Rigidity of morphisms for log schemes

Logarithmically Regular Morphisms

We consider the stack $\mathcal{L}og_X$ parametrizing log schemes over a log scheme $X$, and weak and strong properties of log morphisms via $\mathcal{L}og_X$, as defined by Olsson. We give a concrete combinatorial presentation of $\mathcal{L}og_X$, and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato's chart criterion of logarithmic smoothness.

Heuristic formula for logarithm of the Frobenius morphism

We show that the logarithm $\log_q$ of the Frobenius morphism $x\to x^q$ is given by the formula $x\to x\log x$ (the natural logarithm). In particular, it does not depend on $q$. This is the explicit (although heuristical) formula for the operator conjectured by Hilbert whose eigenvalues coincide with the zeroes of the zeta function.

Moduli of morphisms of logarithmic schemes

We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the algebraicity of the stack of stable logarithmic maps without restriction on the logarithmic structure of the target.

关于log canonical Calabi-Yau对的准阿贝尔 morphisms

On quasi-Albanese morphisms for log canonical Calabi-Yau pairs

本文研究了log canonical Calabi-Yau对的准阿贝尔 morphisms，提出了toric pairs的特征化结果。

Compactifications of Log Morphisms

We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.

Homomorphisms of algebraic groups: representability and rigidity

Given two algebraic groups $G$, $H$ over a field $k$, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not étale. When $G$ is linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$ is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting…

阿贝尔簇上有理点像的加法刚性 I：简单情形

Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case

研究阿贝尔簇群律与射影空间下像的加法结构之间的相互作用，利用一致Mordell-Lang猜想证明简单阿贝尔簇上有限秩子群像的加法能量和和集满足二次增长，并推广到一般阿贝尔簇。

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics

We prove a relative Kawamata Viehweg vanishing type theorem for birational morphisms. We use this to prove a Grauert Riemenschneider theorem over log canonical threefolds without zero dimensional log canonical centers, in residue characteristic p greater than five. In large enough residue characteristics, we prove a Grauert Riemenschneider theorem over threefold log canonical singularities with standard coefficients. These vanishing theorems can also be used to study the depth of log canonical singularities at non log canonical centers, as well as the singularities of the log canonical centers…

On a generalized canonical bundle formula for generically finite morphisms

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then applied to obtain a subadjunction formula for log canonical centers of generalized pairs. As another application, we show that the image of an anti-nef log canonical generalized pair has the structure of a numerically trivial log canonical generalized pair. This readily implies a result of Chen--Zhang. Along the way we prove that the Shokurov type convex se…

关于带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性

On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

本文研究了带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性，证明了在特定条件下，Albanese映射的两个一般纤维是birationally等价的。

Exactness, integrality, and log modifications

In this paper we discuss log blow-up's, introduced by Kazuya Kato, and define the concept of log modifications. Using this concept we prove that any morphism f: X ---> Y of locally noetherian fs log schemes with underlying structures of f and Y quasi-compact can be modified to an exact morphism, and moreover to an integral morphism. By a well-known fact on the underlying structure of an integral morphism this result can be considered as a weak log-version of flattening theorem by Raynaud and Gruson.

Local uniqueness of approximations and finite determinacy of log morphisms

I prove a local finite determinacy result of singular fibres in families. As an application, I generalize finite determinacy results on the fibres of log morphisms and nearby and vanishing cycles sheaves by Illusie and Kisin.

环射影空间丛的仿射局部判据

An affine local criterion for toric projective space bundles

该研究探讨等维环态射成为射影空间丛的条件，证明仿射刚性定理并将其应用于长极射线相关的环收缩，得到无需ℚ-因子性的射影空间丛定理。

Noncommutative rigidity

In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Ostvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the …

对数动机吉辛同构

Log motivic Gysin isomorphisms

研究为 fs 对数概型在公理动机设置下构建吉辛同构，通过制定对数光滑态射的纯度变换，证明特定非严格态射的纯度变换是同构。

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…

高维对数 Riemann--Hurwitz 不等式与覆盖的刚性

A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers

本文证明了对数 Riemann--Hurwitz 不等式及其严格版本，并应用于曲线模空间，得出带层级结构的模空间的有限满射自态射必为同构。

Bounds for log canonical thresholds with applications to birational rigidity

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a suitable smooth morphism. This in turn is based on an inequality relating the log canonical threshold and the Samuel multiplicity, generalizing our previous result from math.AG/0205171. We then give a lower bound for the log canonical threshold of an affine scheme defined by homogeneous equations of the same degree in terms of the dimension of the non log te…

Albanese morphism of log smooth klt compact Kähler manifold with nef log anticanonical divisor

Let $(X, ω)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-β_j) Y_j=\sum (1-β_j) [s_j=0]$ a divisor with simple normal crossings with $β_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.

