Takuro Abe, Masahiko Yoshinaga
The purpose of this paper is to study the structure of reflexive sheaves over projective spaces through hyperplane sections. We give a criterion for a reflexive sheaf to split into a direct sum of line bundles. An application to the theory of free hyperplane arrangements is also given.
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We study the cohomology of reflexive rank 2 sheaves on smooth projective threefolds. Applications are given to the moduli space of reflexive sheaves.
In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any $m$-tail reflexive sheaf $\mathcal{E}$ on $\mathcal{P}^n$ is greater or equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on $\mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail reflexive sheaves of higher rank.
Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along certain globally generated torsion-free sheaves on $D$. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the $μ_L$-(semi)stability of such reflexive sheaves under certain conditions.
We use Serre construction and deformation to construct stable bundles and reflexive sheaves on Calabi-Yau threefolds.
We investigate the reflexive sheaves on $\PP^3$ spanned in codimension 2 with very low first Chern class $c_1$. We also give the sufficient and necessary conditions on numeric data of such sheaves for indecomposabiity. As a by-product we obtain that every reflexive sheaf on $\PP^3$ spanned in codimension 2 with $c_1=2$ is spanned.
In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.
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.
It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\FF$ which is generated by $H^0\FF$ on …
We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…
In this note, we discuss the concept of pseudoeffective vector bundle and also introduce pseudoeffective torsion-free sheaves over compact Kähler manifolds. We show that a pseudoeffective reflexive sheaf over a compact Kähler manifold with vanishing first Chern class is in fact a numerically flat vector bundle. A proof is obtained through a natural construction of positive currents representing the Segre classes of pseudoeffective vector bundles.
Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves
本文证明idyll上拟阵范畴是proto-exact和proto-abelian，并由此推出热带环面自反层范畴也具有这些性质，进而研究其稳定性与Harder-Narasimhan滤过。
Moduli spaces of slope-semistable sheaves with reflexive Seshadri graduations
本文研究了在光滑项目ive variety上具有反射或局部自由Seshadri梯度的斜率半稳定sheaves的模空间，并证明了其在特征为零时具有良好的模空间结构。
Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,σ), σ\in H^0(\sF), and the functor which deforms the corresponding pair (X,ξ) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without a…
On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of PT-stable objects - under the following assumptions: no two of the stability vectors are collinear, and the degree and rank of the objects are relatively prime. We also describe a close relation between the intersection of the moduli spaces of PT-stable and dual-PT-stable objects, and the moduli of reflexive sheaves.
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。
We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\mathscr{G}$ is a subfoliation of a codimension one distribution $\mathscr{F}$ with isolated singularities, the…
In this work we construct global resolutions for general coherent equivariant sheaves over toric varieties. For this, we use the framework of sheaves over posets. We develop a notion of gluing of posets and of sheaves over posets, which we apply to construct global resolutions for equivariant sheaves. Our constructions give a natural correspondence between resolutions for reflexive equivariant sheaves and free resolutions of vector space arrangements.
We study moduli spaces $\mathcal{N}$ of rank 2 stable reflexive sheaves on $\mathbb{P}^3$. Fixing Chern classes $c_1$, $c_2$, and summing over $c_3$, we consider the generating function $\mathsf{Z}^{\mathrm{refl}}(q)$ of Euler characteristics of such moduli spaces. The action of the torus $T$ on $\mathbb{P}^3$ lifts to $\mathcal{N}$ and we classify all sheaves in $\mathcal{N}^T$. This leads to an explicit expression for $\mathsf{Z}^{\mathrm{refl}}(q)$. Since $c_3$ is bounded below and above, $\mathsf{Z}^{\mathrm{refl}}(q)$ is a polynomial. We find a simple formula for its leading term when $c_…
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion. As an application, when (X,L) is a polarized toric orbifold of dimension n, we relate stable equivariant reflexive sheaves on (X, L) to stable equivar…
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.
Reconstruction of a surface from the category of reflexive sheaves
本文定义了余维2饱和曲面，证明任何正规曲面有唯一余维2饱和模型，并从自反层加性范畴重构该模型，引入弱局部化Serre子范畴处理范畴问题。
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. A…
We prove existence of reflexive sheaves on singular surfaces and threefolds with prescribed numerical invariants and study their moduli.
Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is dev…
We construct a moduli space of slope-semistable pure sheaves, building upon previous work of Le Potier and Jun Li on torsion-free sheaves over smooth surfaces. In particular, our construction provides a compactification of the Simpson moduli space of slope-stable reflexive sheaves. We also prove an effective restriction theorem for slope-(semi)stable pure sheaves following an approach due to Langer.
The Weil Decoration of the Horrocks-Mumford Bundle
本文研究了正常代数簇中反射性秩一sheaf与Weil除子之间的关系，并将其推广到任意秩的反射性sheaf及其所谓的Weil装饰，同时定义并研究了Horrocks-Mumford丛的一个自然推广。
We define reflexive sheaves on a singular quadric Q that generalize the spinor bundles on smooth quadrics, using matrix factorizations of the equation of Q. We study the first properties of these spinor sheaves, give a Horrocks-type criterion, and show that they are semi-stable, and indeed stable in some cases.
Acyclic toric sheaves
本文基于Perlman和Smith的定理，证明了在光滑射影的环形变种上，通过Weil装饰给出典范sheaf无环的显式充分条件。
Toric sheaves, stability and fibrations
本文研究了极化环面簇上等变反射层沿环面纤维化拉回的斜率稳定性，证明了在绝热极化下稳定性得以保持，给出了严格半稳定情形的充要条件，并展示了其在切层稳定扰动及模空间单射映射中的应用。
Intersection theory and Chern classes on normal varieties
本文研究正规簇上自反层的相交理论与陈类，将Mumford理论推广至更高维，定义第二陈类，并证明正特征下的Bogomolov型不等式及有界性结果。
We introduce the notion of $T$-stability for torsion-free Higgs sheaves as a natural generalization of the notion of $T$-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of torsion-free Higgs sheaves are important in the definition of $T$-stability. Using this, we show that this notion is preserved under dualization and tensor product with an arbitrary Higgs line bundle. Then, we prove that for a torsion-free Higgs sheaf over a compact Kähler manifold…
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.
In this paper we study the cohomology of the de Rham complex of sheaves of reflexive differential forms on a normal complex space. First, we prove that the complex is exact in degree one under suitable conditions on the underlying topological space, but that exactness in general depends on the complex structure. Second, we show exactness in high degrees for holomorphically contractible spaces under mild assumptions on the nature of singularities, e.g. klt singularities. Subsequently, the exactness of the de Rham complex of reflexive differential forms is related to the Lipman-Zariski conjectur…
We give an explicit proof of a Bogomolov-type inequality for $c_3$ of reflexive sheaves on $\mathbb{P}^3$. Then, using resolutions of rank-two reflexive sheaves on $\mathbb{P}^3$, we prove that some strata of the moduli of rank-two complexes that are both PT-stable and dual-PT-stable are quotient stacks. Using monads, we apply the same techniques to $\mathbb{P}^2$ and show that some strata of the moduli of Bridgeland-stable complexes are quotient stacks.
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $μ(F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ(F)$. We show that these canonical subsheaves behave well with respect to the pul…
We provide an effective version of Katz' criterion for finiteness of the monodromy group of a lisse, pure of weight zero, $\ell$-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf
Coherent sheaves on subvarieties in Hopf manifolds
本文证明了对于具有中心为x的可逆全纯收缩γ的正规复解析簇X，GAGA定理的一个版本，并研究了X的仿射簇结构以及γ-等变凝聚层的代数化，进而通过商簇X₀/γ嵌入Hopf流形，给出了Hopf流形中正规子簇的结构定理和凝聚层的滤过性。
Toric sheaves and polyhedra
本文从多面体视角研究光滑射影环面簇上的环面层，通过多面体包含/排除序列构造nef线丛的泛扩张，并建立双复形与谱序列以计算环面层上同调。
Semi-rigid stable sheaves: a criterion and examples
本文提出半刚性稳定她af 的准则，并通过线包和拉格朗日子变种上的例子展示其应用。
We shall show that $q$-semipositivity of the vector bundle $E$ over a Kähler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on the base manifold.
