Kontsevich–Kollár–Xu conjecture on dual boundary complexes of log Calabi–Yau varieties

A log Calabi–Yau variety is a smooth complex quasi-projective variety admitting a log compactification whose boundary is a simple normal crossing divisor and whose logarithmic canonical class is trivial. Its dual boundary complex is the dual complex of the irreducible components of the boundary, recording their intersections. Kontsevich–Kollár–Xu conjecture. The dual boundary complex of any log Calabi–Yau variety is a finite quotient of a sphere.

This conjecture predicts a strong topological restriction on dual boundary complexes, contrasting with the fact that arbitrary finite simplicial complexes can occur in general. The supplied text attributes the conjecture to M. Kontsevich and independently to J. Kollár and C. Xu, but gives no evidence of a resolution.