Fabrice Rosay
We prove that the neutral component of the group of derived autoequivalences of a smooth projective variety is the semi-direct product of the neutral component of its Picard group and its group of automorphisms. We use this result to prove several results concerning pairs of derived equivalent varieties.
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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.
Autoequivalences of Derived Categories of Bielliptic Surfaces
研究双椭圆面导出范畴自同构群的生成元，证明与此类面导出等价的代数簇与其同构。
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.
Central derived autoequivalences of K3 surfaces
本文应用 Bridgeland 稳定性条件理论，描述了复射影 K3 曲面有界导出自等价群的中心。
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.
We describe the group of exact autoequivalences of the bounded derived category of coherent sheaves on a bielliptic surface. We achieve this by studying its action on the numerical Grothendieck group of the surface.
We show that on an Abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to an Abelian variety over a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as an autoequivalence. We also compare the deformation space of these two types of deformations.
We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y admitting a full exceptional sequence. Applications include the case in which X is Calabi-Yau and either X is a hypersurface in Y (this extends a previous result by the author and R.L. Karp, where Y was a weighted projective space) or Y is a hypersurface in X. The proof uses a resolution of the diagonal of Y constructed from the exceptional sequence.
We show that for every smooth projective surface X and every $n\ge 2$ the push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the $S_n$-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial and n=2 this autoequivalence coincides with the known EZ-spherical twist induced by the boundary of the Hilbert scheme. We also generalise the 16 spherical objects on the Kummer surface given by the ex…
Given a smooth stacky Calabi-Yau hypersurface X in a weighted projective space, we consider the functor G which is the composition of the following two autoequivalences of D^b(X): the first one is induced by the spherical object O_X, while the second one is tensoring with O_X(1). The main result of the paper is that the composition of G with itself w times, where w is the sum of the weights of the weighted projective space, is isomorphic to the autoequivalence "shift by 2". The proof also involves the construction of a Beilinson type resolution of the diagonal for weighted projective spaces, v…
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.
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 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 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 …
Derived autoequivalences of length 2 flops via GIT
本文利用 GIT 和窗口理论研究长度 2 坍缩的理查德曲线的导出自同构，并证明弦性凯勒模空间与长度 2 三维流形的描述一致，恢复了 Donovan-Wemyss 的单束作用。
Let X be a smooth complex projective variety. The group of autoequivalences of the derived category of X acts naturally on its singular cohomology H(X, Q) and we denote by Geq(X) the Zariski closure of its image in Gl(H(X, Q)). We study the relation of the Lie algebra LieGeq(X) and the Neron-Severi Lie algebra gNS(X) in case X has trivial canonical line bundle. At the same time for mirror symmetric families of (weakly) Calabi-Yau varieties we consider a conjecture of Kontsevich on the relation between the monodromy of one family and the group Geq(X) for a very general member X of the other fam…
We show that the autoequivalence group of the derived category of any smooth projective toric surface is generated by the standard equivalences and spherical twists obtained from -2-curves. In many cases we give all relations between these generators. We also prove a close link between spherical objects and certain pairs of exceptional objects.
Let $X$ be a smooth projective variety and $\Aut (D(X))$ the group of autoequivalences of the derived category of $X$. In this paper we show that $X$ has no Fourier-Mukai partner other than $X$ when $\Aut (D(X))$ is generated by shifts, automorphisms and tensor products of line bundles.
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.
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…
To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. It is important to evaluate the value of the entropy at zero in relation to the topological entropy. In this paper, we study the entropy at zero of an exact autoequivalence of the derived category of a complex smooth projective curve, and prove that it coincides with the natural logarithm of the spectral radius of the induced automorphism on its numerical Grothendieck g…
We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of $ρ(X)=1$, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence…
Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank 1. We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences, and is contractible.
On a Weierstrass elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of $\mathrm{Stab}(X)$, and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects.
