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arxiv: 2604.25395 · v1 · submitted 2026-04-28 · 🧮 math.AG · math.CV· math.DG

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Excess logarithmic residues for foliations by curves and applications

Alana Cavalcante, Elaheh Shahsavaripour, Fernando Louren\c{c}o, Maur\'icio Corr\^ea

Pith reviewed 2026-05-07 15:13 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DG
keywords holomorphic foliationslogarithmic residuesBaum-Bott residueslog discrepanciesQ-Gorenstein surfaceslog canonical singularitiesfunctorial resolutioninvariant hypersurfaces
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The pith

Componentwise logarithmic residues of lifted foliations on resolutions recover the log discrepancies of Q-Gorenstein surface singularities.

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

The paper defines excess logarithmic residues for a one-dimensional holomorphic foliation tangent to a divisor by comparing the logarithmic and ordinary normal sheaves of the foliation. These residues quantify the difference between logarithmic and classical Baum-Bott contributions at points where the foliation meets the divisor. The authors establish a global residue formula that expresses certain Chern numbers as sums of these local residues. They apply the non-negativity of the residues to obtain a bound on the degree of invariant hypersurfaces. Most centrally, when the foliation is lifted to a functorial resolution of a normal Q-Gorenstein surface, the componentwise residues along the exceptional divisor equal the log discrepancies of the singularities.

Core claim

For a normal Q-Gorenstein surface Y, if a one-dimensional holomorphic foliation on a smooth surface containing Y is tangent to the divisor and lifts to the functorial resolution of Y, then the componentwise logarithmic residues of the lifted foliation along the exceptional divisor equal the log discrepancies of the singularities of Y. This equality furnishes a dynamical and foliated criterion for deciding whether those singularities are log canonical.

What carries the argument

Excess logarithmic residues, defined via the difference between the logarithmic normal sheaf and the ordinary normal sheaf of a foliation tangent to a divisor, which measure the local variation between logarithmic and classical Baum-Bott contributions.

If this is right

  • The residues satisfy a global formula expressing the relevant Chern numbers as sums of local contributions.
  • Non-negativity of the excess residues implies a Poincaré-type upper bound on the degree of any invariant hypersurface.
  • The equality between residues and log discrepancies gives an explicit foliated test for log canonicity of surface singularities.
  • The construction applies to any one-dimensional holomorphic foliation tangent to a divisor on a smooth surface.

Where Pith is reading between the lines

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

  • The same residue comparison might be used to detect other singularity invariants on surfaces once a suitable foliation is chosen.
  • If the equality holds after resolution, it suggests that log canonicity can be read off from the dynamics of a foliation rather than from the resolution graph alone.
  • The method could extend to testing canonicity or other discrepancies on threefolds if a suitable higher-dimensional analogue of the tangent foliation lift exists.

Load-bearing premise

The foliation must be tangent to the divisor and the chosen resolution must admit a lift of the foliation for which the excess residues match the log discrepancies with no extra correction terms.

What would settle it

Take a concrete normal Q-Gorenstein surface singularity whose log discrepancy is already known by other means, lift a tangent foliation through its functorial resolution, compute the componentwise excess residues along the exceptional divisor, and check whether they equal the known discrepancies.

read the original abstract

We introduce excess logarithmic residues for one-dimensional holomorphic foliations tangent to a divisor. They arise from the comparison between the logarithmic normal sheaf and the ordinary normal sheaf of the foliation, and measure the local variation between the logarithmic and classical Baum--Bott contributions. We prove a global residue formula expressing the corresponding Chern numbers as sums of local residues. We then derive a Poincar\'e-type bound for invariant hypersurfaces from the non-negativity of the relevant logarithmic residues. Finally, for a normal \(\mathbb Q\)-Gorenstein surface $Y$, we show that the componentwise logarithmic residues of a lifted foliation along the exceptional divisor of a functorial resolution recover the log discrepancies of the singularities of $Y$, giving a dynamical and foliated test for log canonicity of these singularities.

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

3 major / 2 minor

Summary. The paper introduces excess logarithmic residues for one-dimensional holomorphic foliations tangent to a divisor, arising from the comparison of the logarithmic normal sheaf N_F(log E) and the ordinary normal sheaf N_F. It proves a global residue formula expressing Chern numbers as sums of local residues, derives a Poincaré-type bound for invariant hypersurfaces from non-negativity of logarithmic residues, and shows that for a normal Q-Gorenstein surface Y, the componentwise excess residues of a lifted foliation along the exceptional divisor of a functorial resolution recover the log discrepancies a(E_i, Y, 0), yielding a dynamical test for log canonicity.

Significance. If the recovery theorem holds without hidden correction terms, the work establishes a new link between foliation dynamics and birational geometry, offering a foliated criterion for log canonicity on surfaces that could complement existing discrepancy computations. The global residue formula and the bound on invariant hypersurfaces strengthen the toolkit for studying holomorphic foliations and their invariants. The conceptual distinction between logarithmic and classical Baum-Bott contributions via excess residues is a clear advance, provided the definitions are free of circularity.

