Bounds on Wahl singularities from symplectic topology

TL;DR

This paper establishes a new upper bound on the length of Wahl singularities in degenerations of minimal surfaces of general type, linking symplectic topology constraints to algebraic geometry.

Contribution

It proves a sharper bound on Wahl singularity length using symplectic embedding restrictions, improving previous bounds significantly.

Findings

Wahl singularity length $ ext{l} ext{ } ext{≤} ext{ } 4K^2 + 7

Symplectic embedding of rational homology balls is constrained, e.g., $p ext{ } ext{≤} ext{ } 12$ in quintic surfaces

Provides partial answers to symplectic versions of questions in algebraic geometry

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 $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). 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 $p \leq 12$, partially answering the symplectic version of a question of Kronheimer.