Integrability of pushforward measures by analytic maps

TL;DR

This paper introduces an invariant measuring the integrability of pushforward measures under analytic maps over local fields, relating it to singularity invariants and providing explicit formulas and bounds.

Contribution

It defines a new invariant for analytic maps that connects measure integrability with singularity theory, including explicit formulas, bounds, and geometric characterizations.

Findings

The invariant $psilon_{}()$ quantifies measure integrability and relates to singularities.

Explicit formula for $psilon_{}(,x)$ when $Y$ is one-dimensional.

Bounds for $psilon_{}(,x)$ in terms of log-canonical thresholds.

Abstract

Given a map $\phi:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $\epsilon_{\star}(\phi)$ which quantifies the integrability of pushforwards of smooth compactly supported measures by $\phi$. We further define a local version $\epsilon_{\star}(\phi,x)$ near $x\in X$. These invariants have a strong connection to the singularities of $\phi$. When $Y$ is one-dimensional, we give an explicit formula for $\epsilon_{\star}(\phi,x)$, and show it is asymptotically equivalent to other known singularity invariants such as the $F$-log-canonical threshold $\operatorname{lct}_{F}(\phi-\phi(x);x)$ at $x$. In the general case, we show that $\epsilon_{\star}(\phi,x)$ is bounded from below by the $F$-log-canonical threshold $\lambda=\operatorname{lct}_{F}(\mathcal{J}_{\phi};x)$ of the Jacobian ideal $\mathcal{J}_{\phi}$ near $x$. If $\dim Y=\dim X$, equality is attained. If $\dim Y<\dim X$, the inequality can be strict; however, for $F=\mathbb{C}$, we establish the upper bound $\epsilon_{\star}(\phi,x)\leq\lambda/(1-\lambda)$, whenever $\lambda<1$. Finally, we specialize to polynomial maps $\varphi:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$-varieties $X$ and $Y$. We geometrically characterize the condition that $\epsilon_{\star}(\varphi_{F})=\infty$ over a large family of local fields, by showing it is equivalent to $\varphi$ being flat with fibers of semi-log-canonical singularities.