Bergman kernel functions associated to measures supported on totally real submanifolds

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Cⁿ is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.