The eigenvector inequality for algebraic Bochner tensors of El-Hasan, Li, Nienhaus, Petersen, Stanfield and Wink holds with the sharp constant 2 in complex dimension two
Abstract
In arXiv:2608.23778, El-Hasan, Li, Nienhaus, Petersen, Stanfield and Wink prove a Tachibana-type characterisation of complex projective space among Kähler–Einstein manifolds by a partial-positivity condition on the Calabi curvature operator. The constant in their condition comes from an estimate ab(SB)²+4ab(B(S))² ≤ c ab(B)²ab(S)² for an algebraic Bochner tensor B and an eigenvector S of its Calabi operator; they prove c ≤ 10/4 and conjecture (their Conjecture 5.2) that c=2 works in every complex dimension. We prove the conjecture in complex dimension two, with the constant 2, and show that the constant is attained. The mechanism is an identity the source does not state: ab(SB)²=8ab(B(S))² for every algebraic Bochner tensor on ℂ² and every SinS^(2,0), with no eigenvector hypothesis. It rests on the fact that the Calabi operator carries the five-dimensional space of algebraic Bochner tensors on ℂ² onto the trace-free real symmetric 3 × 3 matrices, which we write down explicitly without square roots; it fails at n=3 (the quadric on ℂ³ has ratio 4), and, by padding, in every higher dimension. For an eigenvector with eigenvalue τ the left-hand side is therefore exactly 12τ²ab(S)², and the conjecture becomes the eigenvalue bound 3τ² ≤ 2tr(C²) for trace-free symmetric 3 × 3 matrices, which we prove by a Cauchy–Schwarz argument in the Frobenius inner product. Equality holds exactly when the two remaining Calabi eigenvalues coincide, which is the case of the complex quadric Q²congmathbb(CP)¹ × mathbb(CP)¹; padding that example by zeros shows the constant 2 is attained in complex dimension three as well. Complex dimension n ≥ 3 remains open; a numerical search over the 27-dimensional space of algebraic Bochner tensors on ℂ³ found no ratio above 2. Every theorem and proposition below is verified in Lean 4 against Mathlib, with no compiled evaluation; the explicit real model and the spectral form of the equality case are stated in prose, and the numerical work is labelled as such.
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Source snapshot 2026-09-07 03:53 UTC
File fingerprint
978c8fc25c84e5ef79dd1b18ef4f81a76903bd990b26f4dc0492536614ea5d20
Claim ledger
Stated results
BS1known data2026-09-07
Compute-first gate: the frame-free |SB|² used here reduces to the paper's printed equation (2.2) for S diagonal in the frame, and Table 1's quadric cell at n = 3 (ratio (n+2)/n = 5/3) comes out exactly from the definitions
BS2known data2026-09-07
The complex quadric Q² ≅ ℂP¹ × ℂP¹ makes Conjecture 5.2 an equality at n = 2: an integral Bochner tensor and eigenvector with |SB|² + 4|B(S)|² = 384 = 2|B|²|S|²
BS3routine2026-09-07
Negative control: no constant below 2 works at n = 2 – any κ valid for all algebraic Bochner tensors on ℂ² and all eigenvectors satisfies 2 ≤ κ
BS4candidate2026-09-07
New identity in complex dimension two: |SB|² = 8|B(S)|² for EVERY algebraic Bochner tensor on ℂ² and EVERY S ∈ S^(2,0) (no eigenvector hypothesis); false from n = 3 on
BS5candidate2026-09-07
Conjecture 5.2 of arXiv:2608.23778v1 holds in complex dimension two, with the conjectured constant 2: |SB|² + 4|B(S)|² ≤ 2|B|²|S|² for every algebraic Bochner tensor on ℂ² and every eigenvector of its Calabi operator
BS6routine2026-09-07
Normal form and non-vacuity at n = 2: the algebraic Bochner tensors on ℂ² are exactly the 5-real-parameter family ofParams b p c, (b, p, c) ∈ ℝ × ℂ × ℂ
BS7candidate2026-09-07
The exact value and the equality case at n = 2: |SB|² + 4|B(S)|² = 12 τ² |S|² for an eigenvector with eigenvalue τ, so equality in Conjecture 5.2 holds if and only if 3τ² = 2|B_(αβ̄γδ̄)|²
BS8routine2026-09-07
The constant 2 is attained in complex dimension three as well: the n = 2 extremal padded by zeros gives equality 1536 = 2·384·2 on ℂ³
Provenance
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- Source context
- arXiv:2608.23778v1, *A characterization of complex projective space via the Calabi curvature operator*, by Hasan M. El-Hasan, Xiaolong Li, Jan Nienhaus, Peter Petersen, James Stanfield and Matthias Wink (submitted 24 Aug 2026, math.DG, v1 is the only version; the abs page was fetched live on 2026-09-07). Numbering below is from the compiled e-print — tectonic run on the authors' own main-3.tex from /backup/arxiv-src/papers/2608/2608.23778.gz,.aux read (addendum 54); the corpus copy is raw LaTeX with unresolved ref.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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