  Inspired by the log Gromov-Witten (or GW) theory of
Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log
J-holomorphic curve relative to a simple normal crossings symplectic divisor
defined in [FMZ1]. Every such moduli space is characterized by a second
homology class, genus, and contact data. For certain almost complex structures,
we show that the moduli space of stable log J-holomorphic curves of any fixed
type is compact and metrizable with respect to an enhancement of the Gromov
topology. In the case of smooth symplectic divisors, our compactification is
often smaller than the relative compactification and there is a projection map
from the latter onto the former. The latter is constructed via expanded
degenerations of the target. Our construction does not need any modification of
(or any extra structure on) the target. Unlike the classical moduli spaces of
stable maps, these log moduli spaces are often virtually singular. We describe
an explicit toric model for the normal cone (i.e. the space of gluing
parameters) to each stratum in terms of the defining combinatorial data of that
stratum. In [FT2], we introduce a natural set up for studying the deformation
theory of log (and relative) curves and obtain a logarithmic analog of the
space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming
paper, we will prove a gluing theorem for smoothing log curves in the normal
direction to each stratum. With some modifications to the theory of Kuranishi
spaces, the latter will allow us to construct a virtual fundamental class for
every such log moduli space and define relative GW invariants without any
restriction.
