Chern classes of Levi non-degenerate CR manifolds

Abstract

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n\ge 2$ and $\lceil n/4\rceil \le m\le \lfloor n/2\rfloor$, there exists a $(2n+1)$-dimensional example of signature $(\lfloor n/2\rfloor ,\lceil n/2\rceil )$ 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 $m=\lceil n/4\rceil$ gives examples with $c_1^{\lfloor 3n/4\rfloor }\ne 0$. For $n=4$ or $n\ge 6$, 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.

arXiv comment13 pages