Oct 21, 2009

Brett Parker

**Introduction**

The category of exploded manifolds is useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, computing Gromov-Witten invariants using normal crossing degenerations, and degenerations appearing in tropical geometry such as Mikhalkin's higher dimensional pair of pants decomposition of projective hypersurfaces[16]. Moreover, it relates 'tropical' curve counts to counts of holomorphic curves in a significantly more general setting than toric manifolds.

This introductory section sketches how exploded manifolds are useful for studying holomorphic curves. The rest of this paper is dedicated to providing an introduction to exploded manifolds, including definitions and differential geometric properties. The cohomology of exploded manifolds is discussed in [20] and the papers [19], [22] and [21] explore the construction of Gromov-Witten invariants of exploded manifolds.

**Tropical Semiring and Exploded Semiring**

In this paper, the word 'tropical' is often in quotes because some objects which are called tropical are much more general than what is traditionally studied in tropical geometry. For instance, our tropical curves will only satisfy a balancing condition when certain topological conditions hold.

**Exploded Manifolds**

Each exploded manifold B has a tropical part, which can be thought of as a collection of convex polytopes, glued over faces using integral affine maps. Given a complex manifold M with normal crossing divisors, the explosion of M is an exploded manifold Exp1(M). Also, the tropical part of any holomorphic curve in ExplM is a tropical curve in ExplM.

**Stratified Structure**

A second principal functor is called the explosion functor. If M is a complex manifold with a collection N of transversely intersecting complex codimension 1 submanifolds, then the tropical part of Expl(M) has a vertex for each connected component. It also contains a face [0;1)2 for each intersection and an n-dimensional quadrant [0;1)n for each n-fold intersection.

**The Explosion Functor and Log Geometry**

The explosion functor can be viewed as a base change in the language of log geometry, which provides a language for describing some holomorphic exploded manifolds. The link between exploded Gromov-Witten invariants and log Gromov-Witten invariants is explored in the paper [23].

**Tangent Space and More**

In the cases where holomorphic curves in exploded manifolds are suitably tamed by a symplectic form, the moduli space of holomorphic curves is compact and carries a virtual fundamental class. This allows Gromov-Witten invariants to be defined within the exploded setting.

In conclusion, exploded Manifolds offer a substantial benefit in the calculation and definition of Gromov-Witten invariants. These invariants can be computed by considering tropical curves, proving significantly useful in a broad array of mathematical settings.

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