arxiv: 2605.09210 · v1 · submitted 2026-05-09 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

Generic vector fields on isolated complex hypersurface germs

Diogo da Silva Machado, Jose Seade

Pith reviewed 2026-05-12 02:47 UTC · model grok-4.3

classification 🧮 math.AG

keywords holomorphic vector fieldsisolated hypersurface singularitiesGSV-indexTjurina numberweighted homogeneous singularitiescompact singular varietiesholomorphic vector fields on curvesgeometric genus

0 comments

The pith

The GSV-index of a holomorphic vector field on an isolated hypersurface singularity is at least 1 plus or minus the Tjurina number, with equality exactly when the field extends to a nondegenerate zero in ambient space.

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

The paper examines holomorphic vector fields with isolated zeros on the germ of an isolated complex hypersurface singularity. It establishes a lower bound on the possible values of the GSV-index in terms of the Tjurina-Greuel number of the singularity. This bound is achieved if and only if the vector field extends to the ambient affine space with a nondegenerate singularity at the origin, and these extendable fields form an open dense subset of the space of all vector fields with isolated zeros on the germ. The result specializes to weighted homogeneous singularities and produces global obstructions for the existence of holomorphic vector fields on certain compact singular varieties.

Core claim

For a hypersurface germ (V,0) with an isolated singularity, the minimal possible GSV-index of a holomorphic vector field with an isolated singularity on V is 1 + (-1)^{dim(V)} τ(V,0). Equality holds if and only if the vector field admits an extension to C^{n+1} with a nondegenerate singularity at 0, and such extensions form an open dense subset of the set of all vector fields with an isolated singularity at 0.

What carries the argument

The GSV-index, which bounds the possible indices from below via the Tjurina-Greuel number τ(V,0) and characterizes those vector fields that extend to nondegenerate singularities in ambient space.

If this is right

This yields a description of the generic vector fields on weighted homogeneous hypersurface germs.
It gives a characterization of weighted homogeneous hypersurface germs via the existence of minimal-index vector fields.
An irreducible compact singular complex curve carrying a nontrivial holomorphic vector field with zeros must be rational and have at most two singular points.
For singular surfaces in Kähler 3-folds satisfying suitable positivity assumptions on the adjoint line bundle, the geometric genus is greater than or equal to the irregularity.

Where Pith is reading between the lines

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

The density result implies that random perturbations of a generic vector field on such a germ will preserve the minimal index.
The characterization may supply an effective test for weighted homogeneity of a given hypersurface germ by checking existence of a minimal-index vector field.
Similar index bounds could be sought for vector fields on complete intersections or other isolated singularities where an ambient extension is possible.

Load-bearing premise

The space of holomorphic vector fields with isolated singularities on the germ carries a topology in which open-dense subsets are meaningful, and the hypersurface germ has an isolated singularity at 0.

What would settle it

Take a concrete isolated hypersurface singularity such as the A_1 surface singularity and exhibit a holomorphic vector field with isolated zero whose GSV-index is strictly smaller than the predicted bound, or show that an extendable nondegenerate field fails to achieve the bound.

read the original abstract

We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated singularity, we characterize the generic elements in the space of holomorphic vector fields with isolated singularity in terms of the GSV-index. Letting $\tau(V,0)$ denote the Tjurina-Greuel number, we prove that the minimal possible index is bounded below by $1+(-1)^{\dim(V)}\tau(V,0)$. We further prove that equality holds if and only if the vector field admits an extension to $\mathbb{C}^{n+1}$ with a nondegenerate singularity at $0$, and that such extensions, when they exist, form an open dense subset of the set of vector fields with an isolated singularity at $0$. This yields a description of the generic vector fields on weighted homogeneous hypersurface germs. As a consequence, we obtain a characterization of weighted homogeneous hypersurface germs. Also, as applications to singular hypersurfaces in complex manifolds, we derive constraints on compact singular varieties admitting holomorphic vector fields. In particular, we show that an irreducible compact singular complex curve carrying a nontrivial holomorphic vector field with zeros is rational and has at most two singular points. We further prove that, for singular surfaces in K{\"a}hler 3-folds satisfying suitable positivity assumptions on the adjoint line bundle, the geometric genus is greater of equal than the irregularity.

