On Stein rational balls smoothly but not symplectically embedded in CP2$\mathbb {CP}^2$

Abstract

We extend recent work of Brendan Owens by constructing a doubly infinite family of Stein rational homology balls which can be smoothly but not symplectically embedded in ℂℙ 2 . M S C ( 2 0 2 0 ) 57R40 (primary), 57K43, 57R17 (secondary)

Horizontal decompositions, II

“…Remark 1.5. In [11] we constructed infinitely many embeddings B p,q → CP 2 , extending a family of embeddings constructed by Owens [13]. It is not hard to check that our constructions implicitely used horizontal decompositions of CP 2 -in fact, those constructions led us to the discovery of horizontal decompositions.…”

“…Owens [13] showed that infinitely many B p,q 's admit smooth embeddings in CP 2 but by [2] they cannot be symplectically embedded. The authors extended Owens' family [11] and proved the non-existence of almost complex embeddings [10] without relying on [2]. The present paper is a natural continuation of [12], where we introduced certain handlebody decompositions that we call horizontal, classified the closed 4-manifolds with the simplest horizontal decompositions and in doing so we recovered infinitely many of the known smooth embeddings B p,q → CP 2 .This paper consists of two parts.…”

“…It is not hard to check that our constructions implicitely used horizontal decompositions of CP 2 -in fact, those constructions led us to the discovery of horizontal decompositions. At the end of [11,Section 3] we showed that all the balls shown to embed in CP 2 satisfy q 2 ≡ −1 mod p. In view of Remark 1.3, Theorem 1.4 implies that all B p,q 's known so far to embed in CP 2 belong to F 1 ∪ F 2 and that (1, h 1 , h 2 , 1)-decompositions of CP 2 induce infinitely many more embeddings of B p,q 's.…”