Integrability of pushforward measures by analytic maps
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866929499726151680 |
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| author | Glazer, Itay Hendel, Yotam I. Sodin, Sasha |
| author_facet | Glazer, Itay Hendel, Yotam I. Sodin, Sasha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12446 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Integrability of pushforward measures by analytic maps Glazer, Itay Hendel, Yotam I. Sodin, Sasha Algebraic Geometry Classical Analysis and ODEs 14B05 (Primary) 03C98, 14E15, 32B20, 60B15 (Secondary) 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. |
| title | Integrability of pushforward measures by analytic maps |
| topic | Algebraic Geometry Classical Analysis and ODEs 14B05 (Primary) 03C98, 14E15, 32B20, 60B15 (Secondary) |
| url | https://arxiv.org/abs/2202.12446 |