May 16, 2007

Brett Parker

## 1. Introduction

A symplectic manifold is a manifold M_{2n} with a closed, maximally non-degenerate two-form ω, known as the symplectic form. Local coordinates of any such manifold (like x, y) ∈ R_{n} × R_{n} have the symplectic form appear as ∑ dx_{i}∧dy_{i}. This implies symplectic manifolds lack local invariants, hence their study is referred to as symplectic topology.

One primary tool in symplectic topology is the study of holomorphic curves. Given a symplectic manifold (M, ω), it is possible to pick a contractible almost complex structure, J on M, that is tamed by ω. This means that ω(v, Jv) > 0 for any nonzero vector v. With such a choice of J, a holomorphic curve is a function f : (S, j) → (M, J) from a Riemann surface S with a complex structure j so that df∘j = J∘df.

## 2. The Moduli Space of Holomorphic Curves

The moduli space of holomorphic curves includes families of holomorphic curves where a bubble forms. This change in the domain's topology cannot occur in a connected smooth family of maps, so if the behavior above is to be considered 'smooth', we need to redefine 'smooth'. This gives a hint that the smooth manifold category might not be suitable for the holomorphic curves theory.

## 3. Holomorphic Curves and Exploded Fibrations

A second reason for finding an extension of the smooth category with an optimal theory of holomorphic curves is that holomorphic curves, in general, are challenging to find in non-algebraic settings. Many techniques for finding holomorphic curve invariants involve the degeneration of the almost complex structure J making holomorphic curves easier to find in the limit. Here, we discuss two types of such degenerations which result in the breaking of a symplectic manifold into smaller, simpler pieces to compute holomorphic curve invariants.

## 4. Definition and Structure of an Exploded Fibrations

In section two of this paper, we will discuss the structure of exploded fibrations. In the third section, we will define the base's structure, and in the fourth section, we will explain how they fit together. The category of exploded fibrations has well-defined products, so it is possible to deal with degenerations that look like products of this type locally.

## 5. Intersection Theory in Exploded Fibrations

As shown in section seven, exploded fibrations have a good intersection theory, further establishing the relevance of exploded fibrations in the study of holomorphic curves and symplectic topology.

## 6. The Perturbation Theory of Holomorphic Curves in Exploded Fibrations

The eighth section of this paper provides a sketch of the perturbation theory of holomorphic curves in exploded fibrations, broadening the understanding of the role of holomorphic curves within the context of exploded fibrations.

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