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Alexey Bondal

Publications and source records attributed to Alexey Bondal.

13 recordsLinked to original sources
arXiv2026

Highest weight categories via pairs of dual exceptional sequences

In this paper we present criteria in terms of dual pairs of exceptional sequences for an abelian category to be highest weight. The criteria are applied in three situations of geometric origin. We give new proofs for the facts that the category of perverse sheaves of middle perversity on complex-analytic manifolds with suitable conditions on the stratification is highest weight and that the derived coherent category of any Grassmannian has a $t$-structure with highest weight heart. Also we show that the abelian null category of any proper birational morphism of regular surfaces is highest weight. For this null category, we give a geometric description of some special objects related to the highest weight structure, such as standard, costandard and characteristic tilting objects.

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arXiv2025

Categorifying non-commutative deformations

We define the functor $\textrm{ncDef}_{(Z_1,\ldots,Z_n)}$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $\mathcal{Z}$. In our categorified approach, we view the underlying spaces of infinitesimal flat deformations as Deligne finite categories, i.e. finite length abelian categories admitting projective generators, with $n$ isomorphism classes of simple objects. More generally, we define the functor $\textrm{ncDef}_ζ$ of non-commutative deformations of an exact functor $ζ\colon \mathcal{A} \to \mathcal{Z}$ of abelian categories. Here the role of an infinitesimal non-commutative thickening of $\mathcal{A}$ is played by an abelian category $\mathcal{B}$ containing $\mathcal{A}$ and such that $\mathcal{A}$ generates $\mathcal{B}$ by extensions. The functor $\textrm{ncDef}_ζ$ assigns to such $\mathcal{B}$ the set of equivalence classes of exact functors $\mathcal{B} \to \mathcal{Z}$ which extend $ζ$. We prove that an exact functor on an infinitesimal extension is fully faithful if and only if it is fully faithful on the first infinitesimal neighbourhood. We show that if $ζ$ is fully faithful, then the functor $\textrm{ncDef}_ζ$ is ind-represented by the extension closure of the essential image of $ζ$. We prove that for a flopping contraction $f\colon X\to Y$ with the fiber over a closed point $C = \bigcup_{i=1}^n C_i$, where $C_i$'s are irreducible curves, $\{\mathcal{O}_{C_i}(-1)\}$ is the set of simple objects in the null-category for $f$. We conclude that the null-category ind-represents the functor $\textrm{ncDef}_{(\mathcal{O}_{C_1}(-1),\ldots,\mathcal{O}_{C_n}(-1))}$.

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arXiv2024

Abelian envelopes of exact categories and highest weight categories

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional collections. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proved for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality is interpreted as a duality between the right and left abelian envelopes of a thin exact category. The duality for thin exact categories is introduced by means of derived categories and Serre functor on them.

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arXiv2023

Reconstruction of a surface from the category of reflexive sheaves

We define a normal surface $X$ to be codim-2-saturated if any open embedding of $X$ into a normal surface with the complement of codimension 2 is an isomorphism. We show that any normal surface $X$ allows a codim-2-saturated model $\widehat{X}$ together with the canonical open embedding $X\to \widehat{X}$. Any normal surface which is proper over its affinisation is codim-2-saturated, but the converse does not hold. We give a criterion for a surface to be codim-2-saturated in terms of its Nagata compactification and the boundary divisor. We reconstruct the codim-2-saturated model of a normal surface $X$ from the additive category of reflexive sheaves on $X$. We show that the category of reflexive sheaves on $X$ is quasi-abelian and we use its canonical exact structure for the reconstruction. In order to deal with categorical issues, we introduce a class of weakly localising Serre subcategories in abelian categories. These are Serre subcategories whose categories of closed objects are quasi-abelian. This general technique might be of independent interest.

