the contributions of saddle-nodes arising in the desingularization process. As applications we recover results of Brunella and Cavalier-Lehmann as well as a related statement in a unified purely numerical framework, and we obtain intrinsic characterizations of generalized curve foliations in terms of indices and of second type foliations in terms of the discrepancy vector.

What carries the argument

The reduction of singularities of a holomorphic foliation germ together with the additive numerical contributions of each saddle-node in the resolved configuration.

If this is right

  • Results of Brunella and of Cavalier-Lehmann are recovered inside a single numerical framework.
  • Generalized curve foliations admit an intrinsic characterization purely in terms of their indices.
  • Foliations of the second type are characterized by a vanishing or specific form of the discrepancy vector.
  • The formulas apply to arbitrary foliations rather than only the generalized-curve subclass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same saddle-node accounting might simplify index calculations for foliations given by explicit polynomial vector fields.
  • The approach could connect to other numerical invariants studied in singularity theory of vector fields.
  • If the formulas remain valid after further blow-ups, they might yield recursive relations among indices at different resolution stages.

Load-bearing premise

The reduction of singularities exists for every holomorphic foliation germ and the contributions of all saddle-nodes can be isolated and summed using only the resolved configuration.

What would settle it

A concrete holomorphic foliation germ for which the index values computed from the new formulas disagree with independent direct computation or with previously known values for the same germ.

Figures

Figures reproduced from arXiv: 2604.08464 by David marin, Maycol Falla Luza, Percy Fern\'andez S\'anchez.

Figure 1
Figure 1. Figure 1: Symbolic illustration of the correspondence Bi ❀ B˜ i between di￾critical and isolated separatrices and the divisor Bˆ 0 = Bˆ 1 + Bˆ 2 + Bˆ 3. the strict transform of B does not pass through the centers pk, . . . , pn−1 of the remaining blow￾ups and consequently νpi (B) = 0 ≤ νpi (B˜) for i = k, . . . , n − 1. We consider the vectors SB = (0, . . . , 0, 1)T and SB˜ associated to the composition of blow-ups… view at source ↗
read the original abstract

We study numerical invariants associated with the reduction of singularities of holomorphic foliation germs on $(\mathbb{C}^2, 0)$. Building on our previous work on generalized curve foliations, we extend explicit formulas for several fundamental invariants to arbitrary foliations. In particular, we provide general expressions for the discrepancy vector, the Milnor and intrinsic Milnor numbers, and classical indices along a separatrix as Camacho-Sad, Variation, G\'omez-Mont-Seade-Verjovsky and also the Baum-Bott index. These extensions require a careful analysis of the contributions of saddle-nodes arising in the desingularization process. As applications, we recover results of Brunella and Cavalier-Lehmann, as well as a related statement appearing in [8], within a unified and purely numerical framework. Furthermore, we obtain intrinsic characterizations of generalized curve foliations in terms of indices and of second type foliations in terms of the discrepancy vector.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive general formulae for the discrepancy vector, Milnor numbers, intrinsic Milnor numbers, and several indices (Camacho-Sad, Variation, Gómez-Mont-Seade-Verjovsky, Baum-Bott) associated to holomorphic foliation germs in the plane. These are obtained by extending previous results for generalized curve foliations through a detailed accounting of the numerical contributions from saddle-nodes that arise in any reduction of singularities. Applications include recovering known results of Brunella and Cavalier-Lehmann and providing characterizations of generalized curve foliations and second-type foliations.

Significance. Should the derivations be correct, the work supplies a purely numerical method to compute these invariants for general foliations, unifying several classical results in a single framework. The extension beyond generalized curves via saddle-node analysis represents a meaningful advance in the study of holomorphic foliations and their invariants.