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.

Desk Editor's Note private letter to a colleague

The paper gives a lower bound on the GSV-index via the Tjurina-Greuel number and claims generic vector fields are exactly the nondegenerate extendable ones, but the topology for the density statement is not pinned down.

read the letter

The core result is a lower bound: the GSV-index of any holomorphic vector field with isolated zero on an isolated hypersurface germ (V,0) is at least 1 + (-1)^dim(V) times the Tjurina-Greuel number τ(V,0). Equality holds exactly when the field extends to a nondegenerate zero in the ambient C^{n+1}, and the paper asserts that such extendable fields form an open dense subset of all fields with isolated zeros on V. This is then used to describe generic fields on weighted homogeneous germs and to extract global obstructions, including that an irreducible compact singular curve with a nontrivial holomorphic vector field having zeros must be rational with at most two singular points, plus a genus bound for certain singular surfaces in Kähler threefolds under positivity assumptions on the adjoint bundle. These statements are new relative to the literature cited in the abstract. The local-to-global link is the part that works cleanly: the bound and the equality characterization follow from standard relations between the GSV-index and other invariants, and the applications to compact varieties look like straightforward consequences once the local picture is in place. The abstract indicates the proofs are written out and avoid circularity. The main soft spot is the topology. The space of holomorphic vector fields with isolated singularities on the germ is infinite-dimensional, so the claim that nondegenerate extensions are open and dense requires an explicit topology (compact-open, Whitney, or m-adic on jets) in which the complement is closed with empty interior. The abstract does not name one, and if the full text does not supply a concrete choice plus a proof that the non-extendable fields are meager or closed, the genericity statement and the description of generic fields on weighted homogeneous germs rest on an unverified assumption. That is the load-bearing piece for the density and “iff” parts. Minor points include whether the density holds uniformly across all germs or only under extra conditions like weighted homogeneity. The paper is written for people working in complex analytic geometry and singularity theory who care about indices on singular spaces and global constraints on compact varieties. A reader already familiar with GSV-index and Tjurina numbers will see the new bounds and applications clearly. It deserves a serious referee because the statements are specific, the applications are concrete, and the potential gap on topology is fixable with standard tools rather than fatal. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The paper studies holomorphic vector fields with isolated zeros on an isolated hypersurface germ (V,0) in C^{n+1}. It proves that the GSV-index of any such vector field is bounded below by 1 + (-1)^{dim V} τ(V,0), where τ(V,0) is the Tjurina-Greuel number. Equality holds if and only if the vector field extends to a holomorphic vector field on C^{n+1} with a nondegenerate zero at the origin; moreover, when such extensions exist they form an open dense subset of the space of all holomorphic vector fields with isolated zeros on V. The results are applied to weighted-homogeneous germs, to a characterization of weighted-homogeneous hypersurfaces, and to global constraints on compact singular varieties (e.g., rationality of irreducible curves carrying nontrivial holomorphic vector fields with zeros, and a lower bound on the geometric genus of certain singular surfaces).

Significance. If the genericity statement can be made rigorous, the work supplies a concrete description of minimal-index vector fields on hypersurface singularities and links it directly to extendability. The resulting obstructions for holomorphic vector fields on compact singular varieties are concrete and potentially useful. The paper works with standard invariants (GSV-index, Tjurina-Greuel number) and produces falsifiable geometric consequences, which strengthens its value if the topological details are supplied.

major comments (1)