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arXiv2022

Coherent Sheaves, Chern Classes, and Superconnections on Compact Complex-Analytic Manifolds

We construct a twist-closed enhancement of the category ${\mathcal D}^b_{\rm coh}(X)$, the bounded derived category of complexes of ${\mathcal O}_X$-modules with coherent cohomology, by means of the DG-category of $\bar\partial$-superconnections. Then we apply the techniques of $\bar\partial$-superconnections to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.

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arXiv2021

Theory of homotopes in application to mutually unbiased bases, harmonic analysis on graphs and perverse sheaves

The paper is the survey of the modern results and applications of the theory of homotopes. The notion of a well-tempered element in an associative algebra is introduced and it is proven that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued t-structure. Hochschild and global dimensions of the homotopes are calculated. The homotopes constructed by generalized Lapalce operators in Poincare groupoids of graphs are studied. It is shown that they are quotients of Temperley-Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2 dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of sl(n, C) into the sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, to generalized Hadamard matrices are discussed.

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arXiv2019

Flops and spherical functors

We study derived categories of Gorenstein varieties X and X^+ connected by a flop. We assume that the flopping contractions f: X \to Y, f^+: X^+ \to Y have fibers of dimension bounded by 1 and Y has canonical hypersurface singularities of multiplicity 2. We consider the fiber product W=X \times_Y X^+ with projections p: W \to X, q: W \to X^+ and prove that the flop functors F = Rq_* Lp^*: D^b(X) \to D^b(X^+), F^+= Rp_*Lq^*: D^b(X^+) \to D^b(X) are equivalences, inverse to those constructed by M. Van den Bergh. The composite F^+ \circ F: D^b(X) \to D^b(X) is a non-trivial auto-equivalence. When variety Y is affine, we present F^+\circ F as the spherical cotwist associated to a spherical functor Ψ. The functor Ψis constructed by deriving the inclusion of the null-category A_f of sheaves F in \Coh (X) with Rf_*(F)=0 into Coh (X). We construct a spherical pair (D^b(X),D^b(X^+)) in the quotient D^b(W)/K^b, where K^b is the common kernel of the derived push-forwards for the projections to X and X^+, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the L^1f^*f_* vanishing for the Van den Bergh's projective generator. We construct a projective generator in the null-category and prove that its endomorphism algebra is the contraction algebra.

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arXiv2018

Perverse schobers and birational geometry

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a "web of flops" provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g=sl(3) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

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arXiv2017

Canonical tilting relative generators

Given a relatively projective birational morphism $f\colon X\to Y$ of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over $Y$) generators $T_{X,f}$ and $S_{X,f}$ in $\mathcal{D}^b(X)$. We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that $\mathcal{D}^b(X)$ has such a filtration $\mathcal{L}$ where the lattice is the set of all birational decompositions $f \colon X \xrightarrow{g} Z \xrightarrow{h} Y$ with smooth $Z$. The $t$-structures related to $T_{X,f}$ and $S_{X,f}$ are proved to be glued via filtrations left and right dual to $\mathcal{L}$. We realise all such $Z$ as the fine moduli spaces of simple quotients of $\mathcal{O}_X$ in the heart of the $t$-structure for which $S_{X,g}$ is a relative projective generator over $Y$. This implements the program of interpreting relevant smooth contractions of $X$ in terms of a suitable system of $t$-structures on $\mathcal{D}^b(X)$.

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arXiv2015

Orthogonal pairs and mutually unbiased bases

The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of mutually unbiased bases. We describe the main steps of our proof for the existence of 4-dimensional family of orthogonal pairs of Cartan subalgebras in sl(6). As an application, we prove existence of a real 4-dimensional family of pairs of mutually unbiased bases in C^6, thus solving the long standing problem in Quantum Information Theory.

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arXiv2015

Symplectic geometry of unbiasedness and critical points of a potential

The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors.

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arXiv2013

Operations on t-structures and perverse coherent sheaves

The notions of consistent pairs and consistent chains of t-structures are introduced. A theorem that two consistent chains of t-structures generate a distributive lattice is proven. 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 by means of the shift functor. This yields a family of t-structures whose hearts are known as perverse coherent sheaves.

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