Gunnar Floystad
A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
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
Derived category of equivariant coherent sheaves on a smooth toric variety and Koszul duality
本文通过定义光滑环面簇扇上的分次代数层，建立组合Koszul对偶等价，并用模层描述等变凝聚层范畴，最终将完备情形下的对偶解释为Serre函子。
We obtain a linear algebra data presentation of the category of constructible with respect to perverse triangulation sheaves on a finite simplicial complex. We also establish Koszul duality between the above mentioned category and the category of perverse sheaves constructible with respect to the triangulation
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
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.
Let $X$ be a finite connected simplicial complex, and let $δ$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($δ$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,δ)$ and $B(X,δ)$ respectively, and we prove that $A(X,δ)$ and $B(X,δ)$ are Koszul dual to each other. We define the $δ$-perverse topo…
We are going to show that the sheafication of graded Koszul modules $% K_Γ$ over $Γ_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_Γ$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_Γ$up to shift. More importantly, the category $K_Γ$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $Γ$ and over the exterior algebra $Λ.$ This is then used to develop a kind of relative Auslander-Reiten theory for the ca…
We prove a new criterion for the homogeneous coordinate ring of a finite set of points in ${\Bbb P}^n$ to be Koszul. Like the well known criterion due to Kempf it involves only incidence conditions on linear spans of subsets of a given set. We also give a sufficient condition for the Koszul property to be preserved when passing to a subset of a finite set of points in ${\Bbb P}^n$.
Tilting sheaves for real groups and Koszul duality
本文发展了实解析簇上的倾斜层理论，构造了其到Soergel双模范畴实类似的完全忠实嵌入，并以此给出了Bezrukavnikov和Vilonen定理的纯几何证明。
For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category O for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.
Koszul Duality for Coherent Sheaves
研究为无限维 Koszul 代数建立 Koszul 对偶性，应用于二次单项式代数等两类代数，还应用于代数几何，得到交换诺特 Koszul 代数定义的射影概型的有界导出范畴的 Koszul 对偶描述及 BGG 型对应。
We study the cohomology of reflexive rank 2 sheaves on smooth projective threefolds. Applications are given to the moduli space of reflexive sheaves.
This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.
For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical.
In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.
Sheaves of low rank in three dimensional projective space
本研究对三维射影空间中秩不超过4的半稳定层的陈特征进行分类，并证明对应最大第三陈特征的半稳定层模空间具有光滑不可约性。
We study the moduli space of framed flags of sheaves on the projective plane via an adaptation of the ADHM construction of framed sheaves. In particular, we prove that, for certain values of the topological invariants, the moduli space of framed flags of sheaves is an irreducible, nonsingular variety carrying a holomorphic pre-symplectic form.
In this note, we show that the canonical orientation data on the quiver hearts are compatible under the autoequivalence $\_\otimesπ^*O(1)$, and hence glue to give an orientation data for the stack of coherent sheaves on local P2.
Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a complex projective curve and constructed a projective moduli space of such objects. We generalize Ramanathan's notion and construction to the case of higher dimension, allowing also objects which we call semistable principal G-sheaves.
We construct the moduli of twisted sheaves on a projective variety. Then we generalize known results on the moduli space of usual sheaves on a K3 surface to the twisted case. Thus we consider the non-emptyness, the deformation type and the Fourier-Mukai transformation. As an application of our results, Daniel Huybrechts and Paolo Stellari prove Caldararu's conjecture.
We show that there exists a fine moduli space for torsion-free sheaves on a projective surface, which have a "good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.
We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.
The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah…
We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperkähler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.
We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasi-coherent sheaves on the projective line. In this model structure the cofibrant objects are the dg-locally projective complexes. We also describe the fibrations of this model structure and show that the model structure is monoidal. We point out that this model structure is necessarily…
This paper has been withdraw. A fully revised version with two new co-authors has been posted: "ADHM construction of perverse instanton sheaves", arXiv:1201.5657.
Uniform boundedness of semistable pure sheaves on projective manifolds
本文证明了射影流形上半稳定纯层的统界性，包括二维情形对任意紧集成立及高维情形的受限版本，并应用于模空间及其壁-室结构。
We develop a theory of general sheaves over weighted projective lines. We define and study a canonical decomposition, analogous to Kac's canonical decomposition for representations of quivers, study subsheaves of a general sheaf, general ranks of morphisms, and prove analogues of Schofield's results on general representations of quivers. Using these, we give a recursive algorithm for computing properties of general sheaves. Many of our results are proved in a more abstract setting, involving a hereditary abelian category.
We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.
Koszul complexes and spectra of projective hypersurfaces with isolated singularities
该研究推广了非奇异射影超曲面的相关经典结论，利用Koszul上同调与极点阶谱序列，推导了孤立奇点加权齐次射影超曲面的BS多项式根与Jacobian环挠部的关系。
A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions.
