Pith Number

pith:PXGOMFCN

pith:2026:PXGOMFCNF42I3WTD5ZY7O52FJI

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Generic vector fields on isolated complex hypersurface germs

Diogo da Silva Machado, Jose Seade

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.

arxiv:2605.09210 v2 · 2026-05-09 · math.AG

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Claims

C1strongest claim

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, and that such extensions form an open dense subset.

C2weakest assumption

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

C3one line summary

For an isolated hypersurface singularity, generic holomorphic vector fields with isolated zeros have GSV-index at least 1 + (-1)^dim(V) * τ(V,0), with equality exactly when the field extends to a nondegenerate zero in ambient space.

Formal links

2 machine-checked theorem links

Cited by

4 papers in Pith

Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields

A Criteria of Weighted Homogeneity via Logarithmic Vector Fields

Receipt and verification

First computed	2026-07-03T01:17:21.230718Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7dcce6144d2f348dda63ee71f777454a18b87702e03ab245d1686245e2ecd4cc

Aliases

arxiv: 2605.09210 · arxiv_version: 2605.09210v2 · doi: 10.48550/arxiv.2605.09210 · pith_short_12: PXGOMFCNF42I · pith_short_16: PXGOMFCNF42I3WTD · pith_short_8: PXGOMFCN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PXGOMFCNF42I3WTD5ZY7O52FJI \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7dcce6144d2f348dda63ee71f777454a18b87702e03ab245d1686245e2ecd4cc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "90fafe085a4ea40702701d1fd1922b0ed33775cf60915c60508eba18f742d2b0",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-09T23:04:46Z",
    "title_canon_sha256": "31962ab357efb66f95775a4b6dbc37d7f7bd7c423c5cf535ac92145623419f17"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.09210",
    "kind": "arxiv",
    "version": 2
  }
}