Log Structures on Generalized Semi-Stable Varieties

This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms.Comment: 47 pages, doubled side

Stable logarithmic maps to Deligne-Faltings pairs I

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