Inspired by the log Gromov–Witten (or GW) theory of
Gross–Siebert/Abramovich–Chen, we introduce a geometric notion of
log–holomorphiccurve relative to a simple normal crossings symplectic divisor defined by
Tehrani–McLean–Zinger (2018). 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
–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 (ie the space of gluing parameters) to each
stratum in terms of the defining combinatorial data of that stratum. In an
earlier preprint, we introduced a natural set up for studying the deformation
theory of log (and relative) curves and obtained a logarithmic analogue 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.
Keywords
Gromov-Witten theory, normal crossing divisor,
pseudoholomorphic curves, relative compactification
