Related papers: Spherical DG-functors
We construct new examples of derived autoequivalences for a family of higher-dimensional Calabi-Yau varieties. Specifically, we take the total spaces of certain natural vector bundles over Grassmannians G(r,d) of r-planes in a d-dimensional…
We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z)…
We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more…
We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category $B$ admitting countable direct sums and…
We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting…
For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl…
This paper establishes semiorthogonal decompositions for derived Grassmannians of perfect complexes with Tor-amplitude in $[0,1]$. This result verifies the author's Quot formula conjecture [J21a] and generalizes and strengthens Toda's…
We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the…
This paper constructs derived autoequivalences of Gorenstein orders as twists around spherical functors. More precisely, given a Gorenstein order $A$ and a quotient $p \colon A \to B$, then we specify natural conditions on $B$ under which…
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 apply the methods of C{\u{a}}ld{\u{a}}raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a…
Let $\Lambda_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b(\mathrm{rep}^n \Lambda_Q)$ be the bounded derived category of finite dimensional nilpotent $\Lambda_Q$-modules. We define spherical…
We study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these automorphisms are spherical twists, and describe how they result from mutations of…
Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$…
In this work we consider the question of realizing triangulated dg-categories by derived categories of algebraic varieties. For this, we introduce the notion of "system of points" in saturated dg-categories. We show that given such a system…
A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the…
We study a triangulated category $\mathscr S$ that admits a full and strong exceptional sequence of three objects with one-dimensional Hom spaces. We show that the isomorphism classes of exact functors from $\mathscr S$ to another…
We discuss the following problem: how can an arbitrary Fourier-Mukai transform $\phi: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective…
We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the…
Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely,…