The dual complex and RC-invariant functors

Abstract

We extend the dual complex of a simple normal crossings divisor to a homotopy type invariant $\mathcal{D}(X)$ of arbitrary complex varieties, and we prove that it is contractible when $X$ is connected and of klt type. In fact, the dual complex belongs to a broader class of invariants, which we call RC (rationally connected)-invariant functors. We prove a rigidity theorem governing the behavior of such functors with respect to rationally connected fibrations.