Gromov-Witten theory of a locally conformally symplectic manifold

We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holomorphic sky catastrophes, an analogue for pseudo-holomorphic curves of sky catastrophes in dynamical systems originally discovered by Fuller. We are able to rule these out in some situations, particularly for certain $\lcs$ 4-folds, and as one application we show that in dimension 4 the classical Gromov non-squeezing theorem has certain $C ^{0} $ rigidity or persistence with respect to $\lcs$ deformations, this is one version of $\lcs$ non-squeezing a first result of its kind. In a different direction we study Gromov-Witten theory of the $\lcsm$ $C \times S ^{1} $ induced by a contact manifold $(C, \lambda)$, and show that the Gromov-Witten invariant (as defined here) counting certain elliptic curves in $C \times S ^{1} $ is identified with the classical Fuller index of the Reeb vector field $R ^{\lambda} $. This has some non-classical applications, and based on the story we develop, we give a kind of `holomorphic Seifert/Weinstein conjecture' which is a direct extension for some types of $\lcsm$'s of the classical Seifert/Weinstein conjecture. This is proved for $\lcs$ structures $C ^{\infty} $ nearby to the Hopf $\lcs$ structure on $S ^{2k+1} \times S ^{1} $.