B. Toen, M. Vaquie
We prove that a compact complex analytic variety is algebraizable if and only if its bounded derived dg-category of coherent sheaves is saturated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Algebraization of rigid analytic varieties and formal schemes via perfect complexes
本文将Toën和Vaquié的定理扩展到非阿基米德和形式环境，证明了光滑且良好的刚性解析簇代数化当且仅当其完美复形的范畴也是光滑且良好的。
An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties G/P are Frobenius split. The aim of this article is to give in this case a complete and self contained representation theoretic approach to this method. The geometric Frobenius method in (char k=p>0) will here be replaced by Lusztig's Frobenius maps for quantum groups at roots of unity (which exist not only for primes but any odd integer \ell >1).
We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. This conjecture, closely related to the Grothendieck Period Conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over $\bar{\mathbb Q}}$. We also derive the Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieti…
This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequ…
We prove that the derived categories for toric varieties have complete exceptional collections.
We prove two theorems on the derived categories of toric varieties, the existence of an exceptional collection consisting of sheaves for a divisorial extraction and the finiteness of Fourier-Mukai partners.
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.
We prove derived McKay correspondence in special cases and the decomposition of toric K-equivalence into flops.
We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.
Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.
Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties
研究阿贝尔簇上$\mathbb{G}_m$示性类的1-扭曲半齐次向量束，并分类阿贝尔簇扭曲导出范畴中的点对象，进而分类阿贝尔簇的扭曲傅里叶-穆卡伙伴。
We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.
We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.
We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety $G^{r}_{d}(C)$ that parametrizing linear series of degree $d$ and dimension $r$ on a general smooth projective curve $C$ is indecomposable when $d\leq g(C)-1$.
Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories
本文综述了紧致超Kähler流形理论的最新发展，从拉格朗日纤维化、稳定sheaves模空间和导出范畴的角度探讨核心方法与贡献。
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.
Derived categories of generalized Kummer varieties: extended Mukai vector
本文利用扩展 Mukai 向量研究广义 Kummer 型超 Kähler 流形的导出等价，构造并证明了一个在导出等价下不变的整格，并以导出单值群阐述其应用。
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
Smooth categories in a 6 functor formalism and compact generation for nuclear categories in analytic geometry
本文在六 functor 形式主义框架下研究光滑 ∞-范畴，并利用凝聚数学理论和解析堆的结果，证明了刚性解析流形光滑当且仅当其关联的核sheaves范畴光滑，同时将核sheaves范畴的紧生成性与刚性解析流形的代数化联系起来，从而得到一个非原子生成但内部光滑的范畴例子。
Derived categories of Fano varieties of lines
本文研究光滑三次四维簇直线 Fano 簇的导出范畴，针对具有有理 Lagrangian 纤维化的一般情形证明 Galkin 猜想，并建立相关权重二 Hodge 结构的等距关系。
We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational…
It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a fam…
Moduli spaces of stable objects in Enriques categories
本文通过关联K3范畴中的模空间，研究了Enriques范畴中稳定对象的模空间，解决了特定Fano簇Kuznetsov分支模空间的非空性问题，并推导出模超Kähler簇上反辛对合不动点轨迹的非空性。
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.
In this short note, we observe that Theorem 3.1 in arXiv:1508.00682 for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes. Here weak indecomposablility refers to the admissibility of components.
We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary smooth proper varieties and in families. As applications, we construct Hodge classes whose algebraicity would imply period-index bounds; construct new counterexamples to the integral Hodge conjecture on Severi-Brauer varieties; and prove the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces.
The semiregularity theorem for equivariant noncommutative varieties
将Buchweitz-Flenner经典半正则性定理推广到带群作用的非交换代数几何中，应用于扭曲导出范畴，回答了Markman的问题并简化了其关于阿贝尔四重霍奇猜想的证明。
Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived categories of smooth varieties, we construct a similar semiorthogonal decomposition for a smooth $G$-invariant divisor in $X$ (under certain technical assumptions). Combining this procedure with the semiorthogonal decompositions constructed in [PV15], we construct semiorthogonal decompositions of some equivariant derived categories of smooth projective varieties.
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.
