Oleg Viro
For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.
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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。
Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets
本文将凸半代数集代数边界的迭代奇异轨迹分层细化为 Whitney (a) 分层，并基于 Teissier 判据提出利用余正规空间和素分解的计算算法，为对偶凸体代数边界的不可约分量提供候选簇列表。
Whitney fold and cusp for algebraic surfaces, and singularities of discriminants
本文研究代数曲面投影的Whitney型奇点稳定性，通过分析判别式奇点轨迹和Vandermonde/Schur矩阵退化，在温和组合条件下枚举所有稳定的多奇点类型。
In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of Zariski equisingularity, w…
A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has a stratification. This result was extended by Lojasiewicz to real (semi)analytic sets. In this paper we present a short geometric proof of existence of stratifications based on Thom's transversality theorem and Milnor's curve selection lemma and not relying on difficult results of Lojasiewicz.
It is easy to find real algebraic varieties with weakly Whitney regular stratifications which are not Whitney regular, and we give such an example in section 3 below. No examples are known among complex analytic varieties however, so that the natural question arises : do Whitney regularity and weak Whitney regularity coincide in the complex case ? As a test, in this paper we study the well-known Briançon-Speder examples. We investigate systematically all of the (infinitely many) Briançon-Speder examples, and establish in particular that none of these examples are weakly Whitney regular. We det…
We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratify…
We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real sheets, the intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend the definition of this self linking number to the case of parametrized generic …
The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets..
We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $α$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $α$. Whenever $α$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibar-Verjovsky global counterpart of the Le-Teissier formula for the local Euler obstruction.
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.
In this short note we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.
We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups.
Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the past century. Affine rotations are the affine equivalents of Euclidean rotations, and include certain shears as well as Euclidean rotations. Affine rotation surfaces are surfaces invariant under affine rotations. In this paper, we analyze several properties of algebraic affine rotation surfaces and, by using some notions and results from affine differential g…
Affine difference algebraic groups are a generalization of affine algebraic groups obtained by replacing algebraic equations with algebraic difference equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these groups and we establish a Jordan-Hölder type theorem that allows us to decompose any affine difference algebraic group into almost-simple affine difference algebraic groups. We also characterize almost-simple affine difference algebraic groups via almost-simple affine algebraic groups.
We observe algebraic derivations on an affine domain B defined over an algebraically closed field of characteristic 0, which are called locally finite derivations in commutative and non-commutative contexts in other references. We observe the graded ring structure which the algebraic derivation defines on B, and give characterizations of a polynomial ring in two variables in terms of algebraic derivations.
We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or n on a normal affine variety of dimension n.
Let $X$ be a smooth affine algebraic variety over the field of complex numbers which is contractible. Then every algebraic $G$-torsor on $X$ is algebraically trivial if $G$ is a semi-simple algebraic group. We also show that if $X$ is a smooth affine algebraic variety such that $Ω^1_X$ is trivial and $X$ is topologically simply connected with $H^i(X,{\mathbb Z})=0$ for $i=1,2$ and $3$, then every algebraic $G$-torsor on $X$ is algebraically trivial for a semi-simple algebraic group $G$.
We prove that the Picard group of a connected affine algebraic group $G$ is isomorphic to the fundamental group of the derived subgroup of the reductive algebraic group $G/{\mathscr R}_u(G)$, where ${\mathscr R}_u(G)$ is the unipotent radical of $G$.
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.
Corrigendum to the paper "A flexible affine $M$-sextic which is algebraically unrealizable"
本文更正先前证明，确立一类伪全纯可实现的平面仿射实代数 $M$-六次曲线同痕类型代数不可实现，从而完成其同痕分类。
A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.
Let $K$ be a complete non-trivially valued non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group over $K$ is affine if and only if the associated $K$-analytic space is Stein; that is, rigid analytic embeddings of it in an affine space may always be chosen to be given by algebraic functions. Arguably curiously, the corresponding statement over the complex numbers is false.
We study commutative associative polynomial operations $\mathbb{A}^n\times\mathbb{A}^n\to\mathbb{A}^n$ with unit on the affine space $\mathbb{A}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.
Algebraic monoid structures on the affine 3-space
本文研究了仿射3空间上代数独异结构的分类，并探讨了相关代数性质。
We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.
Formulated problems concern the following topics: (1) Birationally nonequivalent linear actions; (2) Cayley degrees of simple algebraic groups; (3) Singularities of two-dimensional quotients.
We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.
The affine line minus one point is the underlying space of the algebraic torus of dimension one. However the fibration of an affine algebraic threefold by the affine line minus one point is not always the quotient morphism of the threefold by the algebraic torus. We investigate the structure of affine algebraic threefolds having fibrations by the affine line minus one point. Our results cover not only algebro-geometric but also topological properties of the threefolds. We also consider when the fibration by the affine line minus one point becomes the quotient morphism by an algebraic torus of …
These notes are based on the lecture courses given at the Ruhr-Universit{ä}t-Bochum (03--08.02.1997) and at the Universit{é} Paul Sabatier (Toulouse, 08-12.01.1996).
