Riemann-Roch for sub-lattices of the root lattice An

Abstract

Recently, Baker and Norine found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice An analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalize the result to other sub-lattices of An. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.