# zbMATH — the first resource for mathematics

Chow quotients of toric varieties as moduli of stable log maps. (English) Zbl 1301.14012
Let $$X$$ be a projective normal toric variety $$X$$ with defining torus $$T$$ and $$\iota: T_0\to T$$ a homomorphism from a subtorus of $$T$$. By regarding $$\iota$$ as an action, there is a natural map from the stack quotient $$T'=[T/T_0]$$ to Kollár’s Chow variety $$C(X)$$ of cycles of dimension and homology class equal to the cycles defined by orbits of points of $$T$$ (see [Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 32, Springer, Berlin, 1996; Zbl 0877.14012]). The Chow quotient of $$X/\!\!/T_0$$ is defined as the image of this map endowed with the reduced structure and in the case when $$\iota$$ is an embedding, it coincides with the Chow quotient defined by M. M. Kapranov, B. Sturmfels and A. V. Zelevinsky [Math. Ann. 290, No. 4, 643–655 (1991; Zbl 0762.14023)].
The aim of the present paper is to relate $$X/\!\!/T_0$$ with the moduli space of stable log maps introduced by Q. Chen [Ann. Math. (2) 180, No. 2, 455–521 (2014; Zbl 1311.14028)] and D. Abramovich and Q. Chen [Asian J. Math. 18, No. 3, 465–488 (2014; Zbl 1321.14025)], and independently by M. Gross and B. Siebert [J. Am. Math. Soc. 26, No. 2, 451–510 (2013; Zbl 1281.14044)].
By compactifying $$\iota$$, one gets a map $$f_{\iota}:\mathbb P^1\to X$$, which can be seen as a stable log map $$f_{\iota}:(\mathbb P^1,\mathcal M_{\mathbb P^1})\to (X, \mathcal M_X)$$, where the log structure $$\mathcal M_{\mathbb P^1}$$ of $$\mathbb P^1$$ is given by the two markings $$\{0,\infty\}$$ and the log structure $$\mathcal M_X$$ of $$X$$ is given by the boundary $$X\setminus T$$. By fixing the curve class $$\beta_0$$ of the stable map $$f_{\iota}$$ and the contact orders $$c_0$$ and $$c_{\infty}$$ of $$0$$ and $$\infty$$ with respect to the toric boundary $$X\setminus T$$, there is a proper moduli stack $$\mathfrak K_{\Gamma_0}(X)$$ parametrizing stable log maps to $$X$$ with discrete data $$\Gamma_0=(0,\beta_0,2,\{c_o,c_\infty\})$$. The authors’s main result is that the normalisation of $$X/\!\!/T_0$$ is the coarse moduli space of $$\mathfrak K_{\Gamma_0}(X)$$. All (non-empty) moduli spaces of two-pointed stable log maps are of the form $$\mathfrak K_{\Gamma_0}(X)$$, which implies that these spaces are therefore always irreducible.
Along the way of proving the main result of the paper, the authors also obtain an alternative modular description for $$\mathfrak K_{\Gamma_0}(X)$$ in terms of the Kontsevich moduli pace of stable maps to $$X$$ with genus $$0$$, curve class $$\beta_0$$ and two marked points, $$\mathfrak M_{0,2}(X,\beta_0)$$; namely $$\mathfrak K_{\Gamma_0}$$ is shown to be the normalisation of the closure of the image of $$T'$$ on $$\mathfrak M_{0,2}(X,\beta_0)$$. Other ingredients of the proof of the main theorem obtained by the authors are the log-smoothness of $$\mathfrak K_{\Gamma}(X)$$ in the genus $$0$$ case and the use of the relation between tropical curves and stable log maps to toric varieties to study the boundary of $$\mathfrak K_{\Gamma_0}(X)$$.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
##### Keywords:
toric; Kontsevich; stable log map; Chow quotient
Full Text: