伴随的对数典范反演

Inversion of adjunction for non-degenerate hypersurfaces

We prove a precise inversion of adjunction formula for the log pair associated to a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded from above by dimension.

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。

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。

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。

Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes

In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a complex manifold M. This adjunction formula is used to study the problem of extending L2-cohomology classes of dbar-closed forms from the singular hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the L2-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of V in M. The…

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.

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。

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。

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.

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.

On a generalized canonical bundle formula and generalized adjunction

In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov.

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…

Sources of log canonical centers

We define the "source" and the "spring" of a log canonical center and use them to solve several problems in higher-codimension adjunction. The main application is to the construction of semi log canonical pairs. Version 2: References updated. Version 3: Many small changes.

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.

The Ascending Chain Condition for log canonical thresholds on l.c.i. varieties

Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties.

Existence of log canonical modifications and its applications

The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable assumptions. It recovers Kawakita's inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As applications, we study lengths of extremal rational curves and so on.

Log canonical thresholds in positive characteristic

In this paper, we study the singularities of a pair (X,Y) in arbitrary characteristic via jet schemes. For a smooth variety X in characteristic 0, Ein, Lazarsfeld and Mustata showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on X. Via this correspondence, one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation. We now extend this result to positive characteristic. In particular, we prove Mustata's log canonical threshold formula avoiding the use of log resolutions, making the formula availa…

Strong $(δ,n)$-complements for semi-stable morphisms

We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fibe…

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。

完美纯性与特征标对

Perfectoid Pure and Signature Pairs

该研究将完美纯性推广到混合特征除子对，建立其与对数典范性等的关联，定义完美纯阈值与特征标，证明相关性质及变换规则，为代数几何相关研究提供新工具。

Semi-log canonical vs $F$-pure singularities

If $X$ is Frobenius split, then so is its normalization and we explore conditions which imply the converse. To do this, we recall that given an $\mathcal{O}_X$-linear map $ϕ: F_* \mathcal{O}_X \to \mathcal{O}_X$, it always extends to a map $\barϕ$ on the normalization of $X$. In this paper, we study when the surjectivity of $\barϕ$ implies the surjectivity of $ϕ$. While this doesn't occur generally, we show it always happens if certain tameness conditions are satisfied for the normalization map. Our result has geometric consequences including a connection between $F$-pure singularities and sem…

Mather discrepancy and the arc spaces

The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By introducing new log-canonical threshold and minimal log-discrepancy by means of Mather discrepancy instead of usual discrepancy of canonical divisors, we obtain the formulas of the new log-canonical threshold in terms of arc spaces, inversion of adjunction for wider class of singularities than the known one, lower seimicontinuity of the new minimal log-discrepancy…

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.

关于三重形上秩为一的叶形的Log Canonical极小模型程序

Log Canonical Minimal Model Program for corank one foliations on Threefolds

本文研究了三重形上秩为一的叶形的Log Canonical极小模型程序，证明了通过运行$(K_\mcF+\Delta)$-MMP可以得到极小模型或Mori纤维空间，并探讨了极小模型之间的连接及有限数量性质。

Positivity criteria for log canonical divisors and hyperbolicity

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion f…