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On the Derived Category of the Cayley Grassmannian
Cornell University
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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 two operations with quadric bundles that might be interesting in themselves.
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
Elagin Alexey, Lunts V., Advances in Mathematics 2021 Vol. 378 Article 107525
We classify triangulated categories that are equivalent to finitely generated thick subcategories $T\subset D^b(cohC)$ for smooth projective curves C over an algebraically closed field. ...
Added: February 25, 2021
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
Kuznetsov A., Perry A., Compositio Mathematica 2018 Vol. 154 No. 7 P. 1362-1406
We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, ...
Added: September 13, 2018
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
Kuznetsov A., / Cornell University. Series math "arxiv.org". 2010. No. 1011.4146.
Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. ...
Added: October 4, 2013
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
Kuznetsov A., Polishchuk A., / Cornell University. Series math "arxiv.org". 2011. No. 1110.5607.
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: October 4, 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
Bondal A. I., Известия РАН. Серия математическая 2013 Т. 77 № 4 С. 5-30
В работе вводится понятие согласованных пар и согласованных цепей t-структур. Доказывается, что две согласованных цепи t-структур порождают дистрибутивную решетку. Развитая техника применяется к случаю пары цепей, полученных из стандартной t-структуры на производной категории когерентных пучков и двойственной к ней применением функтора сдвига. В результате получается семейство t-структур, сердцевины которых известны как превратные когерентные пучки. ...
Added: February 6, 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
Netay I. V., Функциональный анализ и его приложения 2013 Т. 47 № 3 С. 54-74
We describe the syzygy spaces for the Segre embedding~$\bbP(U)\times\bbP(V)\subset\bbP(U\otimes V)$ in terms of representations of $\GL(U)\times \GL(V)$ and construct the minimal resolutions of the sheaves~$\mathscr{O}_{\bbP(U)\times\bbP(V)}(a,b)$ in~$D(\bbP(U\otimes V))$ for~$a\geqslant-\dim(U)$ and~$b\geqslant-\dim(V)$. Also we prove some property of multiplication on syzygy spaces of the Segre embedding. ...
Added: June 21, 2013
A. Kuznetsov, Mathematische Zeitschrift 2014 Vol. 276 No. 3 P. 655-672
Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. ...
Added: December 22, 2013
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
Galkin S., Katzarkov L. V., Mellit A. et al., Advances in Mathematics 2015 Vol. 278 P. 238-253
We conjecture that derived categories of coherent sheaves on fake projective n-spaces have a semi-orthogonal decomposition into a collection of n+1 exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum’s surface). Then by passing to equivariant categories we construct new examples ...
Added: October 20, 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
Kuznetsov A., / Cornell University. Series arXiv "math". 2021.
We introduce the notion of hyperbolic equivalence for quadric bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties.
Furthermore, when the ...
Added: November 23, 2021
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021
В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80
Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...
Added: May 3, 2017
Красноярск : ИВМ СО РАН, 2013
Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...
Added: November 18, 2013
Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18
Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...
Added: November 16, 2020
Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148
In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...
Added: May 17, 2017
Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600
We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...
Added: October 25, 2018