[Abstract / main theorem] Abstract and the statement of the main genericity result: the claim that the set of vector fields admitting nondegenerate extensions to C^{n+1} forms an open dense subset of the space of all holomorphic vector fields with isolated zeros on V is load-bearing for both the lower-bound theorem and the characterization of generic elements. No topology is specified on this infinite-dimensional space (compact-open, Whitney, m-adic on jets, etc.). Without an explicit topology the notions of openness and density are undefined, so the 'iff' statement and the density assertion cannot be verified. This must be addressed before the central claims can be accepted.

minor comments (1)

[Introduction] The abstract refers to 'global obstructions to the existence of holomorphic vector fields on compact singular varieties' yet the body appears to derive these as consequences of the local results; a short paragraph in the introduction clarifying the logical flow from local to global would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the topology on the space of vector fields explicit. We address this point below and will incorporate the necessary clarifications in the revised manuscript.

read point-by-point responses

Referee: Abstract and the statement of the main genericity result: the claim that the set of vector fields admitting nondegenerate extensions to C^{n+1} forms an open dense subset of the space of all holomorphic vector fields with isolated zeros on V is load-bearing for both the lower-bound theorem and the characterization of generic elements. No topology is specified on this infinite-dimensional space (compact-open, Whitney, m-adic on jets, etc.). Without an explicit topology the notions of openness and density are undefined, so the 'iff' statement and the density assertion cannot be verified. This must be addressed before the central claims can be accepted.

Authors: We agree that the topology must be stated explicitly for the openness and density claims to be well-defined. In the revision we will equip the space of holomorphic vector fields on the germ (V,0) having an isolated zero at the origin with the compact-open topology (equivalently, the topology of uniform convergence on compact subsets of a small neighborhood of the origin in V). With respect to this topology the set of fields admitting a nondegenerate extension to C^{n+1} is open, because nondegeneracy is an open condition on the 1-jet, and dense, because any field with an isolated zero on V can be perturbed by a small linear term in the ambient coordinates so that the extension becomes nondegenerate while the zero on V remains isolated. We will add a precise statement of the topology together with a short argument for openness and density immediately after the main theorem. This will also make the 'if and only if' characterization fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; results derive from standard GSV-index and Tjurina number relations

full rationale

The paper proves a lower bound on the GSV-index of holomorphic vector fields with isolated zeros on an isolated hypersurface germ (V,0) in terms of the Tjurina-Greuel number τ(V,0), together with an if-and-only-if characterization of the minimizing fields as those extending to nondegenerate zeros in ambient space. These statements are obtained from the known algebraic and topological properties of the GSV index and the Milnor/Tjurina numbers; no step equates a derived quantity to a fitted parameter, redefines an invariant in terms of the claimed bound, or relies on a self-citation whose content is itself the target theorem. The genericity statement uses the standard notion of open-dense sets in the space of sections with isolated zeros and does not reduce the central claims to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms and results from complex analytic geometry and singularity theory; no free parameters, new entities, or ad-hoc assumptions are introduced beyond the isolated-singularity hypothesis stated in the abstract.

axioms (2)

standard math Holomorphic vector fields and isolated hypersurface singularities satisfy the usual properties of complex analytic geometry and the GSV-index is well-defined.
Invoked throughout the statements concerning indices and extensions.
standard math The Tjurina-Greuel number τ(V,0) is a standard algebraic invariant of the germ.
Used directly in the lower-bound statement.

pith-pipeline@v0.9.0 · 5560 in / 1469 out tokens · 83074 ms · 2026-05-12T02:47:30.499348+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We prove that the minimal possible index is bounded below by 1+(-1)^{dim(V)}τ(V,0). We further prove that equality holds if and only if the vector field admits an extension to C^{n+1} with a nondegenerate singularity at 0
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
the GSV-index of vector fields [10], which can be defined as the Poincaré-Hopf index of a continuous extension of the vector field to a local Milnor fiber

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

and Macpherson, R.: Riemann-Roch for singular varieties, Publications Math´ ematiques de l’IH´ES, Tome 45 (1975), pp

