Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

TL;DR

This paper develops a method to construct unbounded sequences of symplectic embeddings of rational homology balls into surfaces of general type, using flips, mutations, and almost toric structures, expanding understanding of symplectic embedding possibilities.

Contribution

It introduces a novel recipe for constructing unbounded symplectic embeddings of rational homology balls via flips and mutations, connecting algebraic and symplectic geometry.

Findings

Unbounded sequences of symplectic embeddings of $B_{p,q}$ are possible into surfaces of general type.

Flips can be interpreted as mutations of almost toric structures and symplectic form deformations.

The approach links Mori's flips with symplectic and almost toric geometry techniques.

Abstract

The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which $B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded $B_{p,q}$ into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urz\'ua, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.