Chow quotients of toric varieties as moduli of stable log maps.

*(English)*Zbl 1301.14012Let \(X\) 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 KollĂˇr’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 \(\mathfrak K_{\Gamma_0}(X)\).

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 \(\mathfrak K_{\Gamma_0}(X)\).

Reviewer: Margarida Melo (Coimbra)