This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov-Witten invariants.

Abstract. Log Gromov-Witten invariants have recently been defined separately by Gross and Siebert and Abramovich and Chen. This paper provides a dictionary between log geometry and holomorphic exploded manifolds in order to compare Gromov-Witten invariants defined using exploded manifolds or log schemes. The gluing formula for Gromov-Witten invariants of exploded manifolds suggests an approach to proving analogous gluing formulas for log Gromov-Witten invariants.

This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, de Rham cohomology, Poincaré duality, and integration along the fiber. The resulting de Rham cohomology theory of exploded manifolds is used in a separate paper [4] to define Gromov Witten invariants of exploded manifolds.

This paper constructs a Kuranishi structure for the moduli stack of holomorphic curves in exploded manifolds. To avoid some technicalities of abstract Kuranishi structures, we embed our Kuranishi structure inside a moduli stack of curves. The construction also works for the moduli stack of holomorphic curves in any compact symplectic manifold.