23 research outputs found
Averaging t-structures and extension closure of aisles
We ask when a finite set of t-structures in a triangulated category can be
`averaged' into one t-structure or, equivalently, when the extension closure of
a finite set of aisles is again an aisle. There is a straightforward, positive
answer for a finite set of compactly generated t-structures in a big
triangulated category. For piecewise tame hereditary categories, we give a
criterion for when averaging is possible, and an algorithm that computes
truncation triangles in this case. A finite group action on a triangulated
category gives a natural way of producing a finite set of t-structures out of a
given one. If averaging is possible, there is an induced t-structure on the
equivariant triangulated category.Comment: 26 pages, 11 figures. v2: fixed minor mistakes, improved
presentation. Comments still welcome
Partially ample line bundles on toric varieties
In this note we study properties of partially ample line bundles on
simplicial projective toric varieties. We prove that the cone of q-ample line
bundles is a union of rational polyhedral cones, and calculate these cones in
examples. We prove a restriction theorem for big q-ample line bundles, and
deduce that q-ampleness of the anticanonical bundle is not invariant under
flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line
bundles.Comment: 12 pages, 2 figures; v.2: proofs simplified, lots of material added,
new autho
Partial compactification of stability manifolds by massless semistable objects
We introduce two partial compactifications of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced classical stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank 2. For these, our partial compactification can be explicitly described and related to the wall-and-chamber structure of the stability space
Discrete triangulated categories
14 pages, 1 table, v2 minor fixes and changesWe introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions
Discrete derived categories I: homomorphisms, autoequivalences and t-structures
Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobiński et al. (Cent Eur J Math 2:19–49, 2004). In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures
Dimer models and Calabi-Yau algebras
In this thesis we use techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a 'superpotential'. Some examples are Calabi-Yau and some are not. We consider two types of 'consistency' condition on dimer models, and show that a 'geometrically consistent' dimer model is 'algebraically consistent'. Finally we prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras.EThOS - Electronic Theses Online ServiceGBUnited Kingdo