Koszul Duality for Symmetric Algebras and Derived Loop Spaces
本文证明了连通复形对称代数的Koszul对偶，并构造了任意有限型分离概形的导出环路空间与切复形对称代数之间的导出范畴等价，推广了Kapranov的光滑情形结果。
In this paper, we study the divisor theory of the Simpson moduli space of semistable sheaves of dimension 1 on the projective plane. We prove that these spaces are all Mori dream spaces, and calculate their nef cones. We also study the effective cones of these spaces for most choices of numerical invariants. As a consequence, we work out precisely when two such spaces are isomorphic.
We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.
The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.
Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties
本文在多重分次Castelnuovo–Mumford正则性框架下，证明射影簇的通用上同调消失定理，推广前期工作，刻画射影空间乘积线丛的极小正则性，重证Green定理并计算超椭圆曲线典范像的Betti表。
We give a complete classification of semistable rank two sheaves on three-dimensional projective space with maximal third Chern character. This implies an explicit description of their moduli spaces. As an open subset they contain rank two reflexive sheaves with maximal number of singularities. These spaces are irreducible, and apart from a single special case, they are also smooth. This extends a result by Okonek and Spindler to all missing cases and gives a new proof of their result. The key technical ingredient is variation of stability in the derived category.
We show that the moduli space M(r,c) of semistable sheaves on n-dimensional projective space with support of dimension one, with multiplicity r and with Euler characteristic c is isomorphic to M(r,-c).
Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks: Koszul duality
本文证明光滑栈上过滤 $\mathcal{D}$-模与其形式环路空间上的 $S^1$-等变 ind-coherent 层之间存在 Koszul 对偶等价，并描述其特殊纤维与一般纤维。
We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.
We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …
We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott's formula for the projective space to a projective bundle over a scheme of characteristic zero.
For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we can prove that $\splcpx_{X/k}\uet$ is smooth and has a symplectic structure.
Sheaves on non-reduced curves can appear in moduli space of 1-dimensional semistable sheaves over a surface, and moduli space of Higgs bundles as well. We estimate the dimension of the stack $\mathbf{M}_{X}(nC,χ)$ of pure sheaves supported at the non-reduced curve $nC~(n\geq 2)$ with $C$ an integral curve on $X$. We prove that the Hilbert-Chow morphism $h_{L,χ}:\mathcal{M}^H_X(L,χ)\rightarrow |L|$ sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for $X$ Fano or with trivial canonical line bundle and $|L|$ contains integral curves.
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In …
The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points. When the surface is the complex projective plane, we determine all the 1-point genus-0 Gromov-Witten invariants extremal with respect to the Gieseker-Uhlenbeck morphism. The main idea is to understand the virtual fundamental class of the moduli space of stable maps by studying the obstruction sheaf and using a meromorphic 2-form on the Gieseker moduli space.
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.
Universal Koszul Duality for Kac-Moody Groups
本文通过Soergel理论描述，证明了Kac-Moody旗 manifold上真equivariant K-动机与Langlands对偶的constructible monodromic sheaves之间的单调等价，扩展了有限维旗 manifold上的结果。
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the…
We prove that, given integers $m\geq 3$, $r\geq 1$ and $n\geq 0$, the moduli space of torsion free sheaves on $\mathbb P^m$ with Chern character $(r,0,\ldots,0,-n)$ that are trivial along a hyperplane $D \subset \mathbb P^m$ is isomorphic to the Quot scheme $\mathrm{Quot}_{\mathbb A^m}(\mathscr O^{\oplus r},n)$ of $0$-dimensional length $n$ quotients of the free sheaf $\mathscr O^{\oplus r}$ on $\mathbb A^m$.
This paper generalises Mori's famous theorem about "Projective manifolds with ample tangent bundles" to normal projective varieties in the following way: A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the complex projective space.
We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we desc…
Let X be a smooth projective complex variety, and let G be an algebraic reductive complex group. We define the notion of principal G-sheaf, that generalises the notion of principal G-bundle. Then we define a notion of semistability, and construct the projective moduli space of semistable principal G-sheaves on X. This is a natural compactification of the moduli space of principal G-bundles. This is the announcement note presented by the second author in the conference held at Catania (11-13 April 2001), dedicated to the 60th birthday of Silvio Greco. Detailed proofs will appear elsewhere.
We show that instanton bundles of rank $r\le 2n-1$, defined as the cohomology of certain linear monads, on an $n$-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford-Takemoto. Furthermore, we show that rank $r\le n$ linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.
We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics.
In a previous paper, the first two named authors established an isomorphism between the moduli space of framed flags of sheaves on the projective plane and the moduli space of stable representations of a certain quiver. In the present note, we substitute one of the claims made, namely [5, Theorem 17], for a weaker claim regarding the existence of unobstructed points in the quiver moduli space. We also extend some of the results of the cited paper, concerning the maximal stability chamber within which the isomorphism mentioned holds, and the existence of a perfect obstruction theory for the qui…
We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, an…
We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
Rank-two reflexive sheaves on the projective space with second Chern class equal to four
本文研究了$\mathbb{P}^3$上$c_2=4$的秩二反身层，扩展了先前$c_2\le3$的结论，证明了所有未被排除的谱均可实现，并详细刻画了特定条件下的模空间。
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。