Tropical curves, their Jacobians and theta functions.

*(English)*Zbl 1152.14028
Alexeev, Valery (ed.) et al., Curves and abelian varieties. International conference, Athens, GA, USA, March 30–April 2, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4334-5/pbk). Contemporary Mathematics 465, 203-230 (2008).

Tropical Geometry is a new subject lying at the shared frontier of several fields of mathematics, including complex algebraic geometry, toric geometry, and nonarchimedean geometry. As in these three subjects, we know the most about 1-dimensional objects, or curves. The paper under review begins the serious study of tropical curves, drawing much from a close analogy with Riemann surfaces.

Tropical varieties can be defined as the set of solutions to a system of polynomial equations defined over the tropical semifield. However, a compact tropical curve admits a more direct ad hoc description. Let \(C\) be a compact locally finite 1-dimensional simplicial complex. Let \(C^{\circ}\) be the complement of the \(1\)-valent points of \(C\) (i.e., the points with a neighborhood homeomorphic to a half-open interval). Equip \(C^{\circ}\) with a complete metric \(d\) such that a point of \(C \smallsetminus C^{\circ}\) is at infinite distance from any point of \(C^{\circ}\). The pair \((C, d)\) is called a compact tropical curve, loosely denoted \(C\).

The authors define regular and rational functions on \(C\), tropical morphisms, divisors and line bundles, and the vector space of real \(1\)-forms \(\Omega(C)\) (§3, 4). With these objects in hand, they study the Jacobian of a compact tropical curve \(J(C) := \Omega(C)^* / H_1(C, \mathbb Z)\). Tropical Theta functions are discussed in §5. As a compact tropical curve \(C\) can be viewed as a metric graph, all of these geometric objects have discrete interpretations in the language of graph theory. Many of the proofs in the present article rely implicitly on a discretization technique that allows one to reduce to graph theoretic arguments.

Once the geometric framework of tropical curves and Jacobians is established, the authors develop Abel/Jacobi theory for compact tropical curves of genus \(g \geq 1\) (§6). They give a tropical morphism \(C \to J(C)\) that induces a bijection between \(\text{Pic}^d(C)\) and \(J(C)\), where \(\text{Pic}^d(C)\) is the group of line bundles on \(C\) of degree \(d\). An interesting phenomenon that distinguishes this theory from that of Riemann surfaces is that the Abel/Jacobi morphism \(C \to J(C)\) is not injective in general. Discrete versions of some of these results are proved by R. Bacher, P. de la Harpe, and T. Nagnibeda [Bull. Soc. Math. France. 125(2), 167–198 (1997; Zbl 0891.05062)] and also by N. Biggs [Bull. Lond. Math. Soc. 29(6), 641–682 (1997; Zbl 0892.05033)].

Next the authors prove the Riemann/Roch formula for compact tropical curves (§7), following the strategy for combinatorial graphs initiated by M. Baker and S. Norine [Adv. Math. 215, 766–788 (2007; Zbl 1124.05049)]. While the formula itself is not surprising to anyone familiar with the analogous result for Riemann surfaces (they are identical), the fact that such a formula exists in the tropical setting is amazing.

Finally, the authors describe the image of the \((g-1)^{\text{st}}\) symmetric power of \(C\) inside its Jacobian in terms of the tropical theta divisor — this is the tropical counterpart of Riemann’s theorem (§8).

For the entire collection see [Zbl 1143.14001].

Tropical varieties can be defined as the set of solutions to a system of polynomial equations defined over the tropical semifield. However, a compact tropical curve admits a more direct ad hoc description. Let \(C\) be a compact locally finite 1-dimensional simplicial complex. Let \(C^{\circ}\) be the complement of the \(1\)-valent points of \(C\) (i.e., the points with a neighborhood homeomorphic to a half-open interval). Equip \(C^{\circ}\) with a complete metric \(d\) such that a point of \(C \smallsetminus C^{\circ}\) is at infinite distance from any point of \(C^{\circ}\). The pair \((C, d)\) is called a compact tropical curve, loosely denoted \(C\).

The authors define regular and rational functions on \(C\), tropical morphisms, divisors and line bundles, and the vector space of real \(1\)-forms \(\Omega(C)\) (§3, 4). With these objects in hand, they study the Jacobian of a compact tropical curve \(J(C) := \Omega(C)^* / H_1(C, \mathbb Z)\). Tropical Theta functions are discussed in §5. As a compact tropical curve \(C\) can be viewed as a metric graph, all of these geometric objects have discrete interpretations in the language of graph theory. Many of the proofs in the present article rely implicitly on a discretization technique that allows one to reduce to graph theoretic arguments.

Once the geometric framework of tropical curves and Jacobians is established, the authors develop Abel/Jacobi theory for compact tropical curves of genus \(g \geq 1\) (§6). They give a tropical morphism \(C \to J(C)\) that induces a bijection between \(\text{Pic}^d(C)\) and \(J(C)\), where \(\text{Pic}^d(C)\) is the group of line bundles on \(C\) of degree \(d\). An interesting phenomenon that distinguishes this theory from that of Riemann surfaces is that the Abel/Jacobi morphism \(C \to J(C)\) is not injective in general. Discrete versions of some of these results are proved by R. Bacher, P. de la Harpe, and T. Nagnibeda [Bull. Soc. Math. France. 125(2), 167–198 (1997; Zbl 0891.05062)] and also by N. Biggs [Bull. Lond. Math. Soc. 29(6), 641–682 (1997; Zbl 0892.05033)].

Next the authors prove the Riemann/Roch formula for compact tropical curves (§7), following the strategy for combinatorial graphs initiated by M. Baker and S. Norine [Adv. Math. 215, 766–788 (2007; Zbl 1124.05049)]. While the formula itself is not surprising to anyone familiar with the analogous result for Riemann surfaces (they are identical), the fact that such a formula exists in the tropical setting is amazing.

Finally, the authors describe the image of the \((g-1)^{\text{st}}\) symmetric power of \(C\) inside its Jacobian in terms of the tropical theta divisor — this is the tropical counterpart of Riemann’s theorem (§8).

For the entire collection see [Zbl 1143.14001].

Reviewer: Xander Faber (Montreal)