  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} $.
