The zero locus of an admissible normal function

Zero loci of admissible normal functions with torsion singularities

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].

Fargues-Rapoport conjecture for p-adic period domains in the non-basic case

We prove the Fargues-Rapoport conjecture for p-adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide.

Pencils and critical locus on normal surfaces

We study linear pencils of curves on normal surface singularities. Using the minimal good resolution of the pencil, we describe the topological type of generic elements of the pencil and characterize the behaviour of special elements. Then we show that the critical locus associated to the pencil is linked to the special elements. This gives a decomposition of the critical locus through the minimal good resolution and as a consequence, information on the topological type of the critical locus.

具有有限0-位势的实调节映射

Real regulator maps with finite 0-locus

研究双变量拉格朗日多项式在条件下的调节映射，证明其扩展的调节函数和A多项式位势均为有限集。

On the zero locus of normal functions and the étale Abel-Jacobi map

We investigate questions of an arithmetic nature related to the Abel-Jacobi map. We give a criterion for the zero locus of a normal function to be defined over a number field, and we give some comparison theorems with the Abel-Jacobi map coming from continuous étale cohomology.

复解析情形下沿对数典范轨迹的正规对的极小模型纲领

The minimal model program for normal pairs along the log canonical locus in the complex analytic setting

本文在复解析情形下建立了沿对数典范轨迹的正规对的极小模型理论，推广了作者先前的结果。

Fields of definition of Hodge loci

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle together with the connection is defined over k. This is closely related with the theory of absolute Hodge classes. The proof uses the spread of the Hodge locus, and is quite similar to the case of the zero…

Cohomology classes of admissible normal functions

We study the map associating the cohomology class of an admissible normal function on the product of punctured disks, and give some sufficient conditions for the surjectivity of the map. We also construct some examples such that the map is not surjective.

Existence and density of general components of the Noether-Lefschetz locus on normal threefolds

We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.

Grothendieck-Lefschetz Theorem with Base Locus

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers.

沿对数典范轨迹的正规配对极小模型纲领

Minimal model program for normal pairs along log canonical locus

本文研究正规配对在非lc轨迹外的极小模型纲领，在非nef轨迹与非lc轨迹不相交且限制半丰富的条件下，证明了MMP终止及极小模型的丰富猜想。

复与 étale Abel-Jacobi 映射及 étale 正规函数零轨迹的代数性

Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions

本文利用开代数簇的 $p$-进 Hodge 理论，证明了 étale Abel-Jacobi 映射消失蕴含复 Abel-Jacobi 映射消失，并由此推导出 étale 正规函数零轨迹包含于复正规函数零轨迹及其代数性结论。

除正常交叉定位外的解析性及伽罗瓦作用

Resolution Except for the Normal-Crossing Locus and Galois actions

本文在零特征下构造了一个基于加权吹脱的通用解析算法，严格保持正常交叉定位，解决Kollar问题。通过加权吹脱框架和对数细化，实现鲁棒解析算法，应用于正常交叉Deligne-Mumford堆的canonical compactification和子簇与堆的functorial解析。

Singularities of admissible normal functions

In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using M. Saito's mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest.

On the structure of some p-adic period domains

We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge-Newton decomposable.

Augmented base loci and restricted volumes on normal varieties

We extend to normal projective varieties defined over an arbitrary algebraically closed field a result of Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa characterizing the augmented base locus (aka non-ample locus) of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes. We also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the Kodaira map defined by large and divisible multiples of the line bundle is an isomorphism.

关于 Ceresa 正规函数的挠轨迹

On the torsion locus of the Ceresa normal function

本文证明 $g\geq3$ 时模空间中 Ceresa 正规函数挠轨迹的正维数部分非 Zariski 稠密，并刻画其一般 Mumford–Tate 群为 $\mathrm{GSp}_{2g}$ 的有限分支及其 Galois 封闭性。

Noether-Lefschetz theorem with base locus

We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.

On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $π: X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $π$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.

Complex-analytic Neron models for arbitrary families of intermediate Jacobians

Given a family of intermediate Jacobians (for a polarized variation of Hodge structure of weight -1) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered D-modules and M. Saito…

Normal completions of toric varieties over rank one valuation rings and completions of $Γ$-admissible fans

We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $Γ$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $Γ$-admissible fans admit $Γ$-admissible completions. In order to show this we prove a combinatorial analog of noeth…

Nakamaye's theorem on log canonical pairs

We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at most 1. We also generalize Ein-Lazarsfeld-Mustata-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.

