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## Descent theory for semiorthogonal decompositions

Sbornik Mathematics. 2012. Vol. 203. No. 5. P. 645-676.

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.

Математический сборник 2012 Т. 203 № 5 С. 33-64

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived ...

Added: July 27, 2012

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

Journal of the Ramanujan Mathematical Society 2013 Vol. 28A No. Special Issue-2013 dedicated to C.S.Seshadri's 80th birthday P. 409-415

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

Added: June 20, 2013

Rationality and the FML invariant / 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

European Journal of Mathematics 2016 Vol. 2 No. 1 P. 283-290

According to the classical theorem, every algebraic variety
endowed with a nontrivial rational action of a connected linear algebraic
group is birationally isomorphic to a product of another algebraic variety
and the projective space of a positive dimension. We show that the classical proof of this theorem
actually works only in characteristic 0 and we give a characteristic free
proof ...

Added: February 2, 2016

Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

Zürich: European Mathematical Society Publishing house, 2012

The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic ...

Added: October 14, 2013

Zürich: European Mathematical Society Publishing house, 2010

Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the ...

Added: October 11, 2013

Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225

The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: July 31, 2019

Selecta Mathematica, New Series 2019 Vol. 25:45 P. 1-60

In this paper we introduce new categorical notions and give many examples. In an
earlier paper we proved that the Bridgeland stability space on the derived category of
representations of K(l), thel-Kronecker quiver, is biholomorphic toC×Hfor l ≥ 3. In
the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l≥2.
More precisely, to a triangulated category ...

Added: November 1, 2019

Математический сборник 2021 Т. 212 № 3 С. 128-138

Обобщается конструкция А. Е. Миронова, представившего в свое время новые примеры минимальных и гамильтоново минимальных лагранжевых подмногообразий в Cn и CPn. В основе его конструкции лежало рассмотрение неполного торического действия Tk, где k<n, на подпространства, инвариантные относительно естественных антиголоморфных инволюций. Такая ситуация имеет место для достаточно широкого класса алгебраических многообразий: комплексных квадрик, грассманианов, многообразий флагов и т.п., что позволяет построить большое количество новых ...

Added: March 29, 2021

Embeddings of groups Aut(F_n) into automorphism groups of algebraic varieties / Cornell University. Series math "arxiv.org". 2021. No. 2106.02072.

For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the ...

Added: June 7, 2021

Underlying varieties and group structures / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Doklady Mathematics 2018 Vol. 98 No. 2 P. 413-415

The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established. ...

Added: November 13, 2018

Derived categories for complex-analytic manifolds / . 2011.

We construct a twist-closed enhancement of the derived category of coherent sheaves on a smooth compact complex-analytic manifold by means of DG-category of dbar-superconnections. ...

Added: October 30, 2013

Успехи математических наук 2019 Т. 74 № 3(447) С. 63-94

В работе дается краткий обзор результатов, связанных с гомотопической конечностью DG-категорий. Мы излагаем общий план доказательства гомотопической конечности производных категорий когерентных пучков и когерентных матричных факторизаций ...

Added: November 12, 2019

Smoothness of Derived Categories of Algebras / Cornell University. Series arXiv "math". 2018.

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated ...

Added: December 1, 2018

Advances in Mathematics 2018 Vol. 323 P. 226-278

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 S_X,f in D^b(X). We develop a piece of general theory of
strict admissible lattice filtrations in triangulated categories
and show that D^b(X) has such a filtration L where the ...

Added: May 2, 2018

Tokyo: American Mathematical Society, World Scientific, 2017

Preface
The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.
Recently, there have been remarkable developments in algebraic geometry and related fields, ...

Added: July 12, 2017

Transformation Groups 2014 Vol. 19 No. 2 P. 549-568

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: March 17, 2014

Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

On the derived category of IGr(3,8) / Cornell University. Series math "arxiv.org". 2018.

We construct a full exceptional collection of vector bundles in the bounded derived category of
coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in a symplectic vector
space of dimension 8. ...

Added: October 19, 2018

Успехи математических наук 2014 Т. 69 № 4(418) С. 189-190

В работе приведена конструкция некоторых исключительных векторных расслоений на грассманианах, а также построено семейство полуортогональных разложений их ограниченных производных категорий. ...

Added: September 19, 2014

Izvestiya: Mathematics, England 2019 Vol. 83 No. 4 P. 830-859

The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: September 29, 2019