Minimality of symplectic fiber sums along spheres

Abstract

Abstract. In this note we complete the discussion begun in [24] concerning the minimality of symplectic fiber sums. We find that for fiber sums along spheres the minimality of the sum is determined by the cases discussed in [27] and one additional case: If X# V Y = Z# V CP 2 CP 2 with V CP 2 an embedded +4-sphere in class [V CP 2 ] = 2[H] ∈ H 2 (CP 2 , Z) and Z has at least 2 disjoint exceptional spheres E i each meeting the submanifold V Z ⊂ Z positively and transversely in a single point, then the fiber sum is not minimal.Key words. Symplectic manifolds; symplectic fiber sum; minimality.AMS subject classifications. 53D35, 53D45, 57R17.1. Introduction. The symplectic fiber sum has been used to great effect since its discovery to construct new symplectic manifolds with certain properties. In four dimensions this has focused on manifolds with symplectic Kodaira dimension 1 or 2. Moreover, it was shown in [18] and [12] that symplectic manifolds with symplectic Kodaira dimension −∞ are rational or ruled; these are rather well understood. Hence it is reasonable to ask whether this surgery can produce new manifolds of Kodaira dimension 0. As Kodaira dimension is defined on the minimal model of a symplectic manifold, it is first necessary to answer the question under what circumstances the symplectic sum produces a minimal manifold.This question has been researched for fiber sums along submanifolds of genus strictly greater than 0 by M. Usher, see [27]. In this note we complete the discussion for fiber sums along spheres.The symplectic fiber sum is a surgery on two symplectic manifolds X and Y , each containing a copy of a symplectic hypersurface V . The sum X# V Y is again a symplectic manifold. Section 2 provides a brief overview of the symplectic sum construction and minimality of symplectic manifolds. A minimal symplectic manifold contains no exceptional spheres, i.e. no embedded symplectic spheres of self-intersection −1. Furthermore, we review the main tool employed in this paper, namely the symplectic sum formula for Gromov-Witten invariants. Applying this formula in the genus 0 case will involve a detailed look at the behavior of relative curves in relation to the hypersurface V .Symplectic sums along spheres can be classified due to the following result by McDuff: Thm. 1.4 in [20] implies that one of the summands must be contained in the following list:or be a blow-up of one of these examples such that the exceptional curves do not intersect V Y . Section 3 provides a number of examples of manifolds which can be involved in a symplectic sum along a sphere. Emphasis is placed on the case of a −4-sphere as this provides the most intriguing examples.

Kodaira dimension of fiber sums along spheres

“…Hence the numbers n sy and n sm can differ. For example, if X is irrational ruled, then it was shown in [1], that every −4-sphere is produced from the 3 point blow-up of an exceptional sphere. Moreover, there is always a symplectic form making V X symplectic and admitting symplectic disjoint embedded exceptional spheres each transversely intersecting the hypersurface V X in a single positive point.…”

“…The minimality of symplectic fiber sums is described by the following Theorem: [1]). Let M be the symplectic fiber sum X# V Y of the symplectic manifolds (X, ω X ) and (Y, ω Y ) along an embedded symplectic surface V of genus g ≥ 0.…”

“…We will assume that this holds in Y . As described in [1], the symplectic sum along a sphere thus reduces to one of the following four cases for (Y, V Y ) (Thm. 1.4, [24], [6]):…”

“…Minimality of symplectic fiber sums in four-manifolds has been researched in [30] and [1] and precise criteria under which such a sum is non-minimal have been found. Symplectic Kodaira dimension is defined on the minimal model of a symplectic manifold and, given our understanding of the behavior of symplectic sums, it is of interest what the Kodaira dimension of a given symplectic sum is.…”

Lantern substitution and new symplectic 4-manifolds with $b_2{}^{+} = 3$

“…For the proof we refer the reader to [18] and [8]. See also the work of J. Dorfmeister [7,8] who gives a related criteria on symplectic minimality and how the symplectic Kodaira dimension changes under the rational blowdown along a symplectic −4 sphere (see also related work in [9]). Lemma 14.…”