Stability for holomorphic spheres and Morse theory.

*(English)*Zbl 0981.57014
Grove, Karsten (ed.) et al., Geometry and topology, Aarhus. Proceedings of the conference on geometry and topology, Aarhus, Denmark, August 10-16, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 258, 87-106 (2000).

The authors’ abstract: We study the question of when does a closed, simply connected, integral symplectic manifold \((X,\omega)\) 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, \(\varinjlim \text{Hol}_{x_0} (\mathbb{P}^1,X) \simeq\Omega^2X\). This limit will be viewed as a kind of stabilization of \(\text{Hol}_{x_0} (\mathbb{P}^1,X)\). We conjecture that this stability property holds if and only if an evaluation map \(E:\varinjlim \text{Hol}_{x_0} (\mathbb{P}^1,X)\to X\) is a quasifibration. In this paper we prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category of the symplectic action functional on the \(\mathbb{Z}\)-cover of the loop space, \(\widetilde L X\), defined by the symplectic form, has a classifying space that realizes the homotopy type of \(\widetilde LX\). We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We 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”.

For the entire collection see [Zbl 0943.00055].

For the entire collection see [Zbl 0943.00055].

Reviewer: Stanislaw Janeczko (Warszawa)

##### MSC:

57R17 | Symplectic and contact topology in high or arbitrary dimension |

32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |

53D05 | Symplectic manifolds, general |