86 research outputs found
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
Towards a Theory of Logarithmic GLSM Moduli Spaces
In this article, we establish foundations for a logarithmic compactification
of general GLSM moduli spaces via the theory of stable log maps. We then
illustrate our method via the key example of Witten's -spin class. In the
subsequent articles, we will push the technique to the general situation. One
novelty of our theory is that such a compactification admits two virtual
cycles, a usual virtual cycle and a "reduced virtual cycle". A key result of
this article is that the reduced virtual cycle in the -spin case equals to
the r-spin virtual cycle as defined using cosection localization by
Chang--Li--Li. The reduced virtual cycle has the advantage of being
-equivariant for a non-trivial -action. The
localization formula has a variety of applications such as computing higher
genus Gromov--Witten invariants of quintic threefolds and the class of the
locus of holomorphic differentials
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