major comments (2)
  1. [§4] §4 (saddle-node analysis): the claim that local contributions from saddle-nodes (eigenvalues, multiplicity, position relative to the separatrix) sum additively to the discrepancy vector and indices without residual cross-terms is load-bearing for the general formulae, yet the argument appears to assume all saddle-nodes lie on the strict transform or that the resolution tree introduces no non-local interactions; an explicit verification for a configuration with an off-separatrix saddle-node is needed.
  2. [Theorem 5.1] Theorem 5.1 (Baum-Bott index formula): the expression obtained by summing saddle-node terms over the resolved configuration is presented as general, but it is unclear whether the proof accounts for possible dependence on the global structure of the exceptional divisor when multiple saddle-nodes are present; this directly affects the asserted extension from the generalized-curve case.
minor comments (2)
  1. [Abstract] The abstract refers to 'a related statement appearing in [8]' without identifying the reference; the bibliography entry for [8] should be expanded for clarity.
  2. [§2] Notation for the discrepancy vector is used throughout but recalled only by reference to prior work; a brief self-contained definition in §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. We address each major comment below with clarifications on the proof structure and indicate where revisions will be made to enhance clarity and provide additional verification.

read point-by-point responses

  1. Referee: [§4] §4 (saddle-node analysis): the claim that local contributions from saddle-nodes (eigenvalues, multiplicity, position relative to the separatrix) sum additively to the discrepancy vector and indices without residual cross-terms is load-bearing for the general formulae, yet the argument appears to assume all saddle-nodes lie on the strict transform or that the resolution tree introduces no non-local interactions; an explicit verification for a configuration with an off-separatrix saddle-node is needed.

    Authors: The additivity follows from the local nature of the normal forms in each blow-up chart: the discrepancy vector is defined componentwise on the exceptional divisor, and each saddle-node contributes only through its own multiplicity, eigenvalue ratio, and relative position in its chart. The resolution tree ensures that pullbacks of the defining 1-form localize without cross-interactions between distant nodes, as intersections are accounted for via the total transform. We acknowledge that an explicit off-separatrix example would strengthen the exposition. We will add such a verification (a foliation germ with a saddle-node not meeting the strict transform of the separatrix) in the revised §4, computing the invariants both via the formula and by direct resolution to confirm the absence of residual terms. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1 (Baum-Bott index formula): the expression obtained by summing saddle-node terms over the resolved configuration is presented as general, but it is unclear whether the proof accounts for possible dependence on the global structure of the exceptional divisor when multiple saddle-nodes are present; this directly affects the asserted extension from the generalized-curve case.

    Authors: The proof of Theorem 5.1 uses induction over the resolution sequence, where the Baum-Bott index is expressed as a sum of local contributions from each exceptional component and saddle-node; the global topology of the divisor enters only through the total intersection numbers already encoded in the discrepancy vector. Saddle-node terms depend solely on local data (eigenvalue and position relative to the component), so multiple nodes do not introduce dependence beyond what is already summed. This extends the generalized-curve case directly, as the extra terms are precisely the saddle-node corrections. To address the concern, we will insert a clarifying paragraph after the statement of Theorem 5.1 explaining the localization and add a brief remark on the independence from distant configuration. revision: partial

Circularity Check

0 steps flagged

No circularity: formulas derived constructively from standard resolution process

full rationale

The derivation proceeds by applying the known reduction-of-singularities theorem to a holomorphic foliation germ, then computing explicit additive contributions of each resolved singularity (including saddle-nodes) to the discrepancy vector, Milnor numbers, and listed indices. These contributions are obtained by local analytic data at each point of the exceptional divisor and summed along the resolution tree; the resulting expressions are therefore consequences of the geometry of the resolved configuration rather than re-statements of the input foliation or of fitted parameters. The cited prior work on generalized curves supplies background but is not invoked as a uniqueness theorem or load-bearing premise for the saddle-node analysis, which is performed afresh in the present manuscript. No equation is shown to equal its own defining data by construction, and the claims remain externally falsifiable by direct computation on explicit foliation examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a finite reduction of singularities for holomorphic foliation germs (standard in the field) and on the ability to isolate additive contributions from each resolved singularity type.

axioms (1)
  • domain assumption Every holomorphic foliation germ on (C^2,0) admits a finite sequence of blow-ups that resolves all singularities into elementary ones (including saddle-nodes).
    Invoked implicitly when the paper speaks of 'the desingularization process' and 'contributions of saddle-nodes arising in the desingularization'.

pith-pipeline@v0.9.0 · 5465 in / 1319 out tokens · 36939 ms · 2026-05-10T17:06:49.798278+00:00 · methodology

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Reference graph

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