# Simple embeddings of rational homology balls and antiflips

## TL;DR

This paper links simple embeddings of rational homology balls in 4-manifolds to the semi-stable minimal model program in algebraic geometry, revealing new constraints and constructions for such embeddings.

## Contribution

It establishes a connection between simple embeddings of rational homology balls and the MMP, providing a systematic way to find and classify these embeddings via antiflips.

## Key findings

- Simple embeddings are constrained for chains with self-intersection -2.
- Existence of infinitely many disjoint rational homology balls in certain neighborhoods.
- Construction of disjoint pairs of rational homology balls in blown-up manifolds and Milnor fibers.

## Abstract

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequence of ordinary blow-ups from $V$. Then we show that this simple embedding comes from the semi-stable minimal model program (MMP) for $3$-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded $B_{p,q}$'s in $V$ via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of $2$-spheres with self-intersections equal to $-2$. We also show that there are (infinitely many) pairs of disjoint $B_{p,q}$'s smoothly embedded in regular neighborhoods of (almost all) negative chains of $2$-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls $B_{p,q}$ embedded in blown-up rational homology balls $B_{n,a} # \bar{\mathbb{CP}^2}$ (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results in [Khodorovskiy-2014], [H. Park-J. Park-D. Shin-2016], [Owens-2017] by means of a uniform point of view.