Bounds on Wahl singularities from symplectic topology

Abstract

Let X be a minimal surface of general type with positive geometric genus ($b_{+}>1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \le 4K^2+7$. This improves on the current best-known upper bound due to Lee ($\ell \le 400(K^2{)}^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$ embeds symplectically in a quintic surface, then $p\le 12$, partially answering the symplectic version of a question of Kronheimer.

34 pages. v2: minor referee's corrections and bibliographic updates. This version to appear in Algebraic Geometry