Gábor Bodnár, Herwig Hauser, Josef Schicho, Orlando Villamayor
Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that plain varieties are stable under blowup in smooth centers.
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We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{C}^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{C}^*$-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X\times X$ and $Y\times Y$.
Two quasi-projective varieties are called piecewise isomorphic if they can be stratified into pairwise isomorphic strata. We show that the m-th symmetric power $S^m(C^n)$ of the complex affine space $C^n$ is piecewise isomorphic to $C^{mn}$ and the m-th symmetric power $S^m(CP^\infty)$ of the infinite dimensional complex projective space is piecewise isomorphic to the infinite dimensional Grassmannian $Gr(m,\infty)$.
We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.
Prym varieties that are not isomorphic to Jacobian
通过研究具有素数阶自同构的曲线上的分歧双覆盖的Prym簇，利用伽罗瓦理论构造了非同构于雅可比簇的Prym簇。
The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of non-isomorphic principal polarizations on any given abelian variety. We show, for example, that the Jacobian of the genus four underlying curve of Schoen's I-WP minimal surface has at least 9 non-isomorphic principal polarizations. We also explore the Jacobians of Klein's quartic, Fermat's quartic, Bring's curve, and more.
In this paper, we address the following two general problems: given two algebraic varieties in ${\bf C}^n$, find out whether or not they are (1) isomorphic; (2) equivalent under an automorphism of ${\bf C}^n$. Although a complete solution of either of these problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one, and use it to construct numerous examples of isomorphic, but …
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。
The isomorphism problem for Grassmannian Schubert varieties
本文研究了可能不同的Grassmannian中Schubert簇与Richardson簇的同构判定问题，证明了其分别取决于Young图转置相同与偏斜图作为偏序集半同构。
Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.
Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture
本文研究了极小族的有限性问题，证明了在固定点曲线和极小极种变种下，局部平凡族的同构类有限，并在假设条件成立时进一步证明了投影族的有限性。
Let G be a finite group. For each integral representation $ρ$ of G we consider $ρ-$decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with $ρ(G)-$action, of dimension equal to the degree of $ρ$, which admit a decomposition of the lattice for X into two G-invariant sublattices isotropic with respect to $\Im H$, with one of the sublattices $\mathbb{Z}G-$isomorphic to $ρ$. We give a construction for $ρ-$decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves. Conversely…
We show that nef cycle classes on smooth complete spherical varieties are effective, and the products of nef cycle classes are also nef. Let X be a smooth projective spherical variety such that its effective cycle classes of codimension k are nef, where 1<= k <= dim(X)-1. We study the properties of X. And we show that if X is a toric variety, then X is isomorphic to the product of some projective spaces; if X is toroidal, then X is isomorphic to a rational homogeneous space; if X is horospherical, dim(X)>= 3 and k=2, then effective divisors on X are nef; if X is horospherical and effective div…
The Debarre-Voisin hyperkähler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3…
A note on dual abelian varieties
本文显式构造了不与对偶同构的非主极化阿贝尔簇，给出高维下次最大皮卡数的实例，证明极一般非主极化阿贝尔簇不与对偶同构，还给出同构于对偶但无主极化的阿贝尔簇例子。
The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X\times\mathbb{C} and Y\times\mathbb{C} are isomorphic are isomorphic. Iitaka and Fujita established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a famous counter-example using smooth affine surfaces with additive group actions. His construction was further generalized by Fieseler and Wilkens to describe a larger class of affine surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic acti…
We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.
We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a $T$-fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to $\PP^n$ or to the blow-up of $\PP^n$ along a $\PP^{n-2}$. As expected, such results are proved using toric Mori theory due to Reid.
For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, we construct conjugate varieties whose algebras of K-rational (p,p)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when K is contained in an imaginary quadratic number field; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on their second real cohomolog…
The affine cancellation problem, which asks whether complex affine varieties with isomorphic cylinders are themselves isomorphic, has a positive solution for two dimensional varieties whose coordinate rings are unique factorization domains, in particular for the affine plane, but counterexamples are found within normal surfaces Danielewski surfaces and factorial threefolds of logarithmic Kodaira dimension equal to 1. The latter are therefore remote from the affine three-space, the first unknown case where the base of one cylinder is an affine space. Locally trivial Ga-actions play a significan…
This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in K_3^M(k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients.
In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the étale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the étale topology to the combinatorial data describing affine space endowed with a linear torus action. Finally, we provide an effecti…
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is irreducible, we prove that $A$ is isomorphic to the self-product of an elliptic curve and $A/G\simeq \mathbb P^n$. In the general case, assuming $\dim(A^G)=0$, we prove that $A/G$ is isomorphic to a direct product of projective spaces.
Isomorphism of Multiprojective Bundles and Projective Towers
本文研究不同维数光滑射影簇上射影丛的同构问题，证明不同射影空间上的多射影丛除平凡情形外不同构，并给出高度3射影塔顶层簇同构的充要条件。
We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-Fujita, and more generally for varieties which are not dominantly covered by images of the punctured affine line. We show in contrast that cancellation fails in general in every dimension bigger or equal to two.
