1999/04/28 by Ralph L. Cohen, Cohen, Ralph L., John D. S. Jones +3 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.math/9904185
openalex publication_date 1999/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (W,omega) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operators, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim-> Holx0(P1,X) = Omega2 X. This limit will be viewed as a kind of stabilization of Holx0(P1,X). We conjecture that this stability holds if and only if an evaluation map E: lim-> Holx0(P1,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, L~X, defined by the symplectic form, has a classifying space that realizes the homotopy type of L~X. 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].