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

Abstract

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 $2n+1\ge 5$ 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 $D\cong B$ with negative normal bundle in a $\PP^1$-bundle $\PP(E_e)\to B$, 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 $K_B\cong L^2$. More generally, if $K_B\cong L^m$, such a boundary admits a pseudo-Einstein contact form if and only if $(m-1)e=0$, where $e\in H^1(B,L)$ is the extension class; the obstruction is detected on the first infinitesimal neighborhood of $D$.