Integrability of pushforward measures by analytic maps

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.