A counterexample to Lee's conjecture on pseudo-Einstein structures

@misc{yuxin2026counterexample,
  title = {{A counterexample to Lee's conjecture on pseudo-Einstein structures}},
  author = {Dong Yuxin and Wang Haohao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02718},
  eprint = {2611.02718},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02718},
}

TY  - UNPB
TI  - A counterexample to Lee's conjecture on pseudo-Einstein structures
AU  - Dong Yuxin
AU  - Wang Haohao
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02718
UR  - https://arxiv.org/abs/2611.02718
AB  - In 1988, J.~M.~Lee conjectured that a compact strictly pseudoconvex CR manifold admits a pseudo-Einstein contact form if and only if the real first Chern class of its holomorphic tangent bundle vanishes. The necessity of this condition was proved by Lee. We show that the condition is not sufficient: in every real dimension 2n+1≥ 5 there is a compact connected strictly pseudoconvex CR manifold M with c_1(Tone)=0 in H^2(M;Z) which admits no pseudo-Einstein contact form. The examples are boundaries of small strictly pseudoconvex neighborhoods of a section D≅ B with negative normal bundle in a PP^1-bundle PP(E_e)→ B, where B=C^n is a power of a curve of genus two and E_e is a non-split extension of cO_B by an ample line bundle L with K_B≅ L^2. More generally, if K_B≅ L^m, such a boundary admits a pseudo-Einstein contact form if and only if (m−1)e=0, where e∈ H^1(B,L) is the extension class; the obstruction is detected on the first infinitesimal neighborhood of D.
ER  -