We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all generalized Deligne-Faltings log structures (in particular simple normal crossings divisor) from the simplest case with characteristic generated by N (essentially the smooth divisor case).

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We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors.As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics.Punctured Gromov-Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in upcoming work of the second author with

We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X → B with singular fibre over b 0 ∈ B yields a family M (X/B, β) → B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b 0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X b0 is a normal crossings union of two divisors. We exhibit our formulas in explicit examples. Contents 24 Part 2. Practice 34 5. Logarithmic modifications and transversal maps 34 6. Examples 47 References 62

We prove that the moduli space of stable logarithmic maps from logarithmic curves to a fixed target logarithmic scheme is a proper algebraic stack when the target scheme is projective with fine and saturated logarithmic structure. This was previously known only with further restrictions on the logarithmic structure of the target.