scholarly journals A family of vector bundles on P 3 of homological dimension 2 and χ (EndE) = 1

2020 ◽  
Vol 47 (8) ◽  
Author(s):  
Sérgio Mendes ◽  
Rosa M. Miró-Roig ◽  
Helena Soares
Keyword(s):  
Vector Bundles ◽  
Homological Dimension ◽  
Dimension 2

Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sérgio Mendes ◽  
Rosa María Miró-Roig ◽  
Helena Soares
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Homological Dimension ◽  
Linear Resolution ◽  
Integer Solutions ◽  
Dimension 2

Abstract We find the complete integer solutions of the equation X 2 + Y 2 + Z 2 - 4 ⁢ X ⁢ Y - 4 ⁢ Y ⁢ Z + 10 ⁢ X ⁢ Z = 1 X^{2}+Y^{2}+Z^{2}-4XY-4YZ+10XZ=1 . As an application, we prove that, for each solution ( a , b , c ) (a,b,c) such that a > 0 a>0 , b - 2 ⁢ a > 0 b-2a>0 and ( b - 2 ⁢ a ) 2 ≥ 4 ⁢ a (b-2a)^{2}\geq 4a , there is a vector bundle 𝐸 on P 3 \mathbb{P}^{3} defined by a minimal linear resolution 0 → O P 3 ⁢ ( - 2 ) a → O P 3 ⁢ ( - 1 ) b → O P 3 c → E → 0 0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)^{a}\to\mathcal{O}_{\mathbb{P}^{3}}(-1)^{b}\to\mathcal{O}_{\mathbb{P}^{3}}^{c}\to E\to 0 . In particular, 𝐸 satisfies χ ⁢ ( End ⁡ E ) = 1 \chi(\operatorname{End}E)=1 .

Non-embedding theorems for Y-spaces

1967 ◽  
Vol 63 (3) ◽  
pp. 601-612 ◽  
Author(s):  
K. H. Mayer ◽  
R. L. E. Schwarzenberger
Keyword(s):  
Lower Bounds ◽  
Vector Bundles ◽  
Differentiable Manifold ◽  
Characteristic Classes ◽  
Embedding Theorems ◽  
Geometric Properties ◽  
Rank 2 ◽  
Dimension 2

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

scholarly journals Torsion pairs and filtrations in abelian categories with tilting objects

2015 ◽  
Vol 14 (08) ◽  
pp. 1550121
Author(s):  
Jason Lo
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Explicit Description ◽  
Homological Dimension ◽  
Abelian Categories ◽  
Dimension 2 ◽  
Tilting Object ◽  
Category Of Modules ◽  
Torsion Pairs ◽  
Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

scholarly journals Exceptional bundles of homological dimension ${k}$

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Rosa María Miró-Roig ◽  
Helena Soares
Keyword(s):  
Vector Bundles ◽  
Homological Dimension

AbstractWe characterize exceptional vector bundles on

Holomorphic vector bundles on non-algebraic tori of dimension 2

2008 ◽  
Vol 126 (2) ◽  
pp. 143-166 ◽  
Author(s):  
Koichi Kurihara ◽  
Kōta Yoshioka
Keyword(s):  
Vector Bundles ◽  
Algebraic Tori ◽  
Holomorphic Vector Bundles ◽  
Dimension 2

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 $\mathbb {C}$ .

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