Solutions to SU(n + 1) Toda systems with cone singularities via toric curves on compact Riemann surfaces

On a compact Riemann surface X with finite punctures P_1, \cdots, P_k , we define toric curves as multivalued, totally unramified holomorphic maps to \mathbbP^n with monodromy in a maximal torus of \mathrmPSU(n+1) . Toric solutions to \mathrmSU(n+1) Toda systems on X\setminus\P_1,\cdots, P_k\ are recognized by the associated toric curves in \mathbbP^n . We introduce character n-ensembles as n -tuples of meromorphic one-forms with simple poles and purely imaginary periods, generating toric curves on X minus finitely many points. On X , we establish a correspondence between character n -ensembles and toric solutions to the \mathrmSU(n+1) system with finitely many cone singularities. Our approach not only broadens seminal solutions with two cone singularities on the Riemann sphere, as classified by Jost–Wang (Int. Math. Res. Not., 2002, (6): 277–290) and Lin–Wei–Ye (Invent. Math., 2012, 190(1): 169–207), but also advances beyond the limits of Lin–Yang–Zhong’s existence theorems (J. Differential Geom., 2020, 114(2): 337–391) by introducing a new solution class.