Pith Number

pith:JJV5FIZX

pith:2026:JJV5FIZXNWK4UIEOY2ROSL4EOE

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On The Ellipticity of Generalised Monge-Amp\`ere Equations on Vector Bundles

Gao Chen, Kartick Ghosh

When both manifold dimension and bundle rank are at least three, vector bundle versions of the Monge-Ampère, J, and dHYM equations fail to preserve ellipticity along continuity paths, but the σ₂-equation succeeds.

arxiv:2604.21273 v2 · 2026-04-23 · math.DG

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\pithnumber{JJV5FIZXNWK4UIEOY2ROSL4EOE}

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4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We show that when both the dimension of the manifold and the rank of the bundle are greater than or equal to three, these equations do not preserve ellipticity along continuity paths. However, the σ₂-equation does preserve ellipticity along continuity paths.

C2weakest assumption

The specific pointwise definitions of the vector bundle versions of the Monge-Ampère, J, dHYM, and σ_k equations, together with the precise form of the continuity paths and the notion of ellipticity at a point, as stated in the setup of the problem.

C3one line summary

When both manifold dimension and vector bundle rank are at least 3, the Monge-Ampère, J, dHYM, and σ_k equations on holomorphic vector bundles lose ellipticity along continuity paths, while the σ_2 equation preserves it.

Receipt and verification

First computed	2026-05-20T00:04:32.348010Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4a6bd2a3376d95ca208ec6a2e92f847125b2ff9a73863d6b43e0a67eb55a95ce

Aliases

arxiv: 2604.21273 · arxiv_version: 2604.21273v2 · doi: 10.48550/arxiv.2604.21273 · pith_short_12: JJV5FIZXNWK4 · pith_short_16: JJV5FIZXNWK4UIEO · pith_short_8: JJV5FIZX

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JJV5FIZXNWK4UIEOY2ROSL4EOE \
  | 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: 4a6bd2a3376d95ca208ec6a2e92f847125b2ff9a73863d6b43e0a67eb55a95ce

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "47af6ca488b3bb04518b40d71536c49107dcd302a486c64b1995bc83e9c19a37",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-04-23T04:33:16Z",
    "title_canon_sha256": "399ee55c592031290637a6a0bfa465f652064021ae85b480836a8386145a3cfa"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.21273",
    "kind": "arxiv",
    "version": 2
  }
}