1997/11/19 by Miguel Abreu, Abreu, Miguel · 3 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C55 #Secondary 14M25 53C25 58F05 #dg-ga #math.DG #msc:14M25 #msc:53C25 #msc:53C55 #msc:58F05
paper · pdf · doi:10.48550/arxiv.dg-ga/9711014
12 pages, submitted to International Journal of Mathematics
arxiv created 1997/11/19 · arxiv updated 2009/11/30
Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.