The (spherical) Mahler measure of the X-discriminant

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Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere.  Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding , then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.  

In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of the degree of the embedding.  The technical machinery required for this task was developed by Jean-Michel Bismut in several articles in the 1990's 

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