We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these autoequivalences should be mirror to Seidel's Dehn twists about lagrangian $\mathbb P^n$ submanifolds.
We give various connections to spherical objects and spherical twists, and include a simple description of Atiyah and Kodaira-Spencer classes in an appendix.
In this article, we study the group of autoequivalences of derived categories of coherent sheaves on the minimal resolution of $A_n$-singularities on surfaces. Our main result is to find generators of this group.
We study the group of autoequivalences of the derived categories of coherent sheaves on smooth projective elliptic surfaces with non-zero Kodaira dimensions. We find a description of it when each reducible fiber is a cycle of $(-2)$-curves.
Discrete derived categories were studied initially by Vossieck and later by Bobiński, Geiß, Skowroński. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.
We identify a class of autoequivalences of triangulated categories of singularities associated with Calabi-Yau complete intersections in toric varieties. Elements of this class satisfy relations that are directly linked to the toric data.
We study autoequivalence groups of the derived categories on smooth projective surfaces, and show a trichotomy of types according to the maximal dimension of Fourier--Mukai kernels for autoequivalences. This number is $2$, $3$ or $4$, and we also pose a conjecture on the description of autoequivalence groups if it is $2$, and prove it in some special cases.
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 semiorthogonal decompositions. Beyond the GIT setting, we show that all spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner. Motivated by a prediction from mirror symmetry, we refine the recent notion of "grade restriction rules" in equivariant derived categories. We produce additional derived autoequivalences of a GIT quotient …
Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for mapping class groups of closed oriented real surfaces, we study spherical twists for derived categories of algebraic varieties. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. As an application, we compute the center of autoequivalence groups of derived categories of K3 s…
In this article, we construct new derived autoequivalences of generalised Kummer varieties. Together with Huybrechts-Thomas twists around $\mathbb{P}^n$-objects, these are the only known examples of such symmetries.
We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also calculate the group of exact autoequivalences for these categories. The technics of ample sequences in abelian categories is used.
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 vector space, and define endofunctors of the bounded derived categories of coherent sheaves associated to these varieties. In the case r=2 we show that these are autoequivalences using the theory of spherical functors. Our autoequivalences naturally generalize the Seidel-Thomas spherical twist for analogous bundles over projective spaces.
We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi-Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems.
We begin by discussing various ways autoequivalences and stability conditions associated to triangulated categories can interact. Once an appropriate definition of compatibility is formulated, we derive a sufficiency criterion for this compatibility. We next apply this criterion to derived categories associated to Galois covers of the Weierstrass nodal cubic, known as n-gons and denoted by E_n. These are singular non-irreducible genus 1 curves naturally arising in variety of contexts, including as certain degenerations of elliptic curves. In particular, fixing the stability condition to be the…
We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.
We study Fourier--Mukai partners of elliptic ruled surfaces. We also describe the autoequivalence group of the derived categories of ruled surfaces with an elliptic fibration, by using \cite{Ue15}.
We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.
For every smooth quasi-projective surface X we construct a series of P^{n-1}-functors H_{l,n}: D(X x X^[l]) --> D(X^[n+l]) between the derived categories of the Hilbert schemes of points for n>max{l,1} using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P^{n-1}-functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds.
This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of (0,-2)-curves on threefolds, or deforming $\mathbb{P}$-objects introduced by D.Huybrecht and R.Thomas
We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of \cite{AJS}. As an application of the "constant" t-structures…
Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix…
We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition …
E. Segal proved that any autoequivalence of an enhanced triangulated category can be realised as a spherical twist. However, when exhibiting an autoequivalence as a spherical twist one has various choices for the source category of the spherical functor. We describe a construction that realises the composition of two spherical twists as the twist around a single spherical functor whose source category semiorthogonally decomposes into the source categories for the spherical functors we started with. We give a description of the cotwist for this spherical functor and prove, in the special case w…
For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give powerful sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid. Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier-Mukai transforms between the derived categories of algebraic varieties. This is a broad generalisation of the results on spherical objects in [ST01] and…
We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for $\mathbb{P}^2 \times \mathbb{P}^2$ and $\mathbb{P}…
We consider the bounded derived category of $S_k$-equivariant coherent sheaves on $(\mathbb{P}^n)^k$. The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist. The main results of the paper include the construction of a rectangular Lefschetz exceptional collection in the case $k=3$ and in the case $n=1$ when $\mathrm{gcd}(n+1,k)=1$. We also construct minimal Lefschetz exceptional collection for $n=1$ and even $k$, and for $n=2$ and $k=3$.
