Subsheaves of a hermitian torsion free coherent sheaf on an arithmetic variety

Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold

On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $Φ$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $Φ$ to a slope semistable sheaf, up to mod…

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic surfaces

On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted $Z^l$-stability, by varying the polarisation in the definition of Bridgeland stability along a curve in the ample cone of $X$. We show that a slope stable torsion-free sheaf of positive (twisted) degree or a slope stable locally free sheaf is taken by a Fourier-Mukai transform on $D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzero fiber degree can be modified so that its inverse Fourier-Mukai transform is a slope semistable torsion-free sheaf.

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds

We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.

The sheaf $α$ $\bullet$ X

We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $ω\_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $Ω\_{X}^{\bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $Ω\_{X}^{\bullet}/torsion$. This sheaf $α\_{X}^{\bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic d…

关于Kähler微分层的无挠性

On torsion-freeness of Kähler differential sheaves

该研究在特征零代数闭域上的正规代数簇中，给出了Kähler微分层无挠性的等价刻画：其一阶形变理想层的正则截面可从奇点轨迹外延拓至奇点轨迹。

On the torsion of the first direct image of a locally free sheaf

Let $π:M\to B$ be a proper holomorphic submersion between complex manifolds and ${\cal E}$ a holomorphic bundle on $M$. We study and describe explicitly the torsion subsheaf $\mathrm{Tors}(R^1π_*({\cal E}))$ of the first direct image $R^1π_*(\mathcal{E})$ under the assumption $R^0π_*(\mathcal{E})=0$. We give two applications of our results. The first concerns the locus of points in the base of a generically versal family of complex surfaces where the family is non-versal. The second application is a vanishing result for $H^0(\mathrm{Tors}(R^1π_*(\mathcal{E})))$ in a concrete situation related …

A restriction theorem for torsion-free sheaves on some elliptic manifolds

We prove that if $X$ is the total space of an elliptic principal bundle $π:X\ra B$ which is non-Kähler, then the restriction of any torsion-free sheaf on $X$ to the general fiber of $π$ is semi-stable.

扭转增殖

Torsion proliferation

本文研究有限旗簇中舒伯特簇上层的通用性质，证明随着代数群秩增大，几乎所有交上同调层含 $p$-扭转且宇称层非反常，并给出部分旗簇族的显式渐近。

On the Harder-Narasimhan Filtration for Coherent Sheaves on P2: I

Let E be a torsion-free sheaf on P2. We give an effective method which uses the Hilbert function of E to construct a weak version of the Harder-Narasimhan filtration of a torsion-free sheaf on P2 subject only to the condition that E be sufficiently general among sheaves with that Hilbert function. This algorithm uses on a generalization of Davis' decomposition lemma to higher rank.

A GIT interpretration of the Harder-Narasimhan filtration

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.

Hodge-Witt cohomology and Witt-rational singularities

We prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme…

Splitting type, global sections and Chern classes for torsion free sheaves on P^N

In this paper we compare a torsion free sheaf $\FF$ on $\PP^N$ and the free vector bundle $\oplus_{i=1}^n\OPN(b_i)$ having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of $\FF$. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i(\FF(t))$ of twists of $\FF$, only depending on some num…

The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories

This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…

On Gieseker stability for Higgs sheaves

We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective algebraic manifolds. In particular, we show that Gieseker stability for torsion-free Higgs sheaves can be defined using only Higgs subsheaves with torsion-free quotients; and we show that a classical relation between Gieseker stability and Mumford-Takemoto stability extends naturally…

挠率界限与一个有效的最大公约数定理

Bornes de torsion et un théorème effectif du pgcd

本文证明了 Deligne 的“最大公约数定理”的一个有效概率版本，适用于有限域上的光滑射影几何整簇，其核心方法是通过界定 Betti 上同调的挠率、利用大单值性结果及 Frobenius 均匀分布来实现。

Master Spaces for stable pairs

We construct master spaces for oriented torsion free sheaves coupled with morphisms into a fixed reference sheaf. These spaces are projective varieties endowed with a natural $\C^*$-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves and the quot scheme associated with the given data. In the case of curves with trivial reference sheaf, our master spaces compactify the moduli spaces constructed by Bertram, Daskalopoulos and Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf, master spaces provide algebraic analog…

