15 research outputs found
Reconstruction of a surface from the category of reflexive sheaves
We define a normal surface to be codim-2-saturated if any open embedding
of into a normal surface with the complement of codimension 2 is an
isomorphism. We show that any normal surface allows a codim-2-saturated
model together with the canonical open embedding .
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 from the
additive category of reflexive sheaves on . We show that the category of
reflexive sheaves on 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
Coherent Sheaves, Chern Classes, and Superconnections on Compact Complex-Analytic Manifolds
We construct a twist-closed enhancement of the category , the bounded derived category of complexes of -modules
with coherent cohomology, by means of the DG-category of
-superconnections. Then we apply the techniques of
-superconnections to define Chern classes and Bott-Chern classes
of objects in the category, in particular, of coherent sheaves.Comment: 32 page
Exceptional collections of line bundles on projective homogeneous varieties
We construct new examples of exceptional collections of line bundles on the
variety of Borel subgroups of a split semisimple linear algebraic group G of
rank 2 over a field. We exhibit exceptional collections of the expected length
for types A_2 and B_2=C_2 and prove that no such collection exists for type
G_2. This settles the question of the existence of full exceptional collections
of line bundles on projective homogeneous G-varieties for split linear
algebraic groups G of rank at most 2.Comment: 20 pages, comments welcom
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