On almost complex embeddings of rational homology balls

TL;DR

The paper proves that certain Stein rational homology 4-balls cannot be embedded almost complexly into the complex projective plane, highlighting limitations of such embeddings without relying on previous complex symplectic results.

Contribution

It provides elementary proofs that specific rational homology 4-balls do not admit almost complex embeddings into the complex projective plane, extending understanding of embedding restrictions.

Findings

None of the studied rational balls admit almost complex embeddings.

These rational balls do not embed symplectically into the complex projective plane.

Elementary arguments suffice to establish these embedding obstructions.

Abstract

We use elementary arguments to prove that none of the Stein rational homology 4-balls shown by the authors and Brendan Owens to embed smoothly but not symplectically in the complex projective plane admit such almost complex embeddings. In particular, we are able to show that those rational balls admit no symplectic embeddings in the complex projective plane without appealing to the work of Evans-Smith.