Abstract: According to a result of Fujita-Liu the n-dimensional complex projective space may be characterized as the Fano variety which has the largest algebraic degree among all K-semistable Fano varieties. In this talk - which is a non-technical introduction to a joint work with Rolf Andreasson - an arithmetic analog of this result is discussed, where the role of degree is played by Falting's height. The height is formulated in terms of arithmetic intersection theory and depends on the choice of a metric on the anti-canonical line bundle of the corresponding Fano variety X. Our main result establishes an arithmetic analog of Fujita-Liu's result for toric Fano varieties of dimension n ≤ 6. The conjectural extension to any dimension n hinges on a conjectural gap-hypothesis for the algebraic degree.

Speaker: Robert Berman is a professor at Chalmers University of Technology in Sweden, working on analytical aspects of complex algebraic and differential geometry. His research focuses on connections between these fields and probability, statistical mechanics and mathematical physics and also arithmetic (Arakelov) geometry.