On algebraic group 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.

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.

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.

The weight filtration for real algebraic varieties II: Classical homology

We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.

Arithmetic D-modules over algebraic varieties of characteristic p>0

Let $k$ be a field of characteristic $p>0$ not necessarily perfect. Using Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for realizable $k$-schemes of finite type.

On linear dependence over complete differential algebraic varieties

We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.

齐次簇中非常一般超曲面的代数双曲性

Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties

本文推广了Coskun、Riedl和Yeong的技巧，得到了有理齐次簇中非常一般超曲面的代数双曲性的几乎最优次数界，并给出了格拉斯曼簇、正交/辛格拉斯曼簇及旗簇等实例。

代数簇的Flasque拟消解

Flasque quasi-resolutions of algebraic varieties

本文将代数环面的Flasque消解概念推广到含齐性空间在内的更广泛代数簇，并据此得到齐性空间$R$-等价类数量的增强下界。

Estimates of Characteristic numbers of real algebraic varieties

We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $\mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.

Learning Algebraic Varieties from Samples

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety, obtained by pinching a smooth variety along a finite subscheme. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to the Picard variety obtained by this construction.

Divisorial contractions of algebraic varieties

This paper has been withdrawn by the author. It is not true that plt blow-up of toric singularity is toric (in dimension 3 too) and the first main theorem is incorrect also. The error is in deformation argument. I am grateful to Ivan Cheltsov for discussions.

Lie algebras of zero divergence vector fields on complex affine algebraic varieties

For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its dimension depends only on the topology of $X$. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties that admit a nowhere-zero algebraic form of top degree (which plays the role of a volume form).

Algebraic cycles on Fano varieties of some cubics

This note is about cycle-theoretic properties of the Fano variety of lines on a smooth cubic fivefold. The arguments are based on the fact that this Fano variety has finite-dimensional motive. We also present some results concerning Chow groups of Fano varieties of lines on certain cubics in other dimensions.

Homogeneous varieties under split solvable algebraic groups

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.

Elliptic genus of singular algebraic varieties and quotients

We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.

On algebraic varieties with finite polyhedral Mori cone

The fundamental property of Fano varieties with mild singularities is that they have a finite polyhedral Mori cone. Thus, it is very interesting to ask: What we can say about algebraic varieties with a finite polyhedral Mori cone? I give a review of known results. All of them were obtained applying methods which were originated in the theory of discrete groups generated by reflections in hyperbolic spaces with a fundamental chamber of finite volume. This is my report on Fano conference in Torino, October 2002

Resolution of singularities of complex algebraic varieties and their families

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.

Algebraic varieties with quasi-projective universal cover

We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.

实代数簇之间映射的逼近

Approximation of maps between real algebraic varieties

本文刻画了具有逼近性质的实代数簇 Y，即任意与正则映射同伦的 C^inf 映射可在 C^inf 拓扑下被正则映射逼近，并进一步研究了结合插值条件的情形及奇异簇上连续映射的变体。

The weight filtration for real algebraic varieties

Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.

Simplicial homotopy theory of algebraic varieties over real closed fields, Part 1

We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur comparing the real homotopy type with the étale homotopy type. This paper is part one of a sequence of papers on this topic.