Let P be a homogeneous polynomial of degree d in N+1 complex variables. The spherical logarithmic Mahler measure of P, denoted by m(P), is defined as the integral of log∣P∣ over the unit sphere in C^{N+1}. (By contrast, in number theory one typically integrates over the torus, and the polynomials of interest usually have integer coefficients or, more generally, coefficients in a number field K.)

For polynomials P arising as generalized discriminants of polarized manifolds (X,L), recent joint work of Song Sun, Junsheng Zhang, and the speaker establishes estimates of the form m(P)=O(d), where d denotes the degree of a sufficiently large projective embedding of (X,L).

In this talk, the speaker will outline some of the main ingredients in the proof of this estimate, with particular emphasis on its connection to the circle of ideas surrounding the Arithmetic Riemann–Roch theorem of Faltings / Bismut-Gillet-Soule .