Let $g$ be a complex semisimple Lie algebra with adjoint group $G$. Suppose that $σ$ is an involutive automorphism of $g$. Then $σ$ induces uniquely an involution of $G$ also denoted by $σ$, let $K=G^σ$ be a subgroup of $σ$-fixed points. Consider a direct decomposition $g=k+p$ of $g$ into eigenspaces for $σ$. Then $p$ is a $K$-module. Denote by $a\subset p$ any maximal abelian ad-diagonalizable subalgebra. Consider the ``baby Weyl group'' $W=N_K(a)/Z_K(a)$. Let $ψ: C[p]^K\to C[a]^W$ be a restriction map of algebras of invariants. Then the famous Chevalley restriction theorem states that $ψ$ is an isomorphism. The aim of this paper is prove the following Theorem. The restriction map $ψ: C[p\times p]^K\to C[a\times a]^W$ is surjective.
If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples.
We formulate and prove Chevalley's theorem in the setting of affine Nash groups. As a consequence, we show that the semi-direct product of two almost linear Nash groups is still an almost linear Nash group.
We prove the Chevalley restriction theorem for the commuting scheme of symplectic Lie algebras. The key step is the construction of the inverse map of the Chevalley restriction map called the spectral data map. Along the way, we establish a certain multiplicative property of the Pfaffian which is of independent interest.
In this note a proof of a differential analog of Chevalley's theorem \cite{C} on homomorphism extensions is given. An immediate corollary is a condition of finitenes of extensions of differential algebras and several equivalent definitions of a differentially closed field, including Kolchin's Nullstellensatz.
The goal of this paper is to introduce a new constructive geometric proof of the affine version of Chevalley's Theorem. This proof is algorithmic and a verbatim implementation resulted in an efficient code for computing the constructible image of rational maps between affine varieties. Our approach extends the known descriptions of uniform matrix product states to $\operatorname{uMPS}(2,2,5)$
We generalize the classical Chevalley-Shephard-Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme X which is etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli) whose stacky structure is supported on the singular locus of X.
We prove an analogue for algebraic stacks of Hermite-Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley-Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.
A fundamental theorem of Barsotti and Chevalley states that every smooth algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. In this expository article, we explain his proof in the language of modern algebraic geometry.
The classical Chevalley-Weil theorem asserts that for an étale covering of projective varieties over a number field K, the discriminant of the field of definition of the fiber over a K-rational point is uniformly bounded. We obtain a fully explicit version of this theorem in dimension 1.
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.
We obtain a quantitative version of the classical Chevalley-Weil theorem for curves. Let $ϕ: \tilde{C} \to C$ be an unramified morphism of non-singular plane projective curves defined over a number field $K$. We calculate an effective upper bound for the norm of the relative discriminant of the number field $K(Q)$ over $K$ for any point $P\in C(K)$ and $Q\inϕ^{-1}(P)$
We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that similar results hold for differentially closed fields that are saturated, in the sense of model theory. In characteristic p > 0, we obtain related results and establish a differential Nullstellensatz. Analogs…
Chevalley's theorem on the images of morphisms of schemes and the principle of quantifier elimination for the theory of algebraically closed fields are widely understood to be two perspectives on the same theorem. In this paper, we demonstrate that both results can easily be proven simultaneously, using a mixture of geometric and logical techniques. In doing so, we give logical meaning to geometric points of schemes and to finitely presented morphisms thereof, in a manner reminiscent of Spencer Breiner's logical schemes.
We study the moduli stack of degree $0$ semistable $G$-bundles on an irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of $G$ (the $E$-pseudo-Levi subgroups), where each stratum is computed in terms of $H$-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where $E$ has a single cusp (res…
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference analog of this theorem. The approach is based on the philosophy that occasionally one needs to pass to higher powers of $σ$, where $σ$ is the endomorphism defining the difference structure. In other words, we consider difference pseudo fields (which are finite direct products of fields) rather than difference fields. We also prove a result on compatibility o…
Let $f_1,\...,f_r$ be polynomials in $n$ variables over a finite field $F$ of cardinality $q$ and characteristic $p$. Let $f_i$ have total degree $d_i$ and define $d=d_1+\...+d_r$. Write $Z$ for the set of common zeros of the $f_i$, over the field $F$. Warning showed that $#(Z\cap H_1)\equiv#(Z\cap H_2)\mod{p}$ for any two parallel affine hyperplanes $H_1,H_2$ in $F^n$. We prove that the same congruence holds to modulus $q$. Warning also proved that $# Z\ge q^{n-d}$ providing that $Z$ is non-empty. We sharpen this inequality in various ways, assuming that $Z$ is not a linear subspace of $F^n$.
We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest.
In this paper we prove restriction theorems for torsion-free sheaves that are (semi)stable with respect to the truncated Hilbert polynomial over a smooth projective variety. Our results apply in particular to Gieseker-semistable sheaves and generalize the well-known restriction theorems of Mehta and Ramanathan. As an application we construct a moduli space of sheaves in higher dimensions.
We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the $V$-filtration.
We prove that if $X$ is the total space of an elliptic principal bundle $π:X\ra B$ which is non-Kähler, then the restriction of any torsion-free sheaf on $X$ to the general fiber of $π$ is semi-stable.
We consider a Chinese remainder theorem for (labeled) graphs. For $X$ a GKM $T$-variety and $Y$ an invariant subvariety, we use this to give a condition for surjectivity of the restriction map $H^*(X) \to H^*(Y)$. In particular, this applies to certain invariant subvarieties in a smooth toric variety.
We present an algebro-geometric perspective on some generalizations, due to S. Takagi, of the restriction theorem for multiplier ideals. The first version of the restriction theorem for multiplier ideals was discovered by Esnault and Viehweg. In a series of papers S. Takagi has discovered generalizations of the restriction theorem and some formulas for multiplier ideals that follow from the restriction theorem. He uses the technique of tight closure and reduction to positive characteristic. We are able to provide an algebro-geometric proof of generalizations of his restriction theorem and his …
Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.
Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles E which do not enjoy this property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J. 96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew. Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Mat…
The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$, let $X$ be a smooth projective variety over $k$, let $H$ be a very ample line bundle on $X$ and let $E$ be a semistable (resp. stable) principal $G$-bundle on $X$ w.r.t. $H$. The main result of this paper is that the restriction of $E$ to a general smooth curve which is a complete intersection of ample hypersurfaces of sufficiently high degree's is again semi…
We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).
We define certain extensions of Jacobi groups of $A_n$, prove an analogue of Chevalley Theorem for their invariants, and construct a Dubrovin Frobenius structure on it orbit space.
We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.
We provide a generalization of Mehta-Ramanathan theorems to framed sheaves: we prove that the restriction of a $μ$-semistable framed sheaf on a nonsingular projective irreducible variety, of dimension greater or equal than two, to a general hypersurface of sufficiently high degree is again $μ$-semistable. The same holds for $μ$-stability under some additional assumptions.
We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.
We prove that both classical Chevalley-Warning-Ax and Tsen theorems hold for the blowing up of a quintic 3-fold along a line of multiplicity 3. Both proofs, which are of the same spirit than the original ones, involve the description of this blowing-up as a subvariety of a toric variety.
