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Chern classes of Levi non-degenerate CR manifolds

  • Yuya Takeuchi
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Abstract

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every n ≥ 2 and ⌈ n / 4 ⌉ ≤ m ≤ ⌊ n / 2 ⌋, there exists a (2 n + 1)-dimensional example of signature (⌊ n / 2 ⌋, ⌈ n / 2 ⌉) on which all Chern monomials with positive indices at most m and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking m = ⌈ n / 4 ⌉ gives examples with c1⌊ 3 n / 4 ⌋ ≠ 0. For n = 4 or n ≥ 6, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

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Cite

BibTeXRIS
BibTeX
@misc{takeuchi2026chern,
  title = {{Chern classes of Levi non-degenerate CR manifolds}},
  author = {Yuya Takeuchi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12785},
  eprint = {2610.12785},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12785},
}
RIS
TY  - UNPB
TI  - Chern classes of Levi non-degenerate CR manifolds
AU  - Yuya Takeuchi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12785
UR  - https://arxiv.org/abs/2610.12785
AB  - We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every n ≥ 2 and ⌈ n / 4 ⌉ ≤ m ≤ ⌊ n / 2 ⌋, there exists a (2 n + 1)-dimensional example of signature (⌊ n / 2 ⌋, ⌈ n / 2 ⌉) on which all Chern monomials with positive indices at most m and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking m = ⌈ n / 4 ⌉ gives examples with c_1^⌊ 3 n / 4 ⌋ ≠ 0. For n = 4 or n ≥ 6, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.
ER  -