Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2202.12446 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Inhaltsangabe:
- Given a map $ϕ:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $ε_{\star}(ϕ)$ which quantifies the integrability of pushforwards of smooth compactly supported measures by $ϕ$. We further define a local version $ε_{\star}(ϕ,x)$ near $x\in X$. These invariants have a strong connection to the singularities of $ϕ$. When $Y$ is one-dimensional, we give an explicit formula for $ε_{\star}(ϕ,x)$, and show it is asymptotically equivalent to other known singularity invariants such as the $F$-log-canonical threshold $\operatorname{lct}_{F}(ϕ-ϕ(x);x)$ at $x$. In the general case, we show that $ε_{\star}(ϕ,x)$ is bounded from below by the $F$-log-canonical threshold $λ=\operatorname{lct}_{F}(\mathcal{J}_ϕ;x)$ of the Jacobian ideal $\mathcal{J}_ϕ$ 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 $ε_{\star}(ϕ,x)\leqλ/(1-λ)$, whenever $λ<1$. Finally, we specialize to polynomial maps $φ:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$-varieties $X$ and $Y$. We geometrically characterize the condition that $ε_{\star}(φ_{F})=\infty$ over a large family of local fields, by showing it is equivalent to $φ$ being flat with fibers of semi-log-canonical singularities.