Smooth, Nonsymplectic Embeddings of Rational Balls in the Complex Projective Plane

Abstract

We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

On almost complex embeddings of rational homology balls

“…(2) By recent work of Gompf [8, Corollary 1.2] the existence of a topological embedding B p,q ⊂ CP 2 implies, that the (image of the) interior of B p,q is topologically isotopic to a Stein open subset U ⊂ CP 2 . By Corollary 2, in the case of the smooth embeddings B(k, m) ⊂ CP 2 of [11,9] the Stein structure which exists on int(B(k, m)) as a smoothing of a quotient singularity will never be homotopic to the Stein structure pulledback from U by the time-1 map of the isotopy. On the other hand, if the embedding B p,q ⊂ CP 2 is smooth and symplectic as when Condition (ES) is satisfied or, more generally, homotopic to an almost complex embedding, by previous work of Gompf [7,Theorem 2.1], after a smooth ambient isotopy the induced complex structure on B p,q makes it a holomorphically embedded Stein handlebody and determines the original Stein fillable contact structure on the boundary.…”

“…But the rational balls B(k, m) were shown [11,9] to be of the form B p,q with q 2 +9 not divisible by p for each k ≥ 0 and m ≥ 1, hence the statement follows.…”

“…Let p > q ≥ 1 be coprime integers and B p,q the Stein rational homology ball smoothing of the quotient singularity 1 p 2 (pq−1, 1). In [9] the authors extended work of Brendan Owens [11] by exhibiting a subfamily {B(k, m), k ≥ 0, m ≥ 1} ⊂ {B p,q } such that each B(k, m) smoothly embeds in the complex projective plane. Work of Evans-Smith [6] -based on Weimin Chen's adjunction formula for pseudoholomorphic curves in almost complex orbifolds [3] -implies that B(k, m) admits no symplectic embedding in CP 2 .…”

“…Work of Evans-Smith [6] -based on Weimin Chen's adjunction formula for pseudoholomorphic curves in almost complex orbifolds [3] -implies that B(k, m) admits no symplectic embedding in CP 2 . The purpose of this note is to show by elementary arguments independent of [6] that the smooth embeddings constructed in [11,9] are not homotopic to almost complex embeddings. In particular, we deduce that the rational balls B(k, m) admit no symplectic embedding in CP 2 without appealing to [6].…”

“…On the other hand, it is easy to find pairs (p, q) satisfying (2) of Theorem 1 but not (ES) (e.g. (10, 1), (17, 5) and (26,11)). Nevertheless, we were unable to find such a pair with the additional property that B p,q smoothly embeds in CP 2 .…”