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A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds

  • Jie Fu
  • Bin Wang
  • Shihao Wang
  • Zhiwei Zheng
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Abstract

Let X be a smooth compact irreducible hyperkähler manifold of dimension six. We prove that the second Betti number satisfies b2(X) ≤ 23. Furthermore, we show that the third cohomology vanishes whenever b2(X) ≥ 19, and that all odd cohomology vanishes when b2(X) = 23. If the second Chern class lies in the subalgebra generated by H2(X,ℚ), then b2(X) ∈ {5,8,23}. The proof uses the Looijenga--Lunts--Verbitsky decomposition, curvature inequalities for Segre classes, and the Hodge--Riemann bilinear relations on the projective cotangent bundle.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026sharp,
  title = {{A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds}},
  author = {Jie Fu and Bin Wang and Shihao Wang and Zhiwei Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15803},
  eprint = {2610.15803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15803},
}
RIS
TY  - UNPB
TI  - A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds
AU  - Jie Fu
AU  - Bin Wang
AU  - Shihao Wang
AU  - Zhiwei Zheng
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15803
UR  - https://arxiv.org/abs/2610.15803
AB  - Let X be a smooth compact irreducible hyperkähler manifold of dimension six. We prove that the second Betti number satisfies b_2(X) ≤ 23. Furthermore, we show that the third cohomology vanishes whenever b_2(X) ≥ 19, and that all odd cohomology vanishes when b_2(X) = 23. If the second Chern class lies in the subalgebra generated by H^2(X,ℚ), then b_2(X) ∈ {5,8,23}. The proof uses the Looijenga--Lunts--Verbitsky decomposition, curvature inequalities for Segre classes, and the Hodge--Riemann bilinear relations on the projective cotangent bundle.
ER  -