Questions on self maps of algebraic varieties

Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves

We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.

Counting rational points on algebraic varieties

Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and fixed dimension, and are essentially best possible for varieties of degree at least six.

Modular Shimura varieties and forgetful maps

We consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.

Elliptic genera, real algebraic varieties and quasi-Jacobi forms

The first part surveys the push forward formula for elliptic class and various applications obtained in the papers by L.Borisov and the author. In the remaining part we discuss the ring of quasi-Jacobi forms which allow to characterize the functions which are the elliptic genera of almost complex manifolds and extension of Ochanine elliptic genus to certain singular real algebraic varieties.

The standard sign conjecture on algebraic cycles: the case of Shimura varieties

We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur's conjectures plus a bit more). We also indicate what is known (to us) about these statements.

The p-cohomology of algebraic varieties and special values of zeta functions

The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.

Diophantine Approximation on Varieties IV: Derivated algebraic distance and derivative metric Bezout Theorem

The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $θ$ to two properly intersecting cycles in projective space with the derivatives of the algebraic distance of $θ$ to their intersection. This improvement can be used to make algebraic independence criteria, to be proved in a forthcoming paper, more flexible and to refine Approximation results that are proved using the metric Bezout Theorem.

代数簇的Hodge与Frobenius余级

Hodge and Frobenius colevels of algebraic varieties

本文为代数簇的Hodge与Frobenius余级在所有上同调维度上提供了以维数和定义方程多重次数表示的新下界，从而增强了Esnault与第一作者所提问题的肯定回答。

On algebraic group varieties

Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.

A note on the canonical divisor of the generalised affine Stiefel algebraic varieties

In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.

The structure of algebraic varieties

ICM lecture on minimal models and moduli of varieties.

Minkowski sums and Hadamard products of algebraic varieties

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.

Families of algebraic varieties and towers of algebraic curves over finite fields

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.

代数簇的合冲与代数曲线的对称积

Syzygies of algebraic varieties through symmetric products of algebraic curves

本文综述了利用代数曲线对称积的几何方法证明的关于代数簇合冲的若干近期结果，包括亏格猜想、割线簇与切发展曲面的合冲、格林猜想及渐近合冲。

Kodaira dimension of algebraic fiber spaces over abelian varieties

In this short article we provide a proof of the Iitaka conjecture for algebraic fiber spaces over abelian varieties.

Algebraic varieties with semialgebraic universal cover

We study projective varieties whose universal cover is biholomorphic to a semialgebraic open subset of a projective variety.

Exercises in the birational geometry of algebraic varieties

This a collection of about 100 exercises. It could be used as a supplement to the book Kollár--Mori: Birational geometry of algebraic varieties.

Algebraic independence in the Grothendieck ring of varieties

We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.

Algebraic varieties are homeomorphic to varieties defined over number fields

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$

Algebraic vector bundles on the 2-sphere and smooth rational varieties with infinitely many real forms

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

Moduli of algebraic varieties

We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a wide range of moduli problems in algebraic geometry.

On the density of rational and integral points on algebraic varieties

We prove a conjecture of Heath-Brown on the number of rational points of bounded height for a large class of projective varieties.

通过代数减少法研究Kähler变种的丰富性猜想的方法

An approach to the abundance conjecture for Kähler varieties via algebraic reduction

本文通过归纳法建立Kähler变种丰富性猜想的策略，利用代数减少纤维化将问题转化为射影变种的丰富性猜想，特别在四维情况下获得了一些非代数或轨型为零的Kähler四重的丰富性猜想案例。

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in th…

On tangent cones at infinity of algebraic varieties

In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gröbner bases.

复代数簇的相对Lipschitz饱和

Relative Lipschitz saturation of complex algebraic varieties

本文研究复代数簇的相对Lipschitz饱和，解答了Pham和Teissier关于一般代数Lipschitz饱和有限性的公开问题，并给出两代数簇可被闭点上局部双Lipschitz同胚的代数态射连接的代数判据。

具有约化群作用的复代数簇上的实结构

Real structures on complex algebraic varieties with a reductive group action

这是一篇数学代数几何领域的综述，聚焦于研究带有约化群作用的复代数簇上的实结构相关内容。

论代数簇基本形式的消没

On vanishing of fundamental forms of algebraic varieties

本文利用线丛主部层研究代数簇基本形式，建立任意阶消没定理，并揭示其与高阶高斯映射和高阶切簇的联系。

某些广义Prym簇上Hodge类的代数性

Algebraicity of Hodge classes on some generalized Prym Varieties

本文通过重新审视Chad Schoen在特定Prym簇上构造的代数环，利用无分歧几何类域论证明某些广义Prym簇上Hodge类的代数性。

Real Torus Actions on Real Affine Algebraic Varieties

We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.

