Chern classes of Levi non-degenerate CR manifolds

Abstract

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n \geq 2$ and $\lceil n / 4 \rceil \leq m \leq \lfloor n / 2 \rfloor$, there exists a $(2 n + 1)$-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 $c_{1}^{\lfloor 3 n / 4 \rfloor} \neq 0$. For $n = 4$ or $n \geq 6$, 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.