Exploded Fibrations

Abstract

Initiated by Gromov, the study of holomorphic curves in symplectic manifolds has been a powerfull tool in symplectic topology, however the moduli space of holomorphic curves is often very difficult to find. A common technique is to study the limiting behavior of holomorphic curves in a degenerating family of complex structures which corresponds to a kind of adiabatic limit. The category of exploded fibrations is an extension of the smooth category in which some of these degenerations can be described as smooth families. The first part of this paper is devoted to defining exploded fibrations and a slightly more specialized category of exploded torus fibrations. Later sections contain the transverse interesction theory for exploded fibrations and some examples of holomorphic curves in exploded torus fibrations, including a brief discussion of the relationship between tropical geometry and exploded torus fibrations. In the final section, the perturbation theory of holomorphic curves in exploded torus fibrations is sketched.