Autoequivalences of Derived Categories of Moduli Spaces of Vector Bundles
研究固定行列式的稳定向量丛模空间的有界导出范畴的自等价群，结合Bondal-Orlov重构定理和自同构群的描述给出显式刻画，并构造不同秩模空间之间的扩展对应以定义Fourier-Mukai型正合函子。
Autoequivalences and stability conditions on a degenerate K3 surface
本文研究K3曲面III型Kulikov退化中奇异曲面X的导出范畴，证明其支撑稳定性条件空间连通分支单连通，并确定保持该分支的自等价子群同构于Z×Γ₁(3)×Aut(X)。
Half-spherical twists on derived categories of coherent sheaves
本文证明平坦态射纤维上的球状对象诱导纤维导出范畴的自等价，为椭圆曲面奇异纤维和K3曲面退化提供新对称性，并揭示与环面半扭转的联系。
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivale…
We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical functors; to deal with Hilb^n for n > 2 we develop a theory of P-functors. We conjecture that the same construction yields an autoequivalence for any moduli space of sheaves on a K3 surface. In an appendix we give a cohomology and base change criterion which is well-known to experts, but not well-documented.
Fourier-Mukai partners and autoequivalences of moduli spaces of vector bundles
本文分类了稳定向量丛模空间的Fourier-Mukai伙伴，并确定其导出自等价群，证明了亏格≥4时互素模空间导出等价当且仅当同构。
Spherical twists, relations and the center of autoequivalence groups of K3 surfaces
本文研究K3表面导出范畴自同构群的中心，通过球面扭转的交数性质，探讨其与映射类群的类比关系，并分类由两个球面扭转生成的子群。
On pseudo-Anosov autoequivalences
受Thurston成果启发，研究者在Bridgeland稳定性条件存在的前提下，证明三角范畴自等价诱导质量指数增长滤过，提出更广义的伪Anosov自等价定义，构造新实例并证明部分箭图3-Calabi-Yau范畴上的该类自等价作用具有双曲性。
Group Actions from Algebraic Flops
本文通过双模锥为代数翻折收缩构造导出自等价，证明其为局部双边倾斜复形，并由此在导出范畴上实现无限超平面排列基本群的群作用，推广并简化了已有结果。
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.
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 associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito--Takebayshi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of $2$-Calabi…
Categorical Dynamics of Abelian Varieties
本文研究了阿贝尔簇导出范畴自等价的动力学，证明了 Gromov--Yomdin 等式、熵上界、平移数刻画及约化平移数有限性条件，并否证了范畴熵函数的代数性问题。
For any admissible subcategory of the bounded derived category of coherent sheaves on a smooth proper variety, we prove that sections of the canonical bundle impose a strong constraint on the supports of the objects of the subcategory or its semiorthogonal complement. We also show that admissible subcategories are rigid under the actions of topologically trivial autoequivalences. As applications of these results, we prove that the derived category of various minimal varieties in the sense of the minimal model program admit no non-trivial semiorthogonal decompositions, generalizing the result f…
We discuss some mathematical conjectures which have come out of the Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich's homological mirror symmetry conjecture, and "physics proofs" for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category.
A version of the Bondal-Orlov conjecture, proved by Bridgeland, states that if $X$ and $Y$ are smooth complex projective threefolds linked by a flop, then they are derived equivalent. Van den Bergh gave a new proof of Bridgeland's theorem using the notion of a NCCR, which is in particular a ring $A$ together with a derived equivalence between $X$ and $A$. This ring $A$ is constructed as an endomorphism ring of a decomposable module, and hence admits an idempotent $e$. Donovan and Wemyss define the contraction algebra $A_\mathrm{con}$ to be the quotient of $A$ by $e$; it is a finite-dimensional…
Hilbert schemes of elliptic surfaces: group actions and derived categories
针对带整纤维与截面的椭圆曲面，构造$C^{[n]}$上的交换群概型并延拓其作用至$X^{[n]}$，结合导出McKay对应构造$D^b\operatorname{Coh}(X^{[n]})$的正合自等价，证明其与群作用交织，还讨论无截面的情形。
We give an autoequivalence of the derived category of the Ginzburg dg algebra for a mutation loop satisfying the sign stability introduced in [IK21]. We compute the categorical entropies of their restrictions to some subcategories and conclude that they are both given by the logarithm of the cluster stretch factor. Moreover, we discuss the pseudo-Anosovness of them in the sense of [FFHKL19] and [DHKK14].
