The relative symplectic cone and T2-fibrations

Abstract

In this note we introduce the notion of the relative symplectic cone C V M . As an application, we determine the symplectic cone C M of certain T 2 -fibrations. In particular, for some elliptic surfaces we verify the conjecture in [17]: If M underlies a minimal Kähler surface with|e · e > 0 and e · α > 0} for nonzero α ∈ H 2 (M ; R) and P 0 = {e ∈ H 2 (M ; R)|e · e > 0}.

Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

“…A complete description for the symplectic cone of all the 4-manifolds with b + = 1 was subsequently given in [21]. After that, several attempts in different directions were made, e.g., [22] and [8].…”

Stability and existence of surfaces in symplectic 4-manifolds with b +=1

“…The proof of this lemma follows exactly as the proof of Lemma A.2, [11], as the necessary perturbations occur in a neighborhood of a point on V 1 which is not contained in any other V j . Our conditions ensure that such a point exists.…”

“…In this section we wish to extend Theorem 2.13, [11], to the more complicated curve configurations of Def 1.1. The key to the proof of Theorem 2.13 is Lemma 2.14 therein, which provides for the existence of a curve in a given class A ∈ H 2 (M; Z) under certain restrictions on A.…”

“…For such a class, any element u ∈ U will have a point x 0 ∈ Σ such that u(x 0 ) ∈ V i . Therefore, the proof of Lemma A.1, [11], applies as written there, albeit with a different set of underlying almost complex structures. Then J A V is a set of second category in I.…”

“…Now consider only almost complex structures J ∈ G V to understand the behavior of U at points which have a point in the image off of V . The methods of the proof of Lemma A.1, [11] apply in this setting.…”