Covers of surfaces with fixed branch locus

Given a connected smooth projective surface X over the complex numbers, together with a simple normal crossings divisor D on it, we study finite normal covers Y of X that are unramified outside D. Given moreover a fibration of X onto a curve C, we prove that the `height' of Y over C is bounded linearly in terms of the degree of Y over X. We indicate how an arithmetic analogue of this result, if true, can be auxiliary in proving the existence of a polynomial time algorithm that computes the mod-l Galois representations associated to a given smooth projective geometrically connected surface over…

关于一般投影的判别位势

On the discriminant locus of a generic projection

研究一般投影的判别位势的几何性质，证明其与对偶簇的线性截面的对偶关系，并推导判别位势的纯度陈述。

On the locus of smooth plane curves with a fixed automorphism group

In this paper, we study some aspects of the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented by a single "normal form". Henn, and Komiya-Kuribayashi, observed that $\widetilde{…

An Exponential History of Functions with Logarithmic Growth

We survey recent work on normal functions, including limits and singularities of admissible normal functions, the Griffiths-Green approach to the Hodge conjecture, algebraicity of the zero-locus of a normal function, Neron models, and Mumford-Tate groups. Some of the material and many of the examples, esp. in sec. 5-6, are original. The first two sections should be easily accessible to graduate students.

Two observations about normal functions

Two simple observations are made: (1) If the normal function associated to a Hodge class has a zero locus of positive dimension, then it has a singularity. (2) The intersection cohomology of the dual variety contains the cohomology of the original variety, if the degree of the embedding is large.

Singular Q-homology planes of negative Kodaira dimension have smooth locus of non-general type

We continue the program of classification of normal Q-acyclic surfaces defined over the field of complex numbers, so-called 'Q-homology planes'. Here we show that if a Q-homology plane has negative Kodaira dimension then its smooth locus is not of general type. This generalizes an earlier result of Koras-Russell for contractible surfaces.

关于尖点曲线上正规函数的秩

On the Rank of Normal Functions on Pointed Curves

本文研究尖点曲线模空间上正规函数的秩，证明其在阿贝尔微分分层外取极大值，在该分层上恰降一，并由此描述双分歧轨迹的局部几何。

正规簇上可容许的 Hermitian-Yang-Mills 联络

Admissible Hermitian-Yang-Mills connections over normal varieties

本文在正规簇上证明完整版 Donaldson-Uhlenbeck-Yau 定理，建立 Hitchin-Kobayashi 对应，并由此得到 Bogomolov-Gieseker 不等式及其等号刻画，还给出平凡第一 Chern 类正规 Kaehler 簇为环面有限商的新判据。

Focal Loci of Algebraic Varieties I

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map $ε$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($ε$ sends the normal vector $O-x$ to its endpoint $O$), and in this paper we address two general problems : 1) Characterize the "degenerate" case where the focal locus is not a hypersurface 2) …

Normal crossing properties of complex hypersurfaces via logarithmic residues

We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.

从墙结构到闭镜像对称性：$K_{\mathbb{P}^2}$ 的情形——正实轨迹上的重整化周期、来自墙函数的闭 Gromov-Witten 不变量以及热带枚举

From wall structures to closed mirror symmetry. The case of $K_{\mathbb{P}^2}$: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration

本文通过墙函数统一恢复 $K_{\mathbb{P}^2}$ 的闭 Gromov-Witten 不变量与重整化周期，建立 Gross-Siebert 构造到经典枚举镜像对称的直接通道，并给出多项式性定理及热带树和，预期推广至一般 Calabi-Yau 镜像对。

The supersingular locus of the Shimura variety for GU(1,n-1) over a ramified prime

We analyze the geometry of the supersingular locus of the reduction modulo p of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q, in the case that p is ramified. We define a stratification of this locus and show that its incidence complex is closely related to a certain Bruhat-Tits simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety associated to some symplectic group over F_p and some Coxeter element. The closure of each stratum is a normal projective variety with at most isolated singularities. The results are analogous to those of Vollaard/…

关于Kähler微分层的无挠性

On torsion-freeness of Kähler differential sheaves

该研究在特征零代数闭域上的正规代数簇中，给出了Kähler微分层无挠性的等价刻画：其一阶形变理想层的正则截面可从奇点轨迹外延拓至奇点轨迹。

向量丛稳定性的一种函子式方法

A functorial approach to the stability of vector bundles

本文用函子式方法研究向量丛在伽罗瓦覆盖下的μ-稳定性保持问题，证明正规射影簇上该稳定轨迹为开集，并在亏格≥2的光滑射影曲线上证明其补集余维数至少为2。

Admissible metrics on compact Kähler varieties

Let $X$ be a normal compact Kähler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the existence of admissible Hermitian-Yang-Mills metrics on it. The existence will hold if one can prove a uniform Sobolev inequality on singular spaces.

