Pseudo-Einstein Structures on Links of Isolated Singularities

Abstract

We study Jack Lee's conjecture that, for a compact strictly pseudoconvex CR manifold $M$, the condition $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{R})$ implies the existence of a pseudo-Einstein structure. For links of isolated singularities, we construct counterexamples in every odd dimension at least three, with $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{Z})$ and $b_1(M)=0$. For isolated algebraic singularities of log canonical type and complex dimension at least three, we prove that the real Chern class condition is equivalent to $\mathbb{Q}$-Cartierness of the canonical divisor, log canonicity, and the existence of a pseudo-Einstein structure. Finally, we construct a Du Bois threefold singularity whose link satisfies the real Chern class condition but admits no pseudo-Einstein structure. The initial versions of the proofs were generated with AI assistance and were subsequently revised and verified by the authors.