On almost complex embeddings of rational homology balls

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.

Horizontal decompositions, I

“…In [16] we extend Owens' family of smooth embeddings to a two-parameter family {B (k, m)} ⊂ {B p,q } such that B (k, m) cannot be symplectically embedded in CP 2 . Moreover, in [15] we prove the non-existence of almost complex embeddings B (k, m) ⊂ CP 2 without relying on [5].…”