scholarly journals The null-cone and cohomology of vector bundles on flag varieties

2016 ◽  
Vol 20 (17) ◽  
pp. 482-498
Author(s):  
Kari Vilonen ◽  
Ting Xue
Keyword(s):  
Vector Bundles ◽  
Null Cone ◽  
Flag Varieties

scholarly journals Moduli of toric vector bundles

2008 ◽  
Vol 144 (5) ◽  
pp. 1199-1213 ◽  
Author(s):  
Sam Payne
Keyword(s):  
Linear Group ◽  
Group Action ◽  
Chern Class ◽  
Vector Bundles ◽  
Closed Subscheme ◽  
Flag Varieties ◽  
Moduli Stack ◽  
Rank Conditions ◽  
Partial Flag Varieties

AbstractWe give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles.

scholarly journals Four-dimensional Fano quiver flag zero loci

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180791
Author(s):  
Elana Kalashnikov
Keyword(s):  
Electronic Supplementary Material ◽  
Vector Bundles ◽  
Flag Variety ◽  
Flag Varieties ◽  
Joint Work ◽  
Genus Zero ◽  
Fano Manifolds ◽  
Ample Cone ◽  
Quiver Flag Variety ◽  
Supplementary Material

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.

scholarly journals Some results on equivariant contact geometry for partial flag varieties

2016 ◽  
Vol 27 (08) ◽  
pp. 1650066 ◽  
Author(s):  
Peter Crooks ◽  
Steven Rayan
Keyword(s):  
Vector Bundles ◽  
Contact Structures ◽  
Flag Varieties ◽  
Contact Manifolds ◽  
Projective Varieties ◽  
Simply Connected ◽  
Complex Projective ◽  
Special Case ◽  
Partial Flag Varieties

We study equivariant contact structures on complex projective varieties arising as partial flag varieties [Formula: see text], where [Formula: see text] is a connected, simply-connected complex simple group of type ADE and [Formula: see text] is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby’s classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of [Formula: see text]-equivariant vector bundles on [Formula: see text]. A byproduct of our argument is a canonical, global description of the unique [Formula: see text]-invariant contact structure on the isotropic Grassmannian of 2-planes in [Formula: see text].

Restriction properties for the Krull–Schmidt decomposition of vector bundles

2018 ◽  
Vol 29 (03) ◽  
pp. 1850022
Author(s):  
Mihai Halic
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
The Other ◽  
Flag Varieties ◽  
Schmidt Decomposition ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Splitting Criterion ◽  
Irreducible Components ◽  
Smooth Projective Varieties

We obtain decomposability criteria for vector bundles on smooth projective varieties [Formula: see text] by comparing the Krull–Schmidt decomposition on [Formula: see text], on one hand, and along the vanishing locus of a section in an ample vector bundle over [Formula: see text], on the other hand. We determine effective bounds for the amplitude of and also genericity conditions for its sections which ensure that the irreducible components of and those of its restriction correspond bijectively. Moreover, we get a simple splitting criterion for vector bundles on partial flag varieties.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals Fano 3-folds from homogeneous vector bundles over Grassmannians

2021 ◽  
Author(s):  
Lorenzo De Biase ◽  
Enrico Fatighenti ◽  
Fabio Tanturri
Keyword(s):  
Vector Bundles ◽  
Homogeneous Vector

AbstractWe rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

Tropical flag varieties

2021 ◽  
Vol 384 ◽  
pp. 107695
Author(s):  
Madeline Brandt ◽  
Christopher Eur ◽  
Leon Zhang
Keyword(s):  
Flag Varieties
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