58 research outputs found
Semiorthogonal decompositions of derived categories of equivariant coherent sheaves
Let X be an algebraic variety with an action of an algebraic group G. Suppose
X has a full exceptional collection of sheaves, and these sheaves are invariant
under the action of the group. We construct a semiorthogonal decomposition of
bounded derived category of G-equivariant coherent sheaves on X into
components, equivalent to derived categories of twisted representations of the
group. If the group is finite or reductive over the algebraically closed field
of zero characteristic, this gives a full exceptional collection in the derived
equivariant category. We apply our results to particular varieties such as
projective spaces, quadrics, Grassmanians and Del Pezzo surfaces.Comment: 28 pages, uses XY-pi
Symmetries and invariants of twisted quantum algebras and associated Poisson algebras
We construct an action of the braid group B_N on the twisted quantized
enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In
the classical limit q -> 1 we recover the action of B_N on the polynomial
functions on the space of upper triangular matrices with ones on the diagonal.
The action preserves the Poisson bracket on the space of polynomials which was
introduced by Nelson and Regge in their study of quantum gravity and
re-discovered in the mathematical literature. Furthermore, we construct a
Poisson bracket on the space of polynomials associated with another twisted
quantized enveloping algebra U'_q(sp_{2n}). We use the Casimir elements of both
twisted quantized enveloping algebras to re-produce some well-known and
construct some new polynomial invariants of the corresponding Poisson algebras.Comment: 29 pages, more references adde
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
Deformation theory of objects in homotopy and derived categories I: general theory
This is the first paper in a series. We develop a general deformation theory
of objects in homotopy and derived categories of DG categories. Namely, for a
DG module over a DG category we define four deformation functors \Def
^{\h}(E), \coDef ^{\h}(E), \Def (E), \coDef (E). The first two functors
describe the deformations (and co-deformations) of in the homotopy
category, and the last two - in the derived category. We study their properties
and relations. These functors are defined on the category of artinian (not
necessarily commutative) DG algebras.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, Proposition 7.1 and Theorem 8.1 were correcte
Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics
We discuss a conjecture saying that derived equivalence of smooth projective simply connected varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P5 and the corresponding double cover Y→P2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X]−[Y] is annihilated by the affine line class
Deformation theory of objects in homotopy and derived categories III: abelian categories
This is the third paper in a series. In part I we developed a deformation
theory of objects in homotopy and derived categories of DG categories. Here we
show how this theory can be used to study deformations of objects in homotopy
and derived categories of abelian categories. Then we consider examples from
(noncommutative) algebraic geometry. In particular, we study noncommutative
Grassmanians that are true noncommutative moduli spaces of structure sheaves of
projective subspaces in projective spaces.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, a new part (part 3) about noncommutative Grassmanians was adde
Cohomological descent theory for a morphism of stacks and for equivariant derived categories
In the paper we answer the following question: for a morphism of varieties
(or, more generally, stacks), when the derived category of the base can be
recovered from the derived category of the covering variety by means of descent
theory? As a corollary, we show that for an action of a reductive group on a
scheme, the derived category of equivariant sheaves is equivalent to the
category of objects, equipped with an action of the group, in the ordinary
derived category.Comment: 28 page
Bounded derived categories of very simple manifolds
An unrepresentable cohomological functor of finite type of the bounded
derived category of coherent sheaves of a compact complex manifold of dimension
greater than one with no proper closed subvariety is given explicitly in
categorical terms. This is a partial generalization of an impressive result due
to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2)
and many typos are correcte
A Point's Point of View of Stringy Geometry
The notion of a "point" is essential to describe the topology of spacetime.
Despite this, a point probably does not play a particularly distinguished role
in any intrinsic formulation of string theory. We discuss one way to try to
determine the notion of a point from a worldsheet point of view. The derived
category description of D-branes is the key tool. The case of a flop is
analyzed and Pi-stability in this context is tied in to some ideas of
Bridgeland. Monodromy associated to the flop is also computed via Pi-stability
and shown to be consistent with previous conjectures.Comment: 15 pages, 3 figures, ref adde
Noncommutative Geometry in the Framework of Differential Graded Categories
In this survey article we discuss a framework of noncommutative geometry with
differential graded categories as models for spaces. We outline a construction
of the category of noncommutative spaces and also include a discussion on
noncommutative motives. We propose a motivic measure with values in a motivic
ring. This enables us to introduce certain zeta functions of noncommutative
spaces.Comment: 19 pages. Minor corrections and one reference added; to appear in the
proceedings volume of AGAQ Istanbul, 200
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