摘要
On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1)Todasystems on X\{P_(1);…;P_(k)}are recognized by the associated toric curves in.We introduce character n-ensembles as-tuples of meromorphic one-forms with simple poles and purely imaginary periods,generating toric curves on minus finitelymany points.On X,we establish a correspondence between character-ensembles and toric solutions to the SU(n+1)system with finitely many cone singularities.Our approach not only broadens seminal solutions with two conesingularities on the Riemann sphere,as classified by Jost-Wang(Int.Math.Res.Not.,2002,(6):277-290)andLin-Wei-Ye(Invent.Math.,2012,190(1):169-207),but also advances beyond the limits of Lin-Yang-Zhong’s existencetheorems(J.Differential Geom.,2020,114(2):337-391)by introducing a new solution class.
在有限穿孔的紧致黎曼曲面X\{P_(1);…;P_(k)}上,定义toric曲线为到的多值、完全无分歧全纯映射,且其单值化群含于PSU(n+1)的极大环面。SU(n+1)户田系统在X\{P_(1);…;P_(k)}上的toric解通过其关联到中的toric曲线来识别。我们引入特征n系综作为亚纯一形式的n元组,这些一形式具有简单极点和纯虚周期,且生成去掉有限个点上的toric曲线。在X上,我们建立了特征n系综与带锥奇点SU(n+1)户田系统的toric解之间的对应关系。我们的方法不仅扩展了前人在Riemann球面上带两个锥奇点的经典解分类,还通过引入一个新的解类,超越了Lin-Yang-Zhong存在性定理的界限。
基金
supported by the National Natural Science Foundation of China(11931009,12271495,11971450,and 12071449)
Anhui Initiative in Quantum Information Technologies(AHY150200)
the Project of Stable Support for Youth Team in Basic Research Field,Chinese Academy of Sciences(YSBR-001).
作者简介
Jingyu Mu is currently a freelance in education.He received his Ph.D.degree in Pure Mathematics from the University of Science and Technology of China in 2024.His research mainly focused on SU(n+1)Toda systems on Riemman surface;通讯作者:许斌,E-mail:bxu@ustc.edu.cn。is currently a Full Professor with the School of Mathematical Sciences,University of Science and Technology of China(USTC).He received his B.S.degree in Pure Mathematics from USTC in 1996 and his Ph.D.degree in the same field from the University of Tokyo in 2003.His research mainly focuses on complex geometry,with particular emphasis on singular special Kähler structures,the construction of hyperKähler metrics on algebraic integrable systems,and investigations into the classical Hurwitz existence problem.
