We introduce a new compactification of the space of relative stable maps.
This new method uses logarithmic geoemtry in the sense of Kato-Fontaine-Illusie
rather than the expanded degeneration. The underlying structure of our log
stable maps is stable in the usual sense.Comment: We changed the terminology "log stable maps" to "stable log maps". A
gap in the proof of compatibility with formal completions pointed out by
Professor Bernd Siebert has been fixed, see Section 2.7. Few minor changes in
Lemma 3.3.8 and the proof of Lemma 6.5.1 are mad