On equivariant Serre problem for principal bundles
TL;DR: In this paper, a Γ-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Γ, where G and Γ are complex linear algebraic groups.
Abstract: Let EG be a Γ-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Γ, where G and Γ are complex linear algebraic groups. Suppose X is contracti...
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TL;DR: In this article, the authors proved Sylvester's law of nullity and exercise, which states that the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullness of A.
Abstract: In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30
208 citations
TL;DR: In this paper, the notion of compatible ∑-filtered vector space was introduced, where ∑ denotes the fan of a toric variety and G a reductive algebraic group defined over an algebraically closed field.
Abstract: Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.
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TL;DR: In this paper, the authors define the notion of piecewise linear maps from a fan of a toric vector bundle to the underlying space of the Tits building of a linear algebraic group.
Abstract: We define the notion of a piecewise linear map from a fan $\Sigma$ to $\mathcal{B}(G)$, the underlying space of the Tits building of a linear algebraic group $G$. We show that if $X_\Sigma$ is a toric variety over $\mathbb{C}$ with fan $\Sigma$, the set of integral piecewise linear maps from $\Sigma$ to $\mathcal{B}(G)$ classifies (framed) toric principal $G$-bundles on $X_\Sigma$. In particular, this recovers Klyachko's classification of toric vector bundles on $X_\Sigma$. Moreover, extending the case of line bundles, it is shown that criteria for ampleness and global generation of a toric vector bundle translates to convexity conditions on the associated piecewise linear map.
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TL;DR: In this article , the authors consider the question of what makes a variety of toric vector bundles a Mori dream space, and give an answer to this question utilizing a relationship with the associated full flag bundle FL(E).
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08 Aug 2022
TL;DR: In this paper , a classification of rank r torus equivariant vector bundles E on a toric scheme X over a discrete valuation ring, in terms of piecewise affine maps Φ from the polyhedral complex of X to the extended Bruhat-Tits building of GL( r ), was given.
Abstract: . We give a classification of rank r torus equivariant vector bundles E on a toric scheme X over a discrete valuation ring, in terms of piecewise affine maps Φ from the polyhedral complex of X to the extended Bruhat-Tits building of GL( r ). This is an extension of Klyachko’s classification of torus equivariant vector bundles on a toric variety over a field. We also give a simple criterion for equivariant splitting of E into a sum of toric line bundles in terms of its piecewise affine map.
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TL;DR: The Cartan G-space as mentioned in this paper is a generalization of the Cartan topological group to compact Lie groups, and it is defined by Cartan's basic axiom PF for principal bundles in the Seminaire H. Cartan of 1948-49.
Abstract: If G is a topological group then by a G-space we mean a completely regular space X together with a fixed action of G on X. If one restricts consideration to compact Lie groups then a substantial general theory of G-spaces can be developed. However if G is allowed to be anything more general than a compact Lie group, theorems about G-spaces become extremely scarce, and it is clear that if one hopes to recover any sort of theory, some restriction must be made on the way G is allowed to act. A clue as to the sort of restriction that should be made is to be found in one of the most fundamental facts in the theory of G-spaces when G is a compact Lie group; namely the result, proved in special cases by Gleason 12], Koszul [5], Montgomery and Yang [6] and finally, in full generality, by Mostow [8] that there is a "slice" through each point of a G-space (see 2.1.1 for definition). In fact it is clear from even a passing acquaintance with the methodology of proof in transformation group theory that if G is a Lie group and X a G-space with compact isotropy groups for which there exists a slice at each point, then many of the statements that apply when G is compact are valid in this case also. In ? 1 of this paper we define a G-space X (G any locally compact group) -to be a Cartan G-space if for each point of X there is a neighborhood U such that the set of g in G for which g U n U is not empty has compact closure. In case G acts freely on X (i.e., the isotropy group at each point is the identity) this turns out to be equivalent to H. Cartan's basic axiom PF for principal bundles in the Seminaire H. Cartan of 1948-49, which explains the choice of name. In ? 2 we show that if G is a Lie group then the Cartan G-spaces are precisely those G-spaces with compact isotropy groups for which there is a slice through every point. As remarked above this allows one to extend a substantial portion of the theory of G-space that holds when G is a compact Lie group to Cartan G-spaces (or the slightly more restrictive class of proper G-spaces, also introduced in ? 1) when G is an arbitrary Lie group. Part of this extension is carried out in ? 4, more or less by way of showing what can be done. In particular we prove a generalization of Mostow's equivariant embed-
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Journal Article•
TL;DR: In this article, the authors proved Sylvester's law of nullity and exercise, which states that the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullness of A.
Abstract: In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30
208 citations