Excluded homeomorphism types for dual complexes of surfaces

Abstract

We study an obstruction to prescribing the dual complex of a strict semistable degeneration of an algebraic surface. In particular, we show that if $\Delta$ is a complex homeomorphic to a 2-dimensional manifold with negative Euler characteristic, then $\Delta$ is not the dual complex of any semistable degeneration. In fact, our theorem is somewhat more general and applies to some complexes homotopy equivalent to such a manifold. Our obstruction is provided by the theory of tropical complexes.