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## Derived categories of Keum's fake projective planes

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 of phantom categories with both Hochschild homology and Grothendieck group vanishing.

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

Fonarev A., Известия РАН. Серия математическая 2013 Т. 77 № 5 С. 203-224

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when ...

Added: December 3, 2012

Galkin S., Karzhemanov I., Shinder E., / Cornell University. Series math "arxiv.org". 2016. No. 1602.06107.

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the ...

Added: February 23, 2016

Bondal A. I., Izvestiya: Mathematics, Великобритания 2013 Vol. 77 No. 4 P. 651-674

We introduce the notions of consistent pairs and consistent chains of $ t$-structures and prove that two consistent chains of $ t$-structures generate a distributive lattice. The technique developed is then applied to the pairs of chains obtained from the standard $ t$-structure on the derived category of coherent sheaves and the dual $ t$-structure ...

Added: October 21, 2014

Bodzenta A., Bondal A. I., / Cornell University. Series arXiv "math". 2017.

Given a relatively projective birational morphism f:X→Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators TX,f and SX,f in Db(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that Db(X) has such a filtration L where ...

Added: August 28, 2017

Alexander I. Efimov, Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 350-372

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with ...

Added: January 28, 2015

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

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

Fonarev A., Kuznetsov A., Journal of London Mathematical Society 2018 Vol. 97 No. 2 P. 24-46

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree. ...

Added: November 7, 2017

Elagin Alexey, Lunts V., Moscow Mathematical Journal 2016 Vol. 16 No. 4 P. 691-709

We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is exceptional and full. ...

Added: October 22, 2015

Efimov A., Advances in Mathematics 2017 Vol. 304 P. 179-226

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category Db(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GLk. This in particular implies existence of a full exceptional collection, which is ...

Added: October 29, 2016

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

Bodzenta-Skibinska A., / Cornell University. Series math "arxiv.org". 2013.

Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces. ...

Added: November 5, 2014

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

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

Bodzenta-Skibinska A., Proceedings of the American Mathematical Society 2014

Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we show that the category C has finite dimensional spaces of morphisms. ...

Added: November 5, 2014

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

Pirozhkov Dmitrii, International Mathematics Research Notices 2022 No. 3 P. 2250-2273

Let U be the tautological subbundle on the Grassmannian Gr(k,n). There is a natural morphism Tot(U)→An. Using it, we give a semiorthogonal decomposition for the bounded derived category Dbcoh(Tot(U)) into several exceptional objects and several copies of Dbcoh(An). We also prove a global version of this result: given a vector bundle E with a regular ...

Added: September 29, 2023

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., / 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

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

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

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