Normal Algebraic Variety Research Articles

On torsion‐freeness of Kähler differential sheaves

AbstractLet be a normal algebraic variety over an algebraically closed field . We prove that the Kähler differential sheaf of is torsion‐free if and only if any regular section of the ideal sheaf of the first order deformation of inside , defined outside the singular locus of , extends regularly to the singular locus.

On singularity and normality of regular nilpotent Hessenberg varieties

Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.

CHARACTERIZATION OF n-DIMENSIONAL NORMAL AFFINE SLn-VARIETIES

We show that any normal irreducible affine n-dimensional SLn-variety X is determined by its automorphism group seen as an ind-group in the category of normal irreducible affine varieties. In other words, if Y is an irreducible affine normal algebraic variety such that Aut(Y) ≃ Aut(X) as an ind-group, then Y ≃ X as a variety. If we drop the condition of normality on Y , then this statement fails. In case n ≥ 3, the result above holds true if we replace Aut(X) by \U0001d4b0(X), where \U0001d4b0(X) is the subgroup of Aut(X) generated by all one-dimensional unipotent subgroups. In dimension 2 we have some interesting exceptions.

Decomposition theorem and torus actions of complexity one

We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one.

Cohomology of line bundles on horospherical varieties

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. The main tool in this article is the machinery of Grothendieck–Cousin complexes, and we also prove a Kunneth-like formula for local cohomology.

Arithmetic representations of fundamental groups I

Let $X$ be a normal algebraic variety over a finitely generated field $k$ of characteristic zero, and let $\ell$ be a prime. Say that a continuous $\ell$-adic representation $\rho$ of $\pi_1^{\text{\'et}}(X_{\bar k})$ is arithmetic if there exists a representation $\tilde \rho$ of a finite index subgroup of $\pi_1^{\text{\'et}}(X)$, with $\rho$ a subquotient of $\tilde\rho|_{\pi_1(X_{\bar k})}$. We show that there exists an integer $N=N(X, \ell)$ such that every nontrivial, semisimple arithmetic representation of $\pi_1^{\text{\'et}}(X_{\bar k})$ is nontrivial mod $\ell^N$. As a corollary, we prove that any nontrivial semisimple representation of $\pi_1^{\text{\'et}}(X_{\bar k})$, which arises from geometry, is nontrivial mod $\ell^N$.

Volume and Hilbert functions of R-divisors

Let X be a proper, normal algebraic variety of dimension n over a field K and D an R-divisor on X. The Hilbert function of D is the function H(X,D) : m 7→ h(mD) := dimK H(X,OX(bmDc)); defined for all m ∈ R. If D is an ample Cartier divisor then H(X,D) agrees with the usual Hilbert polynomial whenever m 1 is an integer, but in general H(X,D) is not a polynomial, not even if D is a Z-divisor and m ∈ Z. The simplest numerical invariant associated to the Hilbert function is the volume of D, defined as

Complex-analytic quotients of algebraic $$G$$ G -varieties

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class $\mathscr{Q}_G$ has a realisation as a good quotient, and that every complete algebraic variety in $\mathscr{Q}_G$ is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class $\mathscr{Q}_T$, where $T$ is an algebraic torus, is a toric variety.

On restriction of roots on affine $$T$$ T -varieties

Let \(X\) be a normal affine algebraic variety with a regular action of a torus \(\mathbb {T}\) and \(T\subset \mathbb {T}\) be a subtorus. We prove that each root of \(X\) with respect to \(T\) can be obtained by restriction of some root of \(X\) with respect to \(\mathbb {T}\). This allows to give an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of a subtorus action on an affine toric variety are obtained.

Local volumes of Cartier divisors over normal algebraic varieties

Local volumes of Cartier divisors over normal algebraic varieties

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

LetX n ! P N be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of (X;!FSjX) restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n 1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G.

Modulus of a rational map into a commutative algebraic group

For a rational map $\phi: X \to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\phi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the known theories for curves to higher dimensional varieties.

The singular Riemann–Roch theorem and Hilbert–Kunz functions

In the paper, via the singular Riemann–Roch theorem, it is proved that the class of the eth Frobenius power A e can be described using the class of the canonical module ω A for a normal local ring A of positive characteristic. As a corollary, we prove that the coefficient β ( I , M ) of the second term of the Hilbert–Kunz function ℓ A ( M / I [ p e ] M ) of e vanishes if A is a Q -Gorenstein ring and M is a finitely generated A-module of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the eth Frobenius power F ∗ e O X can be described using the canonical divisor K X .

Solvable Fundamental Groups of Algebraic Varieties and Käler Manifolds

It is shown that if the fundamental group of a normal algebraic variety, respectively Zariski open subset of a compact Kähler manifold, is solvable with a faithful linear representation over Q, respectively polycyclic, then it is virtually nilpotent.

