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Stable logarithmic maps to Deligne-Faltings pairs I. (English) Zbl 1311.14028
The goal of this important paper under review is to develop the relative Gromov-Witten theory from a logarithmic approach. This approach was proposed by B. Siebert “Gromov-Witten invariants in relative and singular cases”, Lecture given in the workshop on Algebraic Aspects of Mirror Symmetry, Universität Kaiserslautern, Germany, Jun 26 (2001)] in 2001, and the original idea was to use tropical geometry and probing the stack by standard log point. In this paper the author applies somewhat different methods. An upshot is the notion of marked graphs associated to log maps, which allows the author to define the right base log structure (the minimal log structure).
The main results of the paper are the following. Let \(X\) be a projective variety, and let \(\mathcal M_X\) be a rank-one Deligne-Faltings log structure. Denote by \(X^{\text{log}} = (X, \mathcal M_X)\) the corresponding log scheme. Denote by \(\mathcal K_{\Gamma}(X^{\text{log}})\) the category fibered over the category of schemes, which for any scheme \(T\) associates the groupoid of minimal stable log maps over \(T\) with numerical data \(\Gamma\). The author proves that \(\mathcal K_{\Gamma}(X^{\text{log}})\) is a proper Deligne-Mumford stack, and that the natural map by removing the log structures from minimal stable log maps is representable and finite (Theorem 1.2.1). Moreover, denote by \(\mathcal M_{K_{\Gamma}(X^{\text{log}})}\) the universal minimal log structure. The author further proves that the pair \((K_{\Gamma}(X^{\text{log}}), \mathcal M_{K_{\Gamma}(X^{\text{log}})})\) defines a category fibered over the category of fine and saturated log schemes, which for any fine and saturated log scheme associates the category of stable log maps over it (Theorem 1.2.3).
Around the same time M. Gross and B. Siebert [J. Am. Math. Soc. 26, No. 2, 451–510 (2013; Zbl 1281.14044)] worked out Siebert’s original approach, building on insights from tropical geometry. Several other approaches to the algebricity and boundedness of moduli of stable log maps have also been explored in recent years, including Kim’s logarithmic stable maps [B. Kim, Adv. Stud. Pure Math. 59, 167–200 (2010; Zbl 1216.14023)] and Parker’s theory of exploded manifolds [B. Parker, Adv. Math. 229, No. 6, 3256–3319 (2012; Zbl 1276.53092), Abh. Math. Semin. Univ. Hamb. 82, No. 1, 43–81 (2012; Zbl 1312.14130)].

MSC:
14H10 Families, moduli of curves (algebraic)
14D23 Stacks and moduli problems
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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