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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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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

Journal ArticleDOI
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.

11 citations

Posted Content
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.

6 citations

Journal ArticleDOI
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).

1 citations

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.

1 citations

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TL;DR: In this article, a reductive complex Lie group acting holomorphically on X = ℂn is considered, and the holomorphic linearization problem is formulated as follows: if there is a holomorphic change of coordinates on X such that the G-action becomes linear.
Abstract: Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label.

12 citations

Posted Content
TL;DR: In this article, the authors classify holomorphic and algebraic torus equivariant principal bundles over a nonsingular toric variety, where the principal is a complex linear algebraic group.
Abstract: We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

10 citations

Journal ArticleDOI
TL;DR: In this article, the authors classify holomorphic and algebraic torus equivariant principal G-bundles over a nonsingular toric variety X, where G is a complex linear algebraic group.
Abstract: We classify holomorphic as well as algebraic torus equivariant principal G-bundles over a nonsingular toric variety X, where G is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

9 citations

Journal ArticleDOI
Abstract: We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle E with a structure group bundle $${{\mathscr {G}}}$$ on a reduced Stein space X, such that the fibre of E is a homogeneous space of the fibre of $${{\mathscr {G}}}$$ , with the complexification $$K^{{\mathbb {C}}}$$ of a compact real Lie group K acting on X, $${{\mathscr {G}}}$$ , and E. Our main result is that the inclusion of the space of $$K^{{\mathbb {C}}} \hbox {-equivariant}$$ holomorphic sections of E over X into the space of $$K\hbox {-equivariant}$$ continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch’s classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

9 citations

Posted Content
TL;DR: In this article, it was shown that a biholomorphism of a reductive complex Lie group acting holomorphically on a Stein manifold is sufficient and sufficient for linearisation.
Abstract: Let $G$ be a reductive complex Lie group acting holomorphically on $X={\mathbb C}^n$. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ${\mathbb C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $\Phi\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $Q_X$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $\Phi$ as above. Then $\Phi$ induces a biholomorphism $\phi\colon Q_X\to Q_V$ which is stratified, i.e., the stratum of $Q_X$ with a given label is sent isomorphically to the stratum of $Q_V$ with the same label. The counterexamples to the Linearisation Problem construct an action of $G$ such that $Q_X$ is not stratified biholomorphic to any $Q_V$. Our main theorem shows that, for most $X$, a stratified biholomorphism of $Q_X$ to some $Q_V$ is sufficient for linearisation. In fact, we do not have to assume that $X$ is biholomorphic to ${\mathbb C}^n$, only that $X$ is a Stein manifold.

7 citations