Pushforwards of Measures on Real Varieties under Maps with Rational Singularities

Abstract

Let $X,Y$ be algebraic varieties defined over $\mathbb{R}$. Assume $Y$ is smooth and $X$ is Gorenstein. Suppose $\phi :X\to Y$ is a flat $\mathbb{R}$-morphism such that all the fibers have rational singularities. We show that the pushforward of any smooth, compactly supported measure on $X$ has a continuous density with respect to any smooth measure with non-vanishing density on $Y$. This extends a result of Aizenbud and Avni from the $p$-adic case to the archimedean case.

39 pages, no figures. arXiv admin note: text overlap with arXiv:1307.0371 by other authors