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## Exceptional collections on isotropic Grassmannians

Cornell University
,
2011.
No. 1110.5607.

Kuznetsov A., Polishchuk A.

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.

Kuznetsov A., Polishchuk A., Journal of the European Mathematical Society 2016 Vol. 18 No. 3 P. 507–574

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups. ...

Added: December 22, 2013

Р.С. Авдеев, Математический сборник 2012 Т. 203 № 11 С. 3–22

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies a simple condition. ...

Added: February 25, 2014

Р.С. Авдеев, Горфинкель Н. Е., Функциональный анализ и его приложения 2012 Т. 46 № 3 С. 1–15

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of representations of G on spaces of regular sections of homogeneous line bundles over G/H. ...

Added: February 25, 2014

Galkin S., Mellit A., Smirnov M., International Mathematics Research Notices 2015 Vol. 2015 No. 18 P. 8847–8859

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: October 20, 2014

Penkov I., Tikhomirov A. S., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 289–323

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow} X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a grassmannian or an isotropic grassmannian (possibly mixing grassmannians and isotropic grassmannians), and the embeddings $\phi_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one ...

Added: October 9, 2014

Kuznetsov A., / Cornell University. Series math "arxiv.org". 2012. No. 1211.4693.

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...

Added: October 4, 2013

Popov V., / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.

We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...

Added: January 9, 2013

Galkin S., Shinder E., Advances in Mathematics 2013 Vol. 224 No. 10 September P. 1033–1050

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and ...

Added: July 5, 2013

Galkin S., Mellit A., Smirnov M., / Cornell University. Series math "arxiv.org". 2014. No. 1405.3857.

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: May 16, 2014

Stanislav Fedotov, Transactions of the American Mathematical Society 2013 Vol. 365 No. 8 P. 4153–4179

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle ...

Added: November 5, 2015

Roman Avdeev, Selecta Mathematica, New Series 2015 Vol. 21 No. 3 P. 931–993

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna’s general classification of arbitrary spherical subgroups restricted ...

Added: July 8, 2015

Kuznetsov A., Journal fuer die reine und angewandte Mathematik 2015 Vol. 2015 No. 708 P. 213–243

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...

Added: December 22, 2013

Р.С. Авдеев, Известия РАН. Серия математическая 2010 Т. 74 № 6 С. 3–26

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical ...

Added: February 25, 2014

Przyjalkowski V., Shramov K., Успехи математических наук 2014 Т. 69 № 6(420) С. 181–182

В Московском математическом обществе.
Сообщения Московского математического общества ...

Added: February 26, 2015

Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5–62

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...

Added: February 25, 2014

Guseva L., / Cornell University. Series arXiv "math". 2022.

We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from ...

Added: September 12, 2022

Lvovsky S., М.: МЦНМО, 2013

Всякое одномерное семейство прямых на плоскости (кроме вырожденных случаев) является семейством касательных к некоторой кривой. В пространстве, однако, это уже совершенно не так; в брошюре объясняется, как, глядя на одномерное семейство прямых в пространстве, определить, является ли оно «касательным». По ходу дела чита- тель знакомится с такими важными понятиями современной математики, как внешняя алгебра и ...

Added: October 3, 2013

Xu J., Zhang S., Elagin Alexey, European Journal of Mathematics 2021 Vol. 7 P. 69–115

We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of Lutz Hille and Markus Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on ...

Added: February 25, 2021

Galkin S., Iritani H., / Cornell University. Series math "arxiv.org". 2015. No. 1508.00719.

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in ...

Added: August 5, 2015

Arzhantsev I., Zaitseva Y., Research in Mathematical Sciences 2024 Vol. 11 No. 2 Article 27

An algebraic variety X is called a homogeneous variety if the automorphism group Aut(X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we ...

Added: March 23, 2024

Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13–25

Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...

Added: November 5, 2022

Galkin S., Golyshev V., Russian Mathematical Surveys 2006 Vol. 61 No. 1 P. 171–173

Added: September 14, 2013

Smirnov E., Journal of Algebra and its Applications 2024

As shown by A. Melnikov, the orbits of a Borel subgroup acting by conjugationon upper-triangular matrices with square zero are indexed by involutions in the symmetricgroup. The inclusion relation among the orbit closures deﬁnes a partial order on involutions.We observe that the same order on involutive permutations also arises while describing theinclusion order on B-orbit ...

Added: July 11, 2024

Arzhantsev I., Indagationes Mathematicae 2023 Vol. 34 No. 4 P. 812–819

We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admit a surjective morphism from an affine space. ...

Added: May 24, 2023