Higher rank sheaves on threefolds and functional equations
本文研究光滑射影三维流形上任意秩稳定无挠层模空间中奇点集为空或0维的开子集，证明其拓扑Euler特征生成函数等于MacMahon函数的幂乘以一个在q↔q⁻¹下不变的Laurent多项式，并通过Quot概型与Pandharipande-Thomas对的墙跨越完成证明。
We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is in the Gorenstein liaison class of a complete intesection and we a…
Moduli of rank two semistable sheaves on rational Fano threefolds of the main series
本文研究了在有理Fano三维流形上秩为二的半稳定sheaves的模空间，证明了其第三陈类的有界性，并构造了新的有理和无理成分的无限系列。
Stable vector bundles on a hyper-Kahler manifold with a rank 1 obstruction map are modular
本文证明不可约全纯辛流形上正秩斜率稳定自反层在障碍映射秩为1时高度模，并构造LLV格向量及三类导出等价来源的例子。
G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves. As an application, we show that the M-regularity of a suitable $\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ample line bundles on abelian varieties.
Torsion free instanton sheaves on the blow-up of $\mathbb{P}^{3}$ at a point
本文推广瞬子丛定义至爆破空间上的无挠层，证明自反瞬子层局部自由，严格无挠者奇点纯一维，构造例子并研究其稳定性，用于紧化模空间分量。
Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M x M there exists a rank 2n-2 reflexive hyperholomorphic sheaf E_M, whose fiber over a non-diagonal point (F,G) is Ext^1(F,G). The sheaf E_M can be deformed along some twistor path to a sheaf E_X over the cartesian square of every Kahler manifold X deformation equivalent to M. We prove that E_X is infinitesimally rigid, and the isomorphism class of the Azumaya algebra End(E_X) is independent of the twistor path chosen. This verifies conjectures in arXiv:1310.5782 and arXiv:150…
Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on $\Q_n\subset \mathbb P^{n+1}$ with the Castelnuovo-Mumford regularity of their extension by zero in $\mathbb P^{n+1}$. We also classify the coherent sheaves with Qregularity $-\infty$. We use our notion of Qregularity in order to prove an extension of Evans-Griffiths criterion to vector bundles on Quadrics. In particular we get a new and simple proof of the Knörrer's characterization of ACM bundles.
We establish a criterion for sheaves on an adically complete DG scheme to be coherent. We deduce a description of coherent sheaves on an adically complete lci singularity in terms of modules for a DG Lie algebra.
A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
本文将Höring–Peternell的平坦性准则从射影簇推广至凯勒空间，证明klt紧凯勒空间上第一陈类为零的伪有效层经有限拟平覆盖后自反拉回局部自由且平坦，核心依赖两项新结果。
Admissible Hermitian-Yang-Mills connections over normal varieties
本文在正规簇上证明完整版 Donaldson-Uhlenbeck-Yau 定理，建立 Hitchin-Kobayashi 对应，并由此得到 Bogomolov-Gieseker 不等式及其等号刻画，还给出平凡第一 Chern 类正规 Kaehler 簇为环面有限商的新判据。
Divisors on coherent schemes and homogeneous spaces
本文研究相干基上齐性空间上除子的正性与可延展性，发展了局部相干概形上的估值除子理论，建立了有效估值除子与秩为一的反射sheaf的精确对应关系，推导出非Noetherian的Ramanujam-Samuel定理。
In this article, we prove an extension property of semipositively metrized ample invertible sheaves on a projective scheme over a complete non-archimedean valued field. As an application, we establish a Nakai-Moishezon type criterion for adelically normed graded linear series.
Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities
本文引入有理曲面奇点部分解消上的特殊 Cohen--Macaulay 层范畴，证明其具有 Frobenius 结构，并建立与特殊 Cohen--Macaulay 模的精确等价及稳定范畴的三角等价，给出几何证明和 Buchweitz 型等价。
Fixed point loci of moduli spaces of sheaves on toric varieties
该研究推广前人工作给出非奇异环面簇上纯等变层的组合描述，借助GIT构造对应模空间，显式表出Gieseker稳定层模空间的环面作用不动点轨迹，构造出对应Gieseker稳定性的等变线丛，并得到μ-稳定自反层模空间不动点轨迹的组合描述，可用于计算相关欧拉示性数生成函数。
We extend Langton's valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results.
Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed Kähler class.
Toric Foliated Minimal Model Program
本文利用Klyachko滤过理论，将Spicer的结果推广至任意秩，证明了环簇叶状极小模型纲领的存在性。
Saito criterion and its avatars
本文提出 Saito 判据的新证明，覆盖仿射空间、射影空间及多导子情形，并推广至多个多项式对应的导子模。
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