Excluded homeomorphism types for dual complexes of surfaces

TL;DR

This paper identifies topological obstructions, based on tropical complex theory, that prevent certain 2-dimensional complexes with negative Euler characteristic from being dual complexes of algebraic surface degenerations.

Contribution

It establishes a new topological obstruction criterion for dual complexes of algebraic surface degenerations using tropical complex theory.

Findings

Complexes homeomorphic to 2D manifolds with negative Euler characteristic cannot be dual complexes of semistable degenerations.

The obstruction applies to complexes homotopy equivalent to such manifolds.

Tropical complexes provide the theoretical framework for the obstruction.

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.