To any model category $\mathcal{M}$, we associate a modular model category, a functor of points $\mathcal{M}[-]:$ Cat $\rightarrow$ Cat, that associates to any small category $\mathcal{C}$ a functor category $\mathcal{M}[\mathcal{C}] = \text{Fun}_{fes}(\mathcal{C}, \mathcal{M})$ of full and essentially surjective functors from $\mathcal{C}$ to $\mathcal{M}$, providing parametrizations of a same model category $\mathcal{M}$ by different small categories. We are in particular interested in using schemes as parameters. We consider $\mathbb{Z}$Sm$/k$ the category of linear combinations of smooth s…
Building on recent results by A. Blanc, M. Robalo and the authors, we present an $\ell$-adic trace formula for smooth and proper dg-categories over a base $\mathbb{E}_{\infty}$-algebra $B$. We also give a variant when $B$ is only an $\mathbb{E}_2$-algebra. As an application of this trace formula, we propose a strategy of proof of Bloch's conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.
In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the a…
We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.
We prove two kinds of $\mathbb{Z}/2$-periodic Koszul duality equivalences for triangulated categories of matrix factorizations associated with $(-1)$-shifted cotangents over quasi-smooth affine derived schemes. We use this result to define $\mathbb{Z}/2$-periodic version of Donaldson-Thomas categories for local surfaces, whose $\mathbb{C}^{\ast}$-equivariant version was introduced and developed in the author's previous paper. We compare $\mathbb{Z}/2$-periodic DT category with the $\mathbb{C}^{\ast}$-equivariant one, and deduce wall-crossing equivalences of $\mathbb{Z}/2$-periodic DT categorie…
We discuss the following problem: how can an arbitrary Fourier-Mukai transform $ϕ: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective spaces of dimensions $a$ and $b$ be expressed in explicit terms as an exact functor between the homotopy categories $\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}( \mathbb{B}^b )$ generated by the full strong exceptional sequences of the line bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and $\mathbb{B}^b = \{\mathca…
We introduce $q$-stability conditions $(σ,s)$ on Calabi-Yau-$\mathbb{X}$ categories $\mathcal{D}_\mathbb{X}$, where $σ$ is a stability condition on $\mathcal{D}_\mathbb{X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname{QStab}_s\mathcal{D}_\mathbb{X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grotendieck group of $\mathcal{D}_\mathbb{X}$ over $\mathbb{Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that the $q$-stability conditions can be identified with the stability conditions on $\mathcal{D}_N$, provid…
The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.
Categorically, we introduce the Calabi-Yau-$\mathbb{X}$ categories $\mathcal{D}_{\mathbb{X}}$ of a graded marked surface $\mathbf{S}^λ$, as a $q$-deformation of the topological Fukaya category $\mathcal{D}_\infty$ of $\mathbf{S}^λ$. We show that $\mathcal{D}_\infty$ can be identified with the cluster-$\mathbb{X}$ category associated to $\mathcal{D}_{\mathbb{X}}$. Geometrically, we construct and identify the space of $q$-quadratic differentials on the logarithm surface $\operatorname{log}_{\mathbf{c}} \mathbf{S}_\vartriangle^λ$ with the space of induced $q$-stability conditions on $\mathcal{D}_…
The notion of Hochschild cochains induces an assignment from $Aff$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under some appropriate finiteness conditions, to a functor $\mathbb H: Aff \to AlgBimod(DGCat)$, where the latter denotes the category of monoidal DG categories and bimodules. Now, any functor $\mathbb A: Aff \to AlgBimod(DGCat)$ gives rise, by taking modules, to a theory of sheaves of categories $ShvCat^{\mathbb A}$. In this paper, we study $ShvCat^{\mathbb H}$. Vaguely speaking, this theory categorifies the theory of D-modules, in the same wa…
In this short survey we give a non-technical introduction to some main ideas of the theory of $\infty$-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides the basic $\infty$-categorical notions leading to presentable $\infty$-categories, we mention the Joyal and Bergner model structures organizing two approaches to a theory of $(\infty,1)$-categories. We also discuss monoidal $\infty$-categories and algebra objects, as well as stable $\infty$-categories. These notions come together in Lurie's treatment of the smash product on spectra, yielding …
We give a new construction, based on categorical logic, of Nori's $\mathbb Q$-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $\mathbb Q$. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $\mathbb Q$-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $\ell$-adic, $p$-adic or motivic cohomology. We prove that the $\mathbb Q$-linear abelian categories of mixed mot…