Picard数为1的Fano流形的切丛的巨大性与动力学刚性（含Jie Liu的附录）

Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)

本文证明了切丛为大的Picard数1的Fano流形的满射态射在非射影空间时为同构，并研究了完全交、del Pezzo和Mukai流形切丛的巨大性及动力学刚性。

对数Kunz定理与Radu–André定理

Logarithmic Kunz and Radu--André Theorems

本文将刻画非奇异性的Kunz定理及其相对版本Radu–André定理，推广到正特征的对数代数几何领域。

Equations of log del Pezzo surfaces of index 2

Del Pezzo surfaces over C with log terminal singularities of index \le 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to describe del Pezzo surfaces of index \le 2 by equations in some weighted projective space. Obtained results are natural completions of similar results of Du Val, and Hidaka and Watanabe, describing del Pezzo surfaces of index 1.

Log homogeneous varieties

Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections. We then show that the Albanese morphism $α$ is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of $D$ are nonsingular, and any partial intersection of them is irreducible. Also, the image of $X$ under the morphism $σ$ associated with $- K_X - D$ is a spherical variety, and the irreducible compon…

On the characterization of abelian varieties for log pairs in zero and positive characteristic

Let $(X,Δ)$ be a pair. We study how the condition $κ(K_X + Δ)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $Δ$ are $\le 1$ and $-(K_X+Δ)$ is semiample, then the Albanese …

三次曲面的对数 Calabi-Yau 模空间的壁穿越

Log Calabi-Yau wall crossing for moduli of cubic surfaces

本文研究无标记三次曲面模空间在对数 Calabi-Yau 阈值处的壁穿越，证明相应模栈存在良好模空间，并明确描述了壁穿越态射：K-模空间到对数 Calabi-Yau 模空间同构，而 KSBA 模空间收缩一个 $\mathbb{P}^3$ 除子到一个点。

A fibered power theorem for pairs of log general type

Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

对数Cartier变换的拓扑斯理论方法

A Topos-Theoretic Approach to the Logarithmic Cartier Transform

本文通过拓扑斯理论方法，将对数Cartier变换推广到对数光滑态射，构造晶体范畴并证明其诱导函子忠实满，从而推广了Ogus和Vologodsky的经典结果。

Log abundance of the moduli b-divisors of lc-trivial fibrations

We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. The result follows from a theorem on the restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms, which allows us to treat $\mathbb{R}$-divisors and proper morphisms possibly with disconnected fibres. We also prove a theorem on extending a finite cover over a closed subvariety to that over a variety in arbitrary characteristic.

Log canonical pairs with good augmented base loci

Let $(X,B)$ be a projective log canonical pair such that $B$ is a $\Q$-divisor, and that there is a surjective morphism $f\colon X\to Z$ onto a normal variety $Z$ satisfying: $K_X+B\sim_\Q f^*M$ for some $\Q$-divisor $M$, and the augmented base locus ${\bf{B_+}}(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.

半稳定族的几何极化对数霍奇结构

Geometric polarized log Hodge structures for semistable families

本文证明了fs对数解析空间上的射影半稳定态射能够以典范方式诱导出极化对数霍奇结构，建立了半稳定族与对数霍奇理论之间的几何联系。

Log Structures on Generalized Semi-Stable Varieties

This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms.

Geometric polarized log Hodge structures with a base of log rank one

We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way.