Some remarks on nef and good divisors on an algebraic variety

Let X be a complete normal algebraic variety defined over a field k algebraically closed and let D be a Cartier divisor on X. Under the assumption that D is nef and good, we describe boundedness properties of base locus of the systems | nD|, algebraic convexity properties of X \\ supp( D), and a criterium for semiampleness.

On actions of reductive groups with one-parameter family

Actions of reductive groups on normal algebraic varieties with one-parameter families of spherical orbits of maximal dimension are studied under the assumption that the categorical quotient for the action is one-dimensional. As an application, the classification of the actions of the group on three-dimensional normal affine varieties is completed. The ground field is assumed to be algebraically closed and of characteristic zero.

Two-orbit varieties with smaller orbit of codimension two

Let G be a connected algebraic group and let X be an irreducible normal algebraic variety on which G acts regularly, where all our objects are defined over the base field 112. If G acts on X with an open orbit f~, then X is called an (algebraic) almost-homogeneous variety. Several approaches have been used to obtain a classification of such varieties under some additional assumptions. For example, a classification - even of complex analytic almost-homogeneous spaces - has been carried out in the case when the comple- ment ofis of dimension smaller or equal to one ((11) and (15)). The work of Luna and Vust ((16)) is fundamental for a good understanding especially in the case when G is a reductive connected group and a Borel subgroup of G acts on X with an open orbit in fL The open orbit f2 is then called a spherical homogeneous space and various results have been obtained for embeddings of such spaces (see e.g. (16), (5), (6), (19), (20)). Two-orbit varieties, i.e. complete almost-homogeneous varieties where the complement E of the open orbit is again an orbit of G, are in some sense the "elementary building blocks" of group actions. Important results for this approach have been obtained by Ahiezer ((1), (2), (3), see also (12)). In his papers, all complete two-orbit varieties X where the smaller orbit is a hypersurface are classified. The classification problem for codim~ E > 1 seems difficult. We hope that progress in the case where E is of codimension two which we present here will lead to some general results. In this paper we consider two-orbit varieties X with smaller orbit E of codimension two such that the acting group G is reductive. Of course, there are two-orbit spaces of this latter kind which arise from one of Ahiezer's examples by blowing down the smaller orbit. Simple examples of these are fibre bundles over homogeneous rational manifolds where the fibre is a Hirzebruch surface. These may be blown down to bundles where the fibre is a Hirzebruch cone. Carefully looking at Ahiezer's classification, it is in principle possible to determine all two-orbit varieties with smaller orbit of codimension two which may be obtained by blowing down the hypersurface orbit in one of Ahiezer's examples. This is why we concentrate on those varieties which don't arise in this way. The orbits of the maximal compact subgroup of G are then of codimension greater than one. We call this the Case 2. We obtain a complete picture of all two-orbit varieties arising in this case (see Theorem A and Theorem B). In short, each of these varieties is a fibre bundle G x Pl Z with fibre Z over the homogeneous rational manifold G/P1, and the construction of this bundle may be explicitly described. Conversely, the precise description of the fibre Z and

Homeomorphism versus isomorphism for varieties

It is well known that a morphism onto a weakly normal algebraic variety that is both birational and a universal homemorphism is an isomorphism of varieties. in this note, the author shows that a birational morphism onto a weakly normal algebraic variety X with a continuous inverse is necessarily an isomorphism of varieties if and only if X had no one-dimensional components. Thus the characterization of the weak normalization of a variety X as the largest variety that is birationally homeomorphic to X is only valid in the absence of one-dimensional components. All varieties are over a fixed algebraically closed field.

On Noetherianness of Nash rings

We introduce a class of rings, called Nash Rings, which generalize the notation of rings of Nash functions. Let k k be any field, X X be a normal algebraic variety in k n {k^n} , and U ⊂ X U \subset X . A Nash ring D D is the algebraic closure of Γ ( X , O X ) \Gamma (X,{\mathcal {O}_X}) in a suitable domain B B such that U U is contained in the maximal spectrum of B B and Γ ( X , O X ) \Gamma (X,{\mathcal {O}_X}) is analytically isomorphic to B B at each x ∈ U x \in U . We show that D D is a ring of fractions of the integral closure of Γ ( X , O X ) \Gamma (X,{\mathcal {O}_X}) in B B . Moreover, if k k is algebraically nonclosed and if every algebraic subvariety V ⊂ X V \subset X intersects U U in a finite number of connected components (in the topology induced by B B ), then D D is noetherian.

Fundamental, Picard, and Class Groups of Rings of Invariants

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],