Zero-cycles on normal varieties

We prove an extension of the Kato-Saito class field theory for smooth projective schemes over a finite field to schemes with singularities. As an application, we obtain Bloch's formula for the Chow groups of 0-cycles on such schemes. We identify the Chow group of 0-cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasi-projective schemes over algebraically closed fields. This completes the missing $p$-torsion part in the torsion theorem of Spiess and Szamuely.

A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic

We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

Singularity categories of normal crossings surfaces, descent, and mirror symmetry

Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix…

Cohomology of normal bundles of special rational varieties

We give a new method for calculating the cohomology of the normal bundles over rational varieties which are smooth projections of Veronese embeddings. The method can be used also when the projections are not smooth, in this case it provides information about the critical locus of maps between projective spaces.

关于分歧素数上 $\mathrm{GU}(2,4)$ 的 Rapoport-Zink 空间

On the Rapoport-Zink space for $\mathrm{GU}(2, 4)$ over a ramified prime

本文研究了分歧素数上 GU(2,4) 对应 Shimura 簇的超奇异轨迹，证明了相关 Rapoport-Zink 空间的平坦性，并显式描述了基本轨迹模 p 约化的不可约分支，将其与辛群和正交群的 Deligne-Lusztig 簇联系起来。

Typical and Admissible ranks over fields

Let $X(\RR)$ be a geometrically connected variety defined over $\RR$ and such that the set of all its (also complex) points $X(\CC)$ is non-degenerate. We introduce the notion of \emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set of points of $X(\CC)$ such that $P\in \langle S \rangle$ that is stable under conjugation. Any set evincing the admissible rank can be equipped with a \emph{label} keeping track of the number of its complex and real points. We show that in the case of generic identifiability there is an open dense euclidean subset of point…

Weak and semi normalization in real algebraic geometry

We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.

紧致Kähler空间上的大和nef上同调类

Big and nef cohomology classes on compact Kähler spaces

本文证明了紧致弱Kähler空间上Demailly-Paun定理的版本，并证明紧致正规Kähler空间上big和nef Bott-Chern类的受限非Kähler轨迹等于通常的非Kähler轨迹。

伪伽利略4-空间 $G_{1}^{4}$ 中若干特殊曲线的研究

The investigation of some special curves in the pseudo-Galilean 4-space $G_{1}^{4} $

本文依据曲率函数刻画伪伽利略4-空间 $G_{1}^{4}$ 中的任意容许曲线，并给出位置曲线、从切曲线、斜螺旋线和球面曲线等特殊曲线的分类及充要条件。

椭圆点之外 A 型群的 Dolbeault 几何 Langlands 对应

The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

本文证明了对 GL_r 及 Langlands 对偶对 SL_r/PGL_r 在 Hitchin 基的包含椭圆点的开集上的 Dolbeault 几何 Langlands 等价，该开集对应至多 A 型奇点的谱曲线。

A note on non-vanishing and applications

Let $X$ be a normal variety over the field of complex numbers with log terminal singularities and the canonical divisor $K_X$ being ${\bf Q}$-Gorenstein. Assume that $L$ is an ample line bundle over $X$ and $ϕ: X\to Y$ is a morphism supported by $K_X+rL$ for some positive rational number $r$. In the present paper we study the evaluation $ϕ^*ϕ_*(L)\to L$ and the locus of points where it is not surjective which we call relative base point locus of $L$. In particular, we prove that, if the dimension of a fiber of $ϕ$ is small with respect to $r$ then the relative base point locus does not meet th…

Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

关于代数曲线的Griffiths无穷小不变量的注记

Remarks on the Griffiths infinitesimal invariant of algebraic curves

本文研究亏格4代数曲线模空间上两个可容许正规函数（含Ceresa正规函数），并给出特定曲线族上其Griffiths无穷小不变量的消失判据。

A note on Flenner's extension theorem

We show that any $p$-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as $p \le \mathrm{codim}_X (X_{\mathrm{sg}}) - 2$, where $c$ is the codimension of the singular locus. A stronger version of this result, allowing no poles at all, is originally due to Flenner. Our proof, however, is not only completely different, but also shorter and technically simpler. We furthermore give examples to show that the statement fails in positive characteristic.