Stability for holomorphic spheres and Morse theory

Abstract

In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, !) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim → Holx0 (P 1 , X) ≃ 2 X. This limit will be viewed as a kind of stabilization of Holx0(P 1 , X). We conjecture that this stabil- ity property holds if and only if an evaluation map E : lim → Holx0(P 1 , X) → X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category (defined in (4)) of the symplectic action functional on the Z - cover of the loop space, ˜ LX, defined by the symplectic form, has a classifying space that realizes the homotopy type of ˜ LX. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen (7).