Semi-Homogeneous Sheaves and Twisted Derived Categories
本文构造了阿贝尔簇挠子与其简单半齐次层模空间间的扭曲导出等价，证明了其逆命题，并推广了Orlov与Polishchuk的导出等价准则。
On finite categories of algebraic varieties
本文证明了特征零域上不可约代数簇的有限生成范畴的有限性是可判定的，并提出了Burnside有限性判据，应用于多个映射的代数动力系统。
Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O'Grady's filtration on the $\mathrm{CH}_0$-group of the $K3$ surface. This solves a conjecture of O'Grady and improves on previous results of Huybrechts, O'Grady, and Voisin. Then we propose a cand…
In this short note we show how results of Orlov and Toën imply that any equivalence between the derived categories of coherent sheaves on two varieties lifts to an equivalence at the level of dg-categories. This establishes the link between the noncommutative geometry practised by the school of Bondal-Orlov, and the variant developed by Kontsevich and Tabuada. As an application we recover Orlov's result that the derived category determines the Chow motive with rational coefficients up to Tate twists.
Derived categories of coherent sheaves on abelian varieties and equivalences between them
本文研究阿贝尔簇上凝聚层的导出范畴，给出两个此类范畴等价的判据，并刻画相应的自等价群。
For a given Fourier-Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier-Mukai equivalences of derived factorization categories of gauged Landau-Ginzburg (LG) models. As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier-Mukai equivalence is linearizable wi…
Generating systems, generalized Thomsen collections and derived categories of toric varieties
本文研究了 toric 变形的生成系统和广义汤姆森集合，验证了 Bondal 关于导出范畴生成的猜想。
Derived categories of symmetric products and moduli spaces of vector bundles on a curve
本文证明曲线对称积的导出范畴可嵌入大秩向量丛模空间，支持其半正交分解预言，并证明低维主极化 Abel 簇等为 Fano visitors。
In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra P: the algebra P is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in proj P, and that their degeneration order coincides with the Jensen-Su-Zimmermann's degener…
In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.
We discuss relations between the motives of two varieties with equivalent derived categories of coherent sheaves.
We show that certain categories of perverse sheaves on a pair of affine toric varieties defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel. The functor expressing this duality is constructed explicitly using a combinatorial model for mixed sheaves on toric varieties.
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic.
Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor. As an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties.
We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying d…
We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves on abelian varieties. As an application we explain how the arising Tannaka groups on abelian varieties can be studied in terms of semiabelian degenerations via the functor of nearby cycles.
We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.
This paper is based on a talk at a conference "JDG 2017: Conference on Geometry and Topology". We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the K-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the K-inequality to the fully faithful embedding.
A counterexample to the Jordan-Hölder property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by Böhning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of the projective space.
Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.
We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties. Finally, we introduce the concept of…
We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories.
Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category…
We present a notion of $Δ$-stability and stability filtration in arbitrary categories which is equivalent to the existence of Harder-Narasimhan (HN) sequences on objects. Indeed it is equivalent to the existence of a zero morphism, a partial order on objects, and a collection of some universal sequences. In additive categories embedded in an ambient triangulated category, we could obtain a numerical polynomial or central charge of objects by calculating the Euler characteristic of slope sequences and objects, inducing a partial order and HN sequences. In the case of normal projective surfaces …
Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.
We extend Orlov's result that certain functors between derived categories of smooth projective varieties are Fourier--Mukai transforms to the case when those varieties are smooth and proper.
Tilting objects in singularity categories of toric Gorenstein varieties
该研究针对特定托里拆利Gorenstein概形的奇点范畴，基于前人工作证明其存在倾斜对象，从而建立与有限维代数完美范畴的三角等价关系。
We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.
Using techniques from motivic homotopy theory, we prove a conjecture of Anthony Blanc about semi-topological K-theory of dg categories with finite coefficients. Along the way, we show that the connective semi-topological K-theories defined by Friedlander-Walker and by Blanc agree for quasi-projective complex varieties and we study étale descent of topological K-theory of dg categories.
Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-equivalent} if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide at least for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.
We propose a conjecture on the existence of a specialization map for derived categories of smooth proper varieties modulo semi-orthogonal decompositions, and verify it for K3 surfaces and abelian varieties.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。