论文标题
通过分析图的推动力措施的整合性
Integrability of pushforward measures by analytic maps
论文作者
论文摘要
鉴于$ f $ - 分析流形之间的地图$ ϕ：x \ rightArrow y $在本地字段$ f $ f $ $ 0 $之间，我们引入了一个不变的$ε_{\ star}（ϕ）$，该$量化了$ ϕ $的平滑压实措施的集成能。我们进一步定义了本地版本$ε_ {\ star}（ϕ，x）$附近$ x \ in x $。这些不变的人与$ ϕ $的奇点有很强的联系。 当$ y $是一维时，我们给出了$ε_ {\ star}（ϕ，x）$的明确公式，并表明它在渐近上等同于其他已知的奇异性不变性，例如$ f $ f $ f $ f $ -log-log-canonicalical thelsonical $ \ pertateArt $ \ perperatate $ \ perperatateRateRateRateNallog \ perperatateRateRateRateRateRateNalloge \ operatateRateRateRateRateRate {lct}; 在一般情况下，我们表明$ε_{\ star}（ϕ，x）$在下面的界限为$ f $ -log -log-canonical threshold $λ= \ operatorName {lct} _ {f} _ {f}（\ mathcal {如果$ \ dim y = \ dim x $，则达到平等。如果$ \ dim y <\ dim x $，则不平等可能是严格的；但是，对于$ f = \ mathbb {c} $，我们建立上限$ε_ {\ star}（ϕ，x）\leqλ/（1-λ）$，只要$λ<1 $。 最后，我们专门研究平滑代数$ \ mathbb {q} $之间的多项式地图$φ：x \ rightarrow y $ - 品种$ x $和$ y $。我们从几何表征$ε_{\ star}（φ_{f}）= \ infty $上的$ε_{\ star}（φ__{f}）= \ infty $上的本地田地，表明它等同于$φ$，与半log-canonical单奇异的纤维平整。
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.
