61 research outputs found
Geometricity for derived categories of algebraic stacks
We prove that the dg category of perfect complexes on a smooth, proper
Deligne-Mumford stack over a field of characteristic zero is geometric in the
sense of Orlov, and in particular smooth and proper. On the level of
triangulated categories, this means that the derived category of perfect
complexes embeds as an admissible subcategory into the bounded derived category
of coherent sheaves on a smooth, projective variety. The same holds for a
smooth, projective, tame Artin stack over an arbitrary field.Comment: 31 page
Categorical measures for finite group actions
Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases, these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary
Differential operators on supercircle: conformally equivariant quantization and symbol calculus
We consider the supercircle equipped with the standard contact
structure. The conformal Lie superalgebra K(1) acts on as the Lie
superalgebra of contact vector fields; it contains the M\"obius superalgebra
. We study the space of linear differential operators on weighted
densities as a module over . We introduce the canonical isomorphism
between this space and the corresponding space of symbols and find interesting
resonant cases where such an isomorphism does not exist
Geometric Phantom Categories
In this paper we give a construction of phantom categories, i.e. admissible
triangulated subcategories in bounded derived categories of coherent sheaves on
smooth projective varieties that have trivial Hochschild homology and trivial
Grothendieck group. We also prove that these phantom categories are phantoms in
a stronger sense, namely, they have trivial K-motives and, hence, all their
higher K-groups are trivial too.Comment: LaTeX, 18 page
The Jacobian Conjecture 2n implies the Dixmier Problem n
The aim of the paper is to describe some ideas, approaches, comments, etc. regarding the Dixmier Conjecture, its generalizations and analogues
Scalar extensions of triangulated categories
Given a triangulated category over a field and a field extension ,
we investigate how one can construct a triangulated category over . Our
approach produces the derived category of the base change scheme if the
category one starts with is the bounded derived category of a smooth projective
variety over and the field extension is finite and Galois. We also
investigate how the dimension of a triangulated category behaves under scalar
extensions.Comment: 15 pages, comments welcom
T-functions revisited: New criteria for bijectivity/transitivity
The paper presents new criteria for bijectivity/transitivity of T-functions
and fast knapsack-like algorithm of evaluation of a T-function. Our approach is
based on non-Archimedean ergodic theory: Both the criteria and algorithm use
van der Put series to represent 1-Lipschitz -adic functions and to study
measure-preservation/ergodicity of these
The non-equivariant coherent-constructible correspondence and a conjecture of King
The coherent-constructible (CC) correspondence is a relationship between coherent sheaves on a toric variety X and constructible sheaves on a real torus T. This was discovered by Bondal and established in the equivariant setting by Fang, Liu, Treumann, and Zaslow. In this paper, we explore various aspects of the non-equivariant CC correspondence. Also, we use the non-equivariant CC correspondence to prove the existence of tilting complexes in the derived categories of toric orbifolds satisfying certain combinatorial conditions. This has applications to a conjecture of King
Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau
hypersurface in projective space, for any d > 2 (for example, d = 3 is the
quintic three-fold). The main techniques involved in the proof are: the
construction of an immersed Lagrangian sphere in the `d-dimensional pair of
pants'; the introduction of the `relative Fukaya category', and an
understanding of its grading structure; a description of the behaviour of this
category with respect to branched covers (via an `orbifold' Fukaya category); a
Morse-Bott model for the relative Fukaya category that allows one to make
explicit computations; and the introduction of certain graded categories of
matrix factorizations mirror to the relative Fukaya category.Comment: 133 pages, 17 figures. Changes to the argument ruling out sphere
bubbling in the relative Fukaya category, and dealing with the behaviour of
the symplectic form under branched covers. Other minor changes suggested by
the referee. List of notation include
The supernatural characters and powers of sacred trees in the Holy Land
This article surveys the beliefs concerning the supernatural characteristics and powers of sacred trees in Israel; it is based on a field study as well as a survey of the literature and includes 118 interviews with Muslims and Druze. Both the Muslims and Druze in this study attribute supernatural dimensions to sacred trees which are directly related to ancient, deep-rooted pagan traditions. The Muslims attribute similar divine powers to sacred trees as they do to the graves of their saints; the graves and the trees are both considered to be the abode of the soul of a saint which is the source of their miraculous powers. Any violation of a sacred tree would be strictly punished while leaving the opportunity for atonement and forgiveness. The Druze, who believe in the transmigration of souls, have similar traditions concerning sacred trees but with a different religious background. In polytheistic religions the sacred grove/forest is a centre of the community's official worship; any violation of the trees is regarded as a threat to the well being of the community. Punishments may thus be collective. In the monotheistic world (including Christianity, Islam and Druze) the pagan worship of trees was converted into the worship/adoration of saints/prophets; it is not a part of the official religion but rather a personal act and the punishments are exerted only on the violating individual
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