This note is based on a talk given at String-Math 2012 in Bonn, on a joint paper with Ed Segal. We exhibit derived equivalences corresponding to certain Grassmannian flops. The construction of these equivalences is inspired by work of Herbst-Hori-Page on brane transport for gauged linear sigma models: in particular, we define 'windows' corresponding to their grade restriction rules. We then show how composing our equivalences produces interesting autoequivalences, which we describe as twists and cotwists about certain spherical functors. Our proofs use natural long exact sequences of bundles o…
We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.
A Remark on Higher Homotopy Sheaves of Derived Arc Spaces
本文探讨导出弧空间在光滑方案与局部完整交方案中的同调sheaf特性，指出其与经典几何空间无异。
This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient stack which admits a stratification of the kind arising in geometric invariant theory. The use of derived algebraic geometry leads to results with pleasingly few hypotheses, even when the stack is not smooth. Using the same methods, we establish a "virtual non-abelian localization theorem" which is a K-theoretic analog of the virtual localization theorem in…
In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.
We present results indicating that the decomposition of a Ricci-flat manifold in its irreducible factors is reflected by the derived category of coherent sheaves. More precisely, we prove that a smooth projective variety that is derived equivalent to an abelian variety resp. an irreducible symplectic variety is of the same type. The paper also contains a proof of a conjecure of Caldararu for manifolds with trivial canonical bundle saying that the modified HKR isomorphism for Hochschild homology is compatible with the module structure.
Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
We give a simple proof for the rigidity of a complex in the bounded derived category of sheaves with constructible cohomology on an abelian variety.
We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.
We discuss several conjectures about derived equivalent varieties, defined over fields of arbitrary characteristics, and implications among them. In particular we show that the (conjectural) derived invariance of the Hasse-Weil Zeta functions of smooth projective varieties over finite fields, implies the derived invariance of the Hodge diamond of complex algebraic varieties.
A Remark on Stability Conditions on Smooth Projective Varieties
本文证明了平滑射影簇的有界导出范畴总能接纳Bridgeland意义下的稳定性条件。
A phantom category is an admissible subcategory with vanishing Grothendieck group of the bounded derived category of coherent sheaves on a smooth projective variety. The goal of this paper is to study the abstract situation when such a category appears and establish some results which provide evidence for the idea that these categories are invisible on the level of Chow motives.
Some remarks about deformation theory and formality conjecture
本文利用代数判据证明了K3曲面导出范畴中普遍可胶合对象及Gushel--Mukai三维簇和四次双实体Kuznetsov分量中多稳定对象的形式性猜想。
Given a nontrivial semi-orthogonal decomposition $\Perf(\X)=\langle \mathcal{A},\mathcal{B}\rangle$, and assume that the base locus of $ω_{\X}$ is a proper closed subset, it was proved by Kotaro Kawatani and Shinnosuke Okawa that all skyscraper sheaves $\k(\x)$ with $\x\notin \mathsf{Bs}\vert ω_{\X}\vert$ belong to exactly one and only one of the components. It is natural to ask which one it is, and whether we can determine this by certain linear invariants. In this note we use Hochschild homology of derived category of coherent sheaves with support to provide another proof that if the $-\n^þ$…
Let $X$ be a projective variety of dimension $n\ge1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki-Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh, Oguiso and Zhang for compact Kähler manifolds.
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