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Alexander Samokhin

Publications and source records attributed to Alexander Samokhin.

13 recordsLinked to original sources
arXiv2026

Semiorthogonal decompositions for families of twisted flag varieties

We show that the derived categories of smooth families of twisted generalized flag varieties admit semiorthogonal decompositions into derived categories of twisted sheaves over the base. In particular, we obtain a categorification of Panin's computation of the Quillen K-theory of twisted flag varieties. The main ingredient is a generalization of the Samokhin-van der Kallen semiorthogonal decomposition for representation categories of parabolic subgroups from split simply connected semisimple groups to arbitrary quasi-split semisimple groups, including non-simply connected cases.

math.AG↗
arXiv2024

Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$

Given a split simply connected and connected algebraic group scheme $\mathbb G$ over $\mathbb Z$ and a split parabolic subgroup scheme $\mathbb P\subset \mathbb G$, this paper constructs semi-orthogonal decompositions of the bounded derived category $D^b(\mathrm {rep}( \mathbb P))$ of noetherian representations of $\mathbb P$ with each semi-orthogonal component being equivalent to the bounded derived category $D^b(\mathrm {rep}( \mathbb G))$ of noetherian representations of $\mathbb G$. The semi-orthogonal components of those decompositions are stable under the monoidal action of $D^b(\mathrm {rep}( \mathbb G))$ on $D^b(\mathrm {rep}( \mathbb P))$. The decompositions depend on an arbitrarily chosen total order on the Weyl group that refines the Bruhat order. The semi-orthogonal decompositions are also compatible with the Bruhat order on cosets of the Weyl group of $\mathbb P$ in the Weyl group of $\mathbb G$. Their construction builds upon the foundational results on $\mathbb B$-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the $ \mathbb T$-equivariant $K$-theory of $ \mathbb G/\mathbb B$. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes $\mathbb G/\mathbb P$ over $\mathbb Z$.

math.AG↗
arXiv2018

Geometry of horospherical varieties of Picard rank one

We study the geometry of non-homogeneous horospherical varieties. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study in detail the case of the horospherical variety associated to the exceptional group $G_2$, and construct a full rectangular Lefschetz exceptional collection in the derived category.

math.AG↗
arXiv2017

The Frobenius morphism on flag varieties, II

In this paper, which is the sequel to arXiv:1410.3742, we study the Frobenius pushforward of the structure sheaf on the adjoint varieties in type ${\bf A}_3$ and ${\bf A}_4$. We show that this pushforward sheaf decomposes into a direct sum of indecomposable bundles and explicitly determine this set that does not depend of the characteristic. In accordance with the results of arXiv:0707.0913, this set forms a strong full exceptional collection in the derived category of coherent sheaves. These computations lead to a natural conjectural answer in the general case that we state at the end.

math.AG↗
arXiv2014

The Frobenius morphism on flag varieties, I

In this paper, given a semisimple algebraic group $\bf G$ of rank 2, we construct a special semiorthogonal decomposition in the derived category of coherent sheaves on the flag variety ${\bf G}/{\bf B}$. These decompositions are defined over the localization ${\mathbb Z}_{\rm S}$, where $\rm S$ is the set of bad primes for $\bf G$, while their block structure is compatible with the Bruhat order on Schubert varieties. The non-standard $t$-structures on ${\rm D}^b({\bf G}/{\bf B})$ defined by these decompositions are self-dual with respect to the duality ${\mathcal RHom}_{{\bf G}/{\bf B}}(-,\omega _{{\bf G}/{\bf B}}^{\frac{1}{2}})$ given by the square root of the canonical sheaf of ${\bf G}/{\bf B}$. For the groups of classical type, this allows to construct an explicit decomposition of the higher Frobenii pushforward bundles ${\sf F}^n_{\ast}{\mathcal O}_{{\bf G}/{\bf B}}$ into a direct sum of indecomposable bundles. When $p>h$, the Coxeer number of the corresponding group, this set of indecomposable bundles forms a full exceptional collection in ${\rm D}^b({\bf G}/{\bf B})$ defined over ${\mathbb Z}_{\rm S}$.

math.AG↗
arXiv2012

Stationary iteration methods for solving 3D electromagnetic scattering problems

Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on the complex plane are obtained. A minimax problem for the determination of optimal complex iteration parameters is formulated. An algorithm of finding an optimal iteration parameter in the case of arbitrary location of the operator spectrum on the complex plane is constructed for the generalized simple iteration method. The results are applied to numerical solution of volume singular integral equations (VSIEs) associated with the problems of the mathematical theory of wave diffraction by 3D dielectric bodies. In particular, the domain of the spectrum location is described explicitly for low-frequency scattering problems and in the general case. The obtained results are discussed and recommendations concerning their applications are given.

math.NA↗
arXiv2010

Tilting bundles via the Frobenius morphism

We show how to construct tilting bundles for a class of smooth projective varieties using characteristic $p$ methods. Given such a variety $X$, reduce it modulo a prime number and consider the direct image of the structure sheaf under the Frobenius morphism. We prove that under suitable restrictions on the characteristic, these direct images are tilting bundles for some toric Fano varieties, Del Pezzo surfaces, and flag varieties $G/B$ of type $A_2$ and $B_2$.

math.AG↗
arXiv2010

A vanishing theorem for sheaves of small differential operators in positive characteristic

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR}) we show that the Frobenius pushforward of the structure sheaf is a tilting bundle on these varieties, provided that $p>h$, the Coxeter number of the corresponding group.

math.AG↗
arXiv2006

Some remarks on the derived categories of coherent sheaves on homogeneous spaces

In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.

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