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Tropical compactification and the Gromov-Witten theory of \(\mathbb {P}^1\). (English) Zbl 1391.14111
Tropical geometry is a systematic method to study algebraic geometry by using piecewise-linear shadows of solutions to polynomial equations. In particular, it provides combinatorially effective tools to approach various enumerative problems in classical algebraic geometry. The first important result in this field is the Tropical Correspodence Theorem of G. Mikhalkin [J. Am. Math. Soc. 18, No. 2, 313–377 (2005; Zbl 1092.14068)], between algebraic curves in \(\mathbb{P}^2\) and tropical curves in \(\mathbb{R}^2\). Later on, this correspondence was generalized also to higher dimensional toric varieties by T. Nishinou and B. Siebert [Duke Math. J. 135, No. 1, 1–51 (2006; Zbl 1105.14073)].
The paper under review establishes a new tropical correspondence theorem for genus \(0\) stable maps to \(\mathbb{P}^1\) relative to two points \(\{ 0,\infty \} \in \mathbb{P}^1\). The significance of the result is that it is established on the level of moduli spaces, providing a deeper connection between the classical and tropical moduli spaces for such maps.
Let \(\mathcal{M}(x^{+},x^{-})\) denote the moduli space of genus \(0\) stable maps to \(\mathbb{P}^1\), with ramification profiles over \(0\) and \(\infty\) specified by tuples of integers \(x^{+} = (x_1,\ldots,x_l) \in \mathbb{Z}^l_{> 0}\) and \(x^{-}=(x_{l+1},\ldots,x_{n}) \in \mathbb{Z}^{n-l}_{< 0}\) respectively, with the additional assumption that \(\sum_{i=1}^{n} x_i=0\). In the article it is shown that there is an embedding \(\mathcal{M}(x^{+},x^{-}) \hookrightarrow T\), where \(T\) is the dense torus orbit of the toric variety \(X_{\sigma}\) associated to the fan \(\sigma\) defined by the tropical moduli space \(\mathcal{\overline{M}^{\mathrm{trop}}}(x^{+},x^{-})\), such that the coarse moduli space \(\mathcal{\overline{M}}(x^{+},x^{-})\) is identified with the tropical compactification of the image of \(\mathcal{M}(x^{+},x^{-})\) in \(X_{\sigma}\). This result is build on various observations, in particular on results of the two papers [M. M. Kapranov, in: I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. 29–110 (1993; Zbl 0811.14043)] and [A. Gibney and D. Maclagan, Algebra Number Theory 4, No. 7, 855–885 (2010; Zbl 1210.14051)]. First, from the article of Kapranov, it follows that moduli space \(\mathcal{M}(0,n)\) of genus zero stable maps with \(n\)-marked points can be embedded into a toric variety X, given by a Chow quetient where the closure of the image gives the compactification \(\mathcal{\overline{M}}(0,n)\). Second, Gibney and Maclagan show that the tropicalization of the moduli space of stable maps under this embedding is given by a subfan of the toric fan for \(X\). The observation that the natural forgetful between \(\mathcal{\overline{M}}(x^{+},x^{-})\) and \(\mathcal{\overline{M}}(0,n)\) induces an isomorphism over the open set \(\mathcal{M}(x^{+},x^{-})\) leads us to an embedding of \(\mathcal{M}(x^{+},x^{-}) \hookrightarrow T\), where \(T\) is the dense torus orbit of the toric variety associated to the fan given by \(\mathcal{\overline{M}}^{\mathrm{trop}}(x^{+},x^{-})\). In the article it is then shown that the closure of \(\mathcal{M}(x^{+},x^{-})\) under this embedding can be identified with the coarse moduli space \(\mathcal{\overline{M}}(x^{+},x^{-})\). As an application of this result, a correspondence between tropical Hurwitz cycles and tropicalizations of classical Hurwitz cycles, as well as a tropical descendant correspondence for genus \(0\) relative invariants of \(\mathbb{P}^1\) is established.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
Full Text: DOI arXiv
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