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Logarithmic Gromov-Witten invariants. (English) Zbl 1281.14044
The paper under review develops the Gromov-Witten theory of logarithmically smooth schemes that occurs in the situations such as non-singular projective varieties containing a normal crossing divisor, semistable degenerations, toroidal pairs etc. The decomposition formula for the ordinary algebraic Gromov-Witten theory was developed by Jun Li using the stacks of expanded degeneration, in which the Gromov-Witten invariants of a target variety is expressed in terms of the relative Gromov-Witten invariants under its semistable degeneration into two components intersecting along a smooth divisor. Using the techniques of this paper this degeneration formula can be recovered and generalized (to the case of normal crossing divisors).
One of the key ingredients of the paper under review is the notion of basic stable log maps replacing the notion of stable maps in the ordinary Gromov-Witten theory. Let \(S\) be a fixed fine saturated log scheme. Another main ingredient is the relation to tropical geometry which gives a way to visualize the stable log maps by the families of tropical curves with values in a piecewise topological space naturally associated to \(X\). The three main results of the paper are as follows:
1) Let \(X\) be a fixed fine saturated log scheme over \(S\). Then the stack \(\mathcal{M}(X/S)\) of basic stable log maps to \(X\) over \(S\) is an algebraic log stack locally of finite type over \(S\), and the forgetful morphism \(\mathcal{M}(X/S)\to M(X/S)\) to the stack of ordinary stable maps is representable.
2) If the morphism between the underlying schemes of \(X\) and \(S\) is projective and \(\beta\) is the degree of the stable log maps satisfying certain finiteness property then \(\mathcal{M}(X/S,\beta)\) is proper over the underlying scheme of \(S\).
3) If furthermore \(X\) is smooth over \(S\) then there exists a virtual fundamental class \([[\mathcal{M}(X/S,\beta)]]\) that gives the corresponding log Gromov-Witten invariants with the expected properties.
The proofs use Olsson’s algebraic stack of fine log schemes, stable reduction, and Olsson’s log cotangent complex. At the end, the paper under review studies the relation to the approach of expanded degenerations.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D20 Algebraic moduli problems, moduli of vector bundles
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