Inversion of adjunction on log canonicity

商奇点的伴随反转

Inversion of adjunction for quotient singularities

本文研究商奇点中的伴随反转问题，建立 klt Cartier 除子的精确公式，并据此证明 klt 超商奇点最小对数偏差的半连续性。

商奇点的伴随反演 II：非线性作用

Inversion of adjunction for quotient singularities II: Non-linear actions

该研究证明了商奇点的精确伴随反演公式，推导了超商奇点极小对数偏差的半连续性，作为此前研究的续篇将相关结果推广至非线性群作用场景。

反向 adjunction 公式对于商奇点 III：半不变情况

Inversion of adjunction for quotient singularities III: semi-invariant case

本文证明了有限线性群商的完全交集变种中半不变方程的反向 adjunction 公式，并证明了这些变种的最小 log 可辨性半连续性，扩展了前文关于不变方程的结果。

伴随的对数典范反演

Log canonical inversion of adjunction

本文是一篇简短注记，研究伴随的对数典范反演问题。

Jet schemes, log discrepancies and Inversion of Adjunction

We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.

Inversion of adjunction for local complete intersection varieties

In this note we build on our previous work with Takehiko Yasuda to prove a precise version of inversion of adjunction for varieties which are local complete intersections.

关于 $F$-纯性的伴随反演

On $F$-pure inversion of adjunction

本文在正特征情形下研究除子对 $F$-纯性的伴随反演，在 $\mathbb{Q}$-Gorenstein 假设下完整解决主除子情形，并刻画了 $(R, Δ+ D)$ 与 $(R_D, \text{Diff}_D(Δ))$ 的 $F$-纯性关系。

On Strongly $F$-Regular Inversion of Adjunction

In this article we give two independent proofs of the positive characteristic analog of the log terminal inversion of adjunction. We show that for a pair $(X, S+B)$ in characteristic $p>0$, if $(S^n, B_{S^n})$ is strongly $F$-regular, then $S$ is normal and $(X, S+B)$ is purely $F$-regular near $S$. We also answer affirmatively an open question about the equality of $F$-Different and Different.

Inversion of adjunction for rational and Du Bois pairs

We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the non-pair setting this asserts that if a family over a smooth base has a special fiber $X_0$ with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near $X_0$.

F-singularities of pairs and Inversion of Adjunction of arbitrary codimension

We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$ of rings $R$ and ideals $\a \subset R$ with real exponent $t > 0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y \subsetneq X$, if the restriction $(Z, Y|_Z)$ to a normal $\Q$-Gorenstein closed subvariety $Z \subsetneq X$ is klt (resp. lc), the…

Inversion of adjunction for $F$-signature

Let $(R,Δ+D)$ be a log $\mathbb{Q}$-Gorenstein pair where $R$ is a Noetherian, $F$-finite, normal, local domain of characteristic $p > 0$, $Δ$ is an effective $\mathbb{Q}$-divisor and $D$ is an integral $\mathbb{Q}$-Cartier divisor. We show that the left derivative of the $F$-signature function $s(R,Δ+ tD)$ at $t = 1$ is equal to $-s(\mathcal{O}_D, \mathrm{Diff}_D(Δ))$. This equality is interpreted as a quantitative form of inversion of adjunction for strong $F$-regularity. As an immediate corollary, we obtain the inequality $s(R,Δ) \geq s(\mathcal{O}_D, \mathrm{Diff}_D(Δ))$. We also discuss t…

关于广义Kaehler对的典范丛公式与伴随性质

On the Canonical Bundle Formula and Adjunction for Generalized Kaehler Pairs

该研究针对Kaehler簇上的广义对，证明了典范丛公式、次伴随及伴随逆定理的类比结果，推导了BDPP猜想与锥定理的关联，并完善了已有相关结论。

The Angehrn-Siu Type Effective Freeness For Quasi-Log Canonical Pairs

We prove the Angehrn-Siu Type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is the inversion of adjunction for quasi-log canonical pairs, which is established in this paper.

关于最小 log 抛物线差异的 PIA 猜想的反例

A counterexample to the PIA conjecture for minimal log discrepancies

本文通过构造反例，推翻了关于最小 log 抛物线差异的 PIA 猜想，并对家庭中的 LSC 猜想也提供了反例。

A vanishing theorem for log canonical pairs

Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld in the smooth case and by Chardin--Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.