Pith Number

pith:NAIHL6D6

pith:2026:NAIHL6D6V5CL3AWVUN25D5OSRV

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Hirzebruch $\chi_{y}$-genus of compact almost K\"{a}hler manifold with negative sectional curvature

Pan Zhang, Teng Huang

If the Nijenhuis tensor is sufficiently small, then the Hirzebruch χ_y-genus of a closed almost Kähler manifold with negative sectional curvature has components satisfying (-1)^{n-p} χ_p(X) ≥ 1 for each p.

arxiv:2604.27423 v2 · 2026-04-30 · math.DG

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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 prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch χ_y-genus satisfy the inequality (-1)^{n-p}χ_p(X)≥1 for all p=0,1,⋯,n. In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies (-1)^n χ(X)≥n+1.

C2weakest assumption

The assumption that the Nijenhuis tensor is 'sufficiently small' (a qualitative rather than quantitative bound) so that the new L² estimates and refined vanishing theorem apply; the abstract gives no explicit threshold or verification that such smallness is compatible with negative sectional curvature on a closed manifold.

C3one line summary

For compact almost Kähler manifolds with negative sectional curvature and sufficiently small Nijenhuis tensor, the Hirzebruch χ_y-genus components satisfy (-1)^{n-p} χ_p(X) ≥ 1 for all p, implying the Hopf conjecture (-1)^n χ(X) ≥ n+1.

Receipt and verification

First computed	2026-05-26T01:03:31.410968Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

681075f87eaf44bd82d5a375d1f5d28d73445c3dd2c5f9b6e8f73dcddf6c5a70

Aliases

arxiv: 2604.27423 · arxiv_version: 2604.27423v2 · doi: 10.48550/arxiv.2604.27423 · pith_short_12: NAIHL6D6V5CL · pith_short_16: NAIHL6D6V5CL3AWV · pith_short_8: NAIHL6D6

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV \
  | 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: 681075f87eaf44bd82d5a375d1f5d28d73445c3dd2c5f9b6e8f73dcddf6c5a70

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b440d9db7201e04050cf428ca511f5bfe032eac61939679cea53411b288bf7b2",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-04-30T04:53:47Z",
    "title_canon_sha256": "460754da47fd28800642990a25ba625b1e7557fd9d29a21afb396ea5d871b7b9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.27423",
    "kind": "arxiv",
    "version": 2
  }
}