Equivariant Ulrich bundles on exceptional homogeneous varieties

Abstract We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane E 6/P 1 and the E 7-adjoint variety E 7/P 1. From this result,we see that a general hyperplane section F 4/P 4 of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.

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A type of the Lefschetz hyperplane section theorem on $${\mathbb{Q}\,}$$ -Fano 3-folds with Picard number one and $${\frac{1}{2}(1,1,1)}$$ -singularities

Geometriae Dedicata ◽

10.1007/s10711-011-9644-6 ◽

2011 ◽

Vol 159 (1) ◽

pp. 41-49

Author(s):

Nam-Hoon Lee

Keyword(s):

Hyperplane Section ◽

Picard Number ◽

Section Theorem

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SPLITTING CRITERIA FOR VECTOR BUNDLES ON MINUSCULE HOMOGENEOUS VARIETIES

Transformation Groups ◽

10.1007/s00031-016-9371-z ◽

2016 ◽

Vol 22 (3) ◽

pp. 753-765 ◽

Cited By ~ 1

Author(s):

M. HALIC

Keyword(s):

Vector Bundles ◽

Homogeneous Varieties

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Residual categories for (co)adjoint Grassmannians in classical types

Compositio Mathematica ◽

10.1112/s0010437x21007090 ◽

2021 ◽

Vol 157 (6) ◽

pp. 1172-1206

Author(s):

Alexander Kuznetsov ◽

Maxim Smirnov

Keyword(s):

Automorphism Group ◽

Cohomology Ring ◽

Quantum Cohomology ◽

Algebraic Groups ◽

Fano Variety ◽

Flag Varieties ◽

Picard Number ◽

Coherent Sheaves ◽

Small Quantum ◽

Exceptional Collection

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$ , that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

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Globally generated vector bundles on the Segre threefold with Picard number two

Mathematische Nachrichten ◽

10.1002/mana.201500077 ◽

2015 ◽

Vol 289 (5-6) ◽

pp. 523-536

Author(s):

Edoardo Ballico ◽

Sukmoon Huh ◽

Francesco Malaspina

Keyword(s):

Vector Bundles ◽

Picard Number

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PICARD GROUPS OF HOMOGENEOUS SPACES OF LINEAR ALGEBRAIC GROUPS AND ONE-DIMENSIONAL HOMOGENEOUS VECTOR BUNDLES

Mathematics of the USSR-Izvestiya ◽

10.1070/im1974v008n02abeh002107 ◽

1974 ◽

Vol 8 (2) ◽

pp. 301-327 ◽

Cited By ~ 37

Author(s):

V L Popov

Keyword(s):

Vector Bundles ◽

Homogeneous Spaces ◽

Algebraic Groups ◽

One Dimensional ◽

Linear Algebraic Groups ◽

Picard Groups ◽

Homogeneous Vector

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SMOOTH HOROSPHERICAL VARIETIES OF PICARD NUMBER ONE AS LINEAR SECTIONS OF RATIONAL HOMOGENEOUS VARIETIES

Journal of the Korean Mathematical Society ◽

10.4134/jkms.2016.53.2.433 ◽

2016 ◽

Vol 53 (2) ◽

pp. 433-446 ◽

Cited By ~ 1

Author(s):

Jaehyun Hong

Keyword(s):

Picard Number ◽

Linear Sections ◽

Homogeneous Varieties

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Homogeneous varieties under split solvable algebraic groups

Indagationes Mathematicae ◽

10.1016/j.indag.2021.04.004 ◽

2021 ◽

Author(s):

Michel Brion

Keyword(s):

Algebraic Groups ◽

Homogeneous Varieties

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EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER–LEFSCHETZ THEOREMS

Communications in Contemporary Mathematics ◽

10.1142/s021919971350003x ◽

2013 ◽

Vol 15 (05) ◽

pp. 1350003 ◽

Cited By ~ 3

Author(s):

G. V. RAVINDRA ◽

AMIT TRIPATHI

Keyword(s):

Characteristic Zero ◽

Deformation Theory ◽

Vector Bundles ◽

Projective Variety ◽

Hyperplane Section ◽

Smooth Projective Variety ◽

Closed Field ◽

Open Set ◽

Higher Rank ◽

Lefschetz Theorems

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

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