We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of $\C^n$ equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.
For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.
Let $Y$ be the underlying variety of a connected affine algebraic group. We prove that two embeddings of the affine line $\mathbb{C}$ into $Y$ are the same up to an automorphism of $Y$ provided that $Y$ is not isomorphic to a product of a torus $(\mathbb{C}^\ast)^k$ and one of the three varieties $\mathbb{C}^3$, $\operatorname{SL}_2$, and $\operatorname{PSL}_2$.
We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\tilde{\tilde{/}}G$ of $G$-orbits is an affine scheme in the following two cases: (I) $G$ is finite; (II) $k$ is a field, and $G$ is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group $G$ over an arbitrary field has an integral. Those super-groups which satisfy the assumptio…
Algebraic models of the line in the real affine plane
研究实仿射直线到实仿射平面的光滑有理闭嵌入，在双有理微分同胚下分类，发现存在非等价类，区别于经典范畴的唯一性。
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).
Let $\AAutH (X)$ be the subgroup of the group $\AutH (X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under $\AAutH (X)$ in terms of the dual graphs of the boundaries $\bX \setminus X$ of their SNC-completions $\bX$.
For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.
Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces
该研究在特征零域上构造了维数不小于4的、与仿射空间不同构的高维$\mathbb{A}^1$-可缩仿射簇代数族，证明其是Zariski消去问题及广义消去问题的反例。
By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on the subject, which emphasizes its analytic aspects and points out some open problems.
We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.
We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani
Algebraizability of Vector Bundles over Real Algebraic Varieties
研究实代数簇上拓扑向量丛能否代数化的问题，利用动机同伦理论，证明\(\dim X\leq 3\)时斯蒂费尔 - 惠特尼类代数性是\(V\)可代数化的充要条件，\(\dim X = 4\)时存在阻碍因素，并给出相关应用。
In this first work dedicated to the generalisation of classic algebraic geometry to non algebraically closed fields and axiomatisable classes of fields, we develop the foundations for equiresidual algebraic geometry (EQAG), i.e. algebraic geometry over any commutative field $k$, algebraically closed or not. This is possible thanks to the existence in non algebraically closed fields of many normic forms, i.e. homogeneous polynomials with no non-trivial zero rational in $k$, and relies on an equiresidual generalisation of Hilbert's Nullstellensatz and the Jacobson radical in finitely presented $…
Equivariant algebraic and semi-algebraic geometry of infinite affine space
本文研究了无限仿射空间中Sym(∞)作用下的代数与半代数几何性质，证明了n=1时的消去量词结果，但该结果在n>1时不成立。
Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras
本文证明了任意仿射代数幺半群均可实现为某个有限维非结合代数的自同态幺半群。
Automorphism groups of affine varieties consisting of algebraic elements
本文证明若仿射代数簇自同构群的中性分支由代数元组成，则其为嵌套结构（代数子群的直极限），并利用连通 ind-group 的相关性质完成证明，改进了先前结论。
We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^1$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^1$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^1$-contractible smooth schemes.
Dimension Polynomials for Affine Partial Difference Algebraic Groups
研究有限个相容的差算子下的差代数群理论，证明差代数群的定义理想是差理想，从而得到任意偏差代数群的维度多项式存在性。
We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere $\mathbb{S}^n$ defined by positively homogeneous $C^1$ functions. This covers the case of $C^1$ varieties in the projective space $\mathbb{P}^n$. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one.
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on de…
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén-Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X-Y is different from T. For toric surfaces we are able to classify those which posses a strong version of the algebraic density property (relative to the singular locus…
In this article, we shall discuss the solution to the Zariski Cancellation Problem in positive characteristic, various approaches taken so far towards the possible solution in characteristic zero, and several other questions related to this problem.
We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of André-Quillen homology. In particular, the relation type is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non existence of affine algebras of embedding dimension three and r…
Work of Laurent and Sarnak, following a conjecture of Lang, shows that the number of torsion points of order n on an algebraic subset of an affine complex torus is polynomial periodic. In this paper, we find bounds on the degree and period of this number as a function of n. Some examples, including the number of n torsion points on Fermat curves, are computed to illustrate the methods.
Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $ω$. We prove that the space of all divergence-free (with respect to $ω$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The pr…
We present a restricted version of some affine Jacobi's residue formula (on an affine algebraic variety) with applications to higher dimensional (and affine) analogues of Wood's (or Reiss's) relations about the interpolation of pieces of analytic manifolds. We also compare algebraic and analytic constructions of residue currents along analytic cycles and sketch some of their main properties. This work extend previous work of the authors in the affine space.
We prove that, under mild hypothesis, every normal algebraic space which satisfies the $1$-resolution property is quasi-affine. More generally, we show that for algebraic stacks satisfying similar hypotheses, the 1-resolution property guarantees the existence of a finite flat cover by a quasi-affine scheme.
We study real rational models of the euclidean plane $\mathbb{R}^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^2$ is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by givin…
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