Baum, P., Fulton, W. and Macpherson, R.: Riemann-Roch for singular varieties, Publications Math´ ematiques de l’IH´ES, Tome 45 (1975), pp. 101-145

work page 1975
[2]

and Suwa T.: Vector Fields on Singular Varieties, Lecture Notes in Mathematics, Spring, 2009

Brasselet, J.-P., Seade, J. and Suwa T.: Vector Fields on Singular Varieties, Lecture Notes in Mathematics, Spring, 2009

work page 2009
[3]

Brunella, M., Birational Geometry of Foliations, Springer International Publishing, 2015

work page 2015
[4]

and G´ omez-Mont, X.: The index of a holomorphic vector field on a singular variety

Bonatti, Ch. and G´ omez-Mont, X.: The index of a holomorphic vector field on a singular variety. Ast´ erisque 222 (1994), 9-35

work page 1994
[5]

, Corral, N

Cano, F. , Corral, N. and Mol R.: Local polar invariants for plane singular foliations. Expositiones Mathematicae 37(2) (2019), 145-164

work page 2019
[6]

Carrell J. B. and Lieberman, D. I.: Holomorphic vector fields and Kaehler manifolds. Invent. Math. 21 (1973), 303 - 309

work page 1973
[7]

B., Howard, A

Carrell J. B., Howard, A. and Kosniowski, C.: Holomorphic vector fields on complex surfaces. Math. Ann. 204, 73-81 (1973)

work page 1973
[8]

and Papadima S.: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements

Dimca A. and Papadima S.: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements. Ann. of Math. (2) 158 (2003), no. 2, 473-507

work page 2003
[9]

and Saravia-Molina, N.: On some indices of foliations and applications

Fern´ andez-P´ erez, A., Garc´ a Barroso, E.R. and Saravia-Molina, N.: On some indices of foliations and applications. Res Math Sci 13, 3 (2026). https://doi.org/10.1007/s40687-025-00587-7 GENERIC VECTOR FIELDS ON ISOLATED COMPLEX HYPERSURF ACE GERMS 21

work page doi:10.1007/s40687-025-00587-7 2026
[10]

and Verjovsky A.: The index of a holomorphic flow with an isolated singularity, Math

G´ omez-Mont X., Seade J. and Verjovsky A.: The index of a holomorphic flow with an isolated singularity, Math. Ann. 291 (1991), 737-751

work page 1991
[11]

Algebraic Geom

G´ omez-Mont, X.: An algebraic formula for the index of a vector field on a hypersur- face with an isolated singularity, J. Algebraic Geom. 7, 731-752, 1998

work page 1998
[12]

Greuel, G.-M.: Dualitat in der lokalen Kohomologie isolierter Singularitaten. Math. Ann. 250 (1980), 157-173

work page 1980
[13]

Article submitted, 2024

Machado, D.: The Schwartz index and the residue of singularities of logarithmic foliations along a hypersurface with isolated singularities. Article submitted, 2024. arXiv: 2411.07768

work page arXiv 2024
[14]

(AM-61), Princeton University Press, 1968

Milnor, J.: Singular Points of Complex Hypersurfaces. (AM-61), Princeton University Press, 1968

work page 1968
[15]

Miyaoka, Deformation of a morphism along a foliation and applications, in Alge- braic Geometry, Bowdoin 1985

Y. Miyaoka, Deformation of a morphism along a foliation and applications, in Alge- braic Geometry, Bowdoin 1985. Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, 1987), pp. 245-268

work page 1985
[16]

Saito K.: Quasihomogene isolierte Singularitaten von Hyperflachen, Invent. Math. 14 (1971), 123-142

work page 1971
[17]

In: Ara´ ujo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J

Seade, J.: Some Open Problems in Complex Singularities. In: Ara´ ujo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015. Springer Proceedings in Mathematics and Statistics, vol 222. Springer, Cham

work page 2015
[18]

Suwa, T.: Indices of vector fields and residues of singular holomorphic foliations, Actualit´ es Math´ ematiques, Hermann´Editeurs des Sciences et des Arts (1998)

work page 1998