Birational geometry of algebraic varieties

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

On Algebraic Hyperbolicity of Log Varieties

Fix a very general hypersurface D in P^n of degree at least 2n + 1 and we show that the complement P^n - D does not contain any algebraic torus C^*.

Algebraic cycles on Prym varieties

This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.

关于代数簇态射纤维的大小

On the size of the fibers of a morphism of algebraic varieties

本文证明代数簇态射中纤维大小等于给定整数或无穷大的点集是可构造集。

Relations between some invariants of algebraic varieties in positive characteristic

We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfaces

On the algebraic fundamental group of smooth varieties in characteristic $p>0$

We define an analog in characteristic $p>0$ of the proalgebraic completion of the topological fundamental group of a complex manifold.

An algebraic-geometric construction of ind-varieties of generalized flags

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in [DP]. These latter ind-varieties have been introduced in terms of the ind-group SL(\infty) (respectively, O(\infty) or Sp(\infty) for isotropic generalized flags), and the…

On the cut\,-\,and\,-\,paste property of algebraic varieties

We show that the above\,-\,named property (after M.\,Larsen and V.\,Lunts) does not hold in general.

Mumford's influence on the moduli theory of algebraic varieties

We give a short appreciation of Mumford's work on the moduli of varieties by putting it into historical context. By reviewing earlier works we highlight the innovations introduced by Mumford. Then we discuss recent developments whose origins can be traced back to Mumford's ideas.

On Endomorphism of Algebraic Varieties

We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism of complex algebraic variety that is injective on the complement of a closed subvariety of codimension at least $2$, is an automorphism.

Virtual Betti numbers of real algebraic varieties

The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).

Geometry on arc spaces of algebraic varieties

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.

实代数簇的有效惠特尼分层

Effective Whitney Stratification of Real Algebraic Varieties

本研究提出计算实代数簇惠特尼分层的新算法，采用余法线或极线技术处理实簇复化并证明其可用实多项式描述，还将方法拓展至全半代数集与实代数映射的分层问题。

Algebraic cobordism of varieties with G-bundles

Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…

Perverse sheaves and the topology of algebraic varieties (2015 PCMI)

Another introduction to perverse sheaves with some exercises. Expanded version of five lectures at the 2015 PCMI.

Connected components of real $CB_{n}$ algebraic varieties

Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for the space of $CB_{n}$-algebraic varieties. We construct $CB_{n}$-polynomials using Young-posets and partitions of integers. Our results establish bounds on the number of connected components for a given set of coefficients. It turns out that this number can achieve an upper bound of $2^{n}+1$ for specific coefficients. We introduce a new method to characteri…

Topological properties of real algebraic varieties: du côté de chez Rokhlin

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

Surjectivity of cycle maps for singular varieties

A theorem of Jannsen asserts that if a smooth projective variety has injective cycle class maps, it has surjective cycle class maps. The object of this note is to present a version of Jannsen's theorem for singular quasi--projective varieties.

Computations with rational maps between multi-projective varieties

We shortly describe the algorithms behind some of the functions provided by the Macaulay2 package MultiprojectiveVarieties, a package for multi-projective varieties and rational maps between them.

Infinite-dimensional algebraic varieties and proof of the Jacobian Conjecture

This paper has been withdrawn by the author, due an error in the proof of Proposion 2.13.

Multiplication maps and vanishing theorems for toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.

Higher Gauss Maps of Veronese Varieties---a generalization of Boole's formula and degree bounds for higher Gauss map images---

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.

On pluricanonical systems of algebraic varieties of general type

We extend Kollar's technique to look for an explicit function h(n) with phi_m birational onto its image for all integers $m\geq h(n)$ and for all n-dimensional nonsingular projective varieties of general type.

Local volumes on normal algebraic varieties

In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.