通过生成元和关系实现一般对数光滑族

Generically log smooth families via generators and relations

本文通过生成元和关系的方法，开发了研究对数光滑族的算法工具，并在Macaulay2中实现，同时探讨了对数光滑结构在toroidal crossing scheme上的应用。

局部对数Cartier变换

Local Logarithmic Cartier Transform

本文推广Ogus-Vologodsky的Cartier变换到对数光滑概形，在Frobenius提升条件下构造了从拟幂零Higgs模到拟幂零可积联络模的完全忠实函子，并证明了对数平坦下降定理。

proper morphisms 的 Stein 度数

Stein degree of proper morphisms

本文探讨了proper morphisms的Stein度数概念，介绍了其在birational geometry中的应用，并提出了关于Stein度数的有界性问题。

Logarithmic Combinatorial Differentials

Given a morphism $X \to S$ of fine log schemes, we develop a geometric description of the sheaves of higher-order differentials $Ω^n_{X/S}$ for $n > 1$, as well as a definition of the de Rham complex in terms of this description.

Chen--Jiang分解的一个对数典范反例

Log Canonical Counterexamples to the Chen--Jiang Decomposition

本文构造了一个对数光滑曲面偶及到椭圆曲线的态射，使其直像为Atiyah不可分解丛，否定了Jiang问题3.4，并通过乘积推广到高直像及阿贝尔基。

A Log PSS morphism with applications to Lagrangian embeddings

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in symplectic cohomology of the complement $X = M \backslash \mathbf{D}$. Under further "topological" assumptions on the pair, these classes can be organized into a Log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of $(M, \mathbf{D})$ to symplectic cohomology. Turning to applications, we show that these metho…

Rational connectedness of log $Q$-Fano varieties

In this paper, we give an affirmative answer to a conjecture in the Minimal Model Program. We prove that log $Q$-Fano varieties of dim $n$ are rationally connected. We also study the behavior of the canonical bundles under projective morphisms.

索引对数 Cartier 变换

Indexed Logarithmic Cartier Transform

本文推广 Cartier 变换到对数情形，通过引入索引结构细化拓扑斯与晶体，利用对数微分算子索引代数的 Azumaya 性质，建立索引晶体范畴的等价，解决 Frobenius 不平坦导致的本质满射问题。

复解析空间之间射影态射的锥与收缩定理

Cone and contraction theorem for projective morphisms between complex analytic spaces

本文在复解析空间中建立正规对的锥与收缩定理，作为复解析对数典范对极小模型纲领的起点，并重点处理奇点比川又对数终端更差的情形。

对数典范Beauville--Bogomolov分解的一个强反例

A strong counterexample to the log canonical Beauville--Bogomolov decomposition

本研究构造了维数不小于4的对数典范K-平凡簇作为对数典范情形下Beauville--Bogomolov分解的强反例，同时给出相关拟射影簇构造，并证明反典范类nef的对数典范配对的Albanese态射为局部稳定族。

Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: Jun Li's original approach, Bumsig Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

Varieties fibred over abelian varieties with fibres of log general type

Let $(X,B)$ be a complex projective klt pair, and let $f\colon X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that $K_X+B$ is relatively big over $Z$. We show that such pairs have good log minimal models.

复分析空间上log canonical对的最小模型程序

Minimal model program for log canonical pairs on complex analytic spaces

本文研究复分析空间上log canonical对的最小模型程序，将Birkar和第二作者的结果推广到Fujino的设定中。

Pull-back of quasi-log structures

We prove that the pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a natural quasi-log structure. We treat an application to log Fano pairs. This paper also contains a proof of the simple connectedness of log Fano pairs with only log canonical singularities by Kento Fujita.

对数方案的Hochschild同调

Hochschild homology for log schemes

将Hochschild同调和循环同调推广到代数空间到代数栈的态射，进而得到Fontaine-Illusie意义下对数方案的这些同调理论。

对数几何与提升有理点

Log geometry and lifting rational points

该研究针对概型态射，给出判断其取值于域等环的点是否可提升的热带准则，推广了Abramovich的成果至含对数稳定映射的更广泛态射类。

The first-order theory of geometric points of schemes: Chevalley's theorem and quantifier elimination

Chevalley's theorem on the images of morphisms of schemes and the principle of quantifier elimination for the theory of algebraically closed fields are widely understood to be two perspectives on the same theorem. In this paper, we demonstrate that both results can easily be proven simultaneously, using a mixture of geometric and logical techniques. In doing so, we give logical meaning to geometric points of schemes and to finitely presented morphisms thereof, in a manner reminiscent of Spencer Breiner's logical schemes.