Yano's conjecture

已知类型的不可约辛簇的超奇异Tate猜想：I

Supersingular Tate conjecture for irreducible symplectic varieties of known types: I

本文证明多种已知类型的超奇异不可约辛簇在特定Artin不变量条件下具有Tate或超奇异阿贝尔Chow motive，进而其乘积满足超奇异Tate猜想。

On the Beauville conjecture

We investigate Beauville's conjecture on the Chow ring of irreducible symplectic varieties. For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations, which proves Beauville's conjecture in many new cases. We further apply the same techniques to reduce Beauville's conjecture to Picard rank two.

Irreducibility properties of Keller maps

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conj…

On a conjecture of Harris

For $d \ge 4$, the Noether-Lefschetz locus $\mathrm{NL}_d$ parametrizes smooth, degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least $2$. A conjecture of Harris states that there are only finitely many irreducible components of the Noether-Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large $d$, but is true for $d \le 5$. She also showed that for $d=6, 7$, there are finitely many \emph{reduced}, irreducible components of $\mathrm{NL}_d$ of non-maximal codimension. In this article, we prove that for any $d \ge 6$, there are …

关于几何Langlands猜想

On the geometric Langlands conjecture

设X是特征为p的域上的光滑、完备、几何连通的曲线。几何朗兰兹猜想指出，对于X上的每个不可约秩n局部系统E，可以在X上的秩n丛的模栈上附加一个逆层（在每个连通分支上不可约），这是关于E的Hecke本征层。本文从一个消失猜想出发，导出了几何Langlands猜想.此外，利用Lafforgue最近的结果，我们证明了这个消失猜想，从而几何Langlands猜想，在情况下，当基域是有限的。

仅含不可约纤维的稳定曲线的函数域上的Bogomolov猜想（2.0版）

Bogomolov conjecture over function fields for stable curves with only irreducible fibers (Version 2.0)

设K是函数域，C是K上亏格g >= 2的非等平凡曲线.本文证明了：如果C有一个只含几何不可约纤维的全局稳定模型，则函数域上的Bogomolov猜想成立。

Supersingular irreducible symplectic varieties

We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …

Pencil of irreducible rational curves and Plane Jacobian conjecture

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.

Two generalizations of the PRV conjecture

Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones.

不可约平面曲线奇点的指数

Exponents of an irreducible plane curve singularity

本文给出了不可约平面曲线奇异性的指数（即位移1的谱）的一个显式公式。作为一个应用程序，我们证明在这种情况下赫特林猜想的方差（即平方的标准差）的指数是有界的最大和最小指数除以12之间的差异。

Fulton一个猜想的几何证明

Geometric Proof of a Conjecture of Fulton

给出了W. Fulton关于GL（r）的不可约表示的张量积中不可约表示的重数。

关于Drinfeld的一个猜想

On a Conjecture of Drinfeld

本文证明了当n和d互质时，Drinfeld猜想中关于摇摆定位为余维数一的结论成立。

有限域上几何不可约平面三次曲线的线性系统

Linear system of geometrically irreducible plane cubics over finite fields

本文研究有限域上所有成员几何不可约的平面三次曲线线性系统的最大维数，通过计算证据推测最大射影维数为3，并证明了存在至多含一个几何可约成员的3维线性系统作为迈向该猜想的步骤。

A short proof of a conjecture of Matsushita

In this note we build on the arguments of van Geemen and Voisin to prove a conjecture of Matsushita that a Lagrangian fibration of an irreducible hyperkähler manifold is either isotrivial or of maximal variation. We also complete a partial result of Voisin regarding the density of torsion points of sections of Lagrangian fibrations.

Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and a cocharacter $μ$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.

A proof of the set-theoretic version of the salmon conjecture

We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).

A note on the plane Jacobian conjecture

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.

An approach to the Jacobian Conjecture in terms of irreducibility

We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements.

行列式不可约性，以及Esterov关于$\mathscr{A}$-判别式的猜想

Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants

本文研究行列式在方阵空间中的不可约性，解决Esterov关于$\mathscr{A}$-判别式的猜想，并推动对判别式组件和余维数的深入描述。

The multiplicity one case of Lusztig's conjecture

We prove the multiplicity one case of Lusztig's conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number.

Progress Towards the Conjecture on APN Functions and Absolutely Irreducible Polynomials

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami…

顶点算子代数、Verlinde猜想和模张量范畴

Vertex operator algebras, the Verlinde conjecture and modular tensor categories

设V是一个单顶点算子代数，满足下列条件：（i）V的权小于0的齐次子空间为0，权0的齐次子空间由真空张成，V'作为V-模与V同构. (ii)每个可由非负整数分次的弱V-模都是完全可约的。(iii)V是C_2-上有限的。给出了V的Verlinde猜想的一个证明，即证明了由不可约V-模之间的融合规则所形成的矩阵被模变换τ \mapsto -1/τ在不可约V-模的特征标空间上的作用所给出的矩阵对角化.讨论了Verlinde猜想的一些结果，包括融合规则的Verlinde公式，由τ \mapsto-1/τ作用量给出的矩阵公式以及该矩阵的对称性.我们还证明了当V满足不等价于V的不可约V-模没有权为0的非零元的条件时，V-模范畴上的辫状张量范畴结构的刚性和非退化性质。特别地，V-模范畴具有模张量范畴的自然结构。