Cubic fourfolds with birational Fano varieties of lines
本文给出了几对非同构三次四重体例子，证明其 Fano 直线簇是有理等价的，为 Fourier-Mukai 伙伴猜想提供新证据，并提出三次四重体与其 Fano 直线簇有理等价性相关的猜想。
The goal of this paper is to define families of toric varieties and to study their properties. These families are locally trivial fibrations over some base, whose fibres are isomorphic to a fixed complete toric variety. The study is motivated by the occurence of such families in the context of gauged sigma models for toric varieties.
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
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.
Fibrations associated to smooth quotients of abelian varieties
本文研究阿贝尔簇被保持原点的有限自同构群作用所得光滑商的结构，将其刻画为阿贝尔簇上以射影空间乘积为纤维的纤维化，并分类纤维粘合方式、证明一般情形为纤维积。
Towards the Global Torelli Theorem
本文证明所有维数下K-平凡簇的有限性结果，利用极化K3曲面的整体托雷利定理建立纤维同构的霍奇理论充分条件，并应用于双有理等价纤维稠密的射影族，得出所有纤维同构的结论。
Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis.
An affine varieties with an action of a semisimple group $G$ is called "small" if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative group $\mathbb{K}^*$ commuting with the $G$-action. We show that $X$ is determined by the $\mathbb{K}^*$-variety $X^U$ of fixed points under a maximal unipotent subgroups $U$ of $G$. Moreover, if $X$ is smooth, then $X$ is a $G$-vector bundle over the quotient $X// G$. If $G$ is of type $A_n$ ($n>1$), $C_n$, $E_6$, $E_7$ or $E_8$, we show that all affine $G$…
A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.
We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.
In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of Demushkin on affine toric varieties.
Let I be a finite set and CI be the algebra of functions on I. For a finite dimensional C algebra A with \CI contained in A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type) varieties. There is a special case of interest in representation theory. Lusztig defined two varieties related to a quiver and gave a bijection between their C-points (citation in article). Savage and Tingley raised the question (citation in article) of whether these varieties are isomorphic as algebraic varieties. This question has been open since Lusztig's o…
In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.
We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and János Kollár.
We give examples of complex hyperplane arrangements for which the top characteristic variety contains positive-dimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.
On non-projective complete toric varieties
本文证明每个Picard数至多为五的光滑非射影完全环面三重簇在经历有限次flop或anti-flip后可变为射影簇。
In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The homology group of $\Zl$ is a representation of a symmetric Kac-Moody Lie algebra $\mathfrak g$, isomorphic to the tensor product $V(λ_1)\otimes...\otimes V(λ_N)$ of integrable highest weight modules. (2) The set of irreducible components of $\Zl$ has a structure of a crystal, isomorphic to that of the $q$-analogue of $V(λ_1)\otimes...\otimes V(λ_N)$. (3) The equiv…
Let $G$ be a connected reductive algebraic group. In this note we prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial $\mathbb{G}_a$-actions that are homogeneous with respect to a Borel subgroup of $G$. As an application we get that a smooth affine $G$-spherical variety that is non-isomorphic to a torus is determined by its automorphism group inside the category of smooth affine irreducible varieties.
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…
We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X and Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic affine 3-spaces.
We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a para…
Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $¶^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $¶^N$ that can be projected isomorphically into $¶^{2n}$ when $N\geq\binom{n+2}{2}-2$.
We consider the distribution of p-power group schemes among the torsion of abelian varieties over finite fields of characteristic p, as follows. Fix natural numbers g and n, and let $ξ$ be a non-supersingular principally quasipolarized Barsotti-Tate group of level n. We classify the F_q-rational forms $ξ^α$ of $ξ$. Among all principally polarized abelian varieties X/F_q of dimension g with p^n-torsion geometrically isomorphic to ξ, we compute the frequency with which X[p^n] is isomorphic to $ξ^α$. The error in our estimate is bounded by $D/\sqrt{q}$ where D depends on g, n and p, but not on $ξ…
We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.
Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.
Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the explanation of the general results of Carrell and Liebermann on the cohomology of T-varities, in the case of toric varieties. We give an explicit isomorphism between Gr A(Z) and Brion's description of the polytope algebra.
We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles $\mathcal O_{\mathbb P^1}(-n)$ for $n \ge 1$.
We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.
Peterson varieties and toric orbifolds associated to Cartan matrices
本研究针对Peterson簇与特定环轨形有理上同调环同构的已知结论，构造了二者间的显式态射，证明该同构并非偶然代数巧合，深化了对Peterson簇与根系相关环轨形关联的理解。
We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional f…
Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms
本文提出一个框架以精确化“由簇参数化的簇的实形式族”概念，并构造了首个具有任意大代数模维度的非同构实形式的拟射影实曲面例子。
Locally Trivial Deformations of Toric Varieties
本文从组合角度研究锥形簇的局部平凡变形，构造了变形函子并证明其与关联的锥形簇的变形函子同构，提出新的判据和计算高阶障碍的公式。
The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.
We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we pro…
Let $K$ be a field of characteristic $0$ and let $G$ and $H$ be connected commutative algebraic groups over $K$. Let $\text{Mor}_0(G,H)$ denote the set of morphisms of algebraic varieties $G \to H$ that map the neutral element to the neutral element. We construct a natural retraction from $\text{Mor}_0(G,H)$ to $\text{Hom}(G,H)$ (for arbitrary $G$ and $H$) which commutes with the composition and addition of morphisms. In particular, if $G$ and $H$ are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If $G$ has no non-trivial unipotent group as a direct factor, w…
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