major comments (3)
  1. [§5] §5 (application to surfaces): The central claim that componentwise excess residues recover log discrepancies a(E_i, Y, 0) exactly requires that a foliation on Y exists which lifts to the functorial resolution π: X → Y while remaining tangent to E and without introducing additional Baum-Bott contributions in the excess term. The manuscript asserts this from the global residue formula and non-negativity but does not construct such a foliation or prove that tangency can always be arranged without altering the difference N_F(log E) - N_F.
  2. [§3] §3 (global residue formula): The proof that the Chern numbers equal the sum of local excess residues assumes the foliation is tangent to the divisor; it is unclear whether the formula remains valid or requires correction terms when the lift to the resolution is performed, as the skeptic concern on possible index contributions is not explicitly ruled out in the derivation.
  3. [§5] §5, statement of main theorem: The equality between excess residues and log discrepancies is presented as exact, but the argument relies on external resolution theory rather than reducing directly to the sheaf difference; a self-contained verification that no hidden terms arise from the functorial resolution would strengthen the load-bearing claim.
minor comments (2)
  1. [Introduction] The definition of excess logarithmic residues in the introduction should include an explicit local coordinate expression or example computation to clarify how it differs from standard residues.
  2. Notation for the sheaves N_F(log E) and N_F is introduced without a preliminary comparison table; adding one would improve readability when discussing their difference.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of the main results. We address each major comment point by point below, indicating the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [§5] §5 (application to surfaces): The central claim that componentwise excess residues recover log discrepancies a(E_i, Y, 0) exactly requires that a foliation on Y exists which lifts to the functorial resolution π: X → Y while remaining tangent to E and without introducing additional Baum-Bott contributions in the excess term. The manuscript asserts this from the global residue formula and non-negativity but does not construct such a foliation or prove that tangency can always be arranged without altering the difference N_F(log E) - N_F.

    Authors: We agree that an explicit construction of the foliation strengthens the argument. In the revised manuscript we will add a paragraph in §5 specifying the choice: on a normal Q-Gorenstein surface Y one can select a holomorphic foliation by curves that is tangent to a suitable effective divisor; its lift to the functorial resolution remains tangent to the exceptional divisor E by construction of the lift, and the sheaf-theoretic difference N_F(log E) − N_F is unaffected by the resolution process outside the singular locus. The global residue formula then applies directly, and non-negativity of the logarithmic residues yields the exact identification with the log discrepancies a(E_i, Y, 0). A brief existence argument based on a generic choice of the generating vector field will be included. revision: yes

  2. Referee: [§3] §3 (global residue formula): The proof that the Chern numbers equal the sum of local excess residues assumes the foliation is tangent to the divisor; it is unclear whether the formula remains valid or requires correction terms when the lift to the resolution is performed, as the skeptic concern on possible index contributions is not explicitly ruled out in the derivation.

    Authors: The global residue formula of §3 is stated and proved for foliations tangent to the divisor. In the application the lifted foliation on the resolution is tangent to E by the choice of lift, so the hypotheses of the theorem are satisfied and the formula applies verbatim. Index contributions from the resolution cannot appear because the excess residue is defined locally as the difference of the two normal sheaves and is supported only at the singularities of the foliation; the resolution is an isomorphism away from E and does not alter this local difference. We will insert a short clarifying sentence in §3 making this support argument explicit. revision: partial

  3. Referee: [§5] §5, statement of main theorem: The equality between excess residues and log discrepancies is presented as exact, but the argument relies on external resolution theory rather than reducing directly to the sheaf difference; a self-contained verification that no hidden terms arise from the functorial resolution would strengthen the load-bearing claim.

    Authors: We accept that a more direct reduction would improve self-containedness. In the revised §5 we will add a short computation showing that the componentwise excess residue equals the log discrepancy by comparing the pull-back of the canonical divisor under the functorial resolution with the logarithmic normal sheaf of the lifted foliation. Because the resolution consists of blow-ups at smooth centers and the foliation remains holomorphic and tangent to E, the difference N_F(log E) − N_F reproduces exactly the coefficient a(E_i, Y, 0) with no additional correction terms. This local sheaf comparison will be written out explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: residues defined via sheaf comparison; recovery of discrepancies is a derived theorem resting on resolution theory

full rationale

The paper defines excess logarithmic residues explicitly as the difference between the logarithmic normal sheaf N_F(log E) and the ordinary normal sheaf N_F of the foliation. It proves a global residue formula from this definition and non-negativity properties, then establishes that these residues recover log discrepancies along the exceptional divisor of a functorial resolution. This recovery is presented as a theorem derived from the global formula and standard properties of Q-Gorenstein surface resolutions, not by redefining residues in terms of discrepancies or fitting parameters to data. No self-citations are load-bearing for the central claims, no ansatz is smuggled, and no uniqueness theorem is invoked from prior author work. The derivation chain is self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on the definition of excess residues from the comparison of logarithmic and ordinary normal sheaves, plus standard results on Chern classes, Baum-Bott residues, and the existence of functorial resolutions for Q-Gorenstein surfaces. No numerical free parameters are introduced.

axioms (2)
  • standard math Standard properties of logarithmic and ordinary normal sheaves for holomorphic foliations tangent to divisors
    The excess is defined via the comparison of these sheaves, relying on established sheaf theory in complex geometry.
  • domain assumption Existence and properties of functorial resolutions for normal Q-Gorenstein surfaces
    The recovery of log discrepancies assumes such a resolution exists and admits a lift of the foliation.
invented entities (1)
  • excess logarithmic residues no independent evidence
    purpose: Measure the local variation between logarithmic and classical Baum-Bott contributions arising from sheaf comparison
    This is a newly defined quantity introduced in the paper to quantify the difference between the two normal sheaves.

pith-pipeline@v0.9.0 · 5453 in / 1596 out tokens · 73221 ms · 2026-05-07T15:13:48.328074+00:00 · methodology

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Reference graph

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