Related papers: $\mathbb P$-objects and autoequivalences of derive…
We construct an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves (modules over an Azumaya algebra) on any small resolution of its relative Jacobian.
Discrete derived categories were studied initially by Vossieck and later by Bobi\'nski, Gei\ss, Skowro\'nski. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and…
We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to…
We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G_2, F_4, as well as the groups…
We extend some of the results of Bondal-Orlov on the equivalence of derived categories to the case of orbifolds by using the category of coherent orbifold sheaves.
In this short note we observe that, for purely formal reasons, any autoequivalence can be constructed as a twist around a spherical functor. As an example, we show how the P-twists constructed by Huybrechts and Thomas can be formulated as…
We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second…
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold…
We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived…
This paper is the first part of a project aimed at understanding deformations of triangulated categories, and more precisely their dg and A infinity models, and applying the resulting theory to the models occurring in the Homological Mirror…
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove…
We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared…
We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves…
We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of…
We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence…
We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that…
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…
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We…
The proposed physical duality known as 3d mirror symmetry relates the geometries of dual pairs of holomorphic symplectic stacks. It has served in recent years as a guiding principle for developments in representation theory. However, due to…
In arXiv:2007.14415 we proved that the "flop-flop" autoequivalence can be realized as the spherical twist around a spherical functor whose source category arises naturally from the geometry. In this companion paper we study in detail some…