$T$-stability for Higgs sheaves over compact complex manifolds

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…

Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some prop…

Torsion-free Sheaves and ACM Schemes

In this paper we study short exact sequences $ 0 \to \mathcal P \to \mathcal N \to \ii_D(k) \to 0 $ with $ \mathcal P, \mathcal N $ torsion--free sheaves and $ D $ closed projective scheme. This is a classical way to construct and study projective schemes (e.g. see \cite{hart-1974}, \cite{hart-2}, \cite{mdp}, \cite{serre-1960}). In particular, we give homological conditions on $ \mathcal P $ and $ \mathcal N $ that force $ D $ to be ACM, without constrains on its codimension. As last result, we prove that if $ \mathcal N $ is a higher syzygy sheaf of an ACM scheme $ X,$ the scheme $ D $ we get…

Moduli of Admissible Pairs for Arbitrary Dimension, II: Functors and Moduli

Admissible pairs $((\widetilde S, \widetilde L), \widetilde E)$ consisting of an $N$-dimensional projective scheme~$\widetilde S$ of certain class with a special ample invertible sheaf $\widetilde L$ and a locally free ${\cal O}_{\widetilde S}$-sheaf $\widetilde E$ are considered. An admissible pair can be produced in the procedure of a transformation (which is called a resolution) of a torsion-free coherent sheaf $E$ on a nonsingular $N$-dimensional projective algebraic variety $S$ to a locally free sheaf $\widetilde E$ on some projective scheme $\widetilde S$. Notions of stability and semist…

任意维数可容许对的模空间，I：消解

Moduli of Admissible Pairs for Arbitrary Dimension, I: Resolution

本文将曲面凝聚层标准消解推广到任意维数，提出将非奇异N维射影代数簇上的无挠凝聚层消解为某类射影概形上局部自由层的方法，并证明其适用于所有无挠层平坦族。

Atiyah class and sheaf counting on local Calabi Yau fourfolds

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class e…

$2$-nilpotent co-Higgs structures

A co-Higgs sheaf is a pair of a torsion-free coherent sheaf $\mathcal{E}$ and a global section of $\mathcal{E}nd(\mathcal{E})\otimes T_X$ with $T_X$ the tangent bundle. We construct $2$-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $\mathbb{P}^3$, using the Hartshorne-Serre correspondence. Then we investigate the non-existence, specially over projective spaces.

Stability of torsion free sheaves on curves and infinite-dimensional Grassmanian manifold

There exists the Krichever map from the set of quintets (C,p,F,t,e) (where C is an integral and complete algebraic curve, p a smooth rational point, F a rank 2 torsion free coherent sheaf on C, t a local formal parameter in p and e a formal trivialization of F around p) to the infinite Grassmanian of $k((z)) \oplus k((z))$. We describe the images of quintets with (semi)stable sheaves F in terms of Plucker coordinates and get some analog of GIT Hilbert-Mumford numerical criterion with respect to actions of some 1-parametric subgroups of the group $SL(2,k[[z]])$ on the determinant bundle of the …

Continuous closure of sheaves

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the étale topologies) and it commutes with field extensions.

三维流形上的高秩层与函数方程

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。

Modular sheaves on hyperkähler varieties

A torsion free sheaf on a hyperkähler variety $X$ is modular if the discriminant satisfies a certain condition, for example if it is a multiple of $c_2(X)$ the sheaf is modular. The definition is taylor made for torsion-free sheaves on a polarized hyperkähler variety (X,h) which deform to all small deformations of (X,h). For hyperkählers deformation equivalent to $K3^{[2]}$ we prove an existence and uniqueness result for slope-stable modular vector bundles with certain ranks, $c_1$ and $c_2$. As a consequence we get uniqueness up to isomorphism of the tautological quotient rank $4$ vector bund…

在 $\mathbb{P}^{3}$ 点爆破上的无挠瞬子层

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。

On the singular scheme of codimension one holomorphic foliations in P^3

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P^3 is locally free if and only the singular scheme is a curve, and that it splits if and only if that curve is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in P^3 is …

Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve associated to Y, and that for every P in C, if z is an equation of C_1 in the local ring of Y at P, then (z^i) is the ideal of C_i. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e. sheaves which are locally i…

Moduli schemes of generically simple Azumaya modules

Let A be an Azumaya algebra over a smooth projective variety X or more generally, a torsion free coherent sheaf of algebras over X whose generic fiber is a central simple algebra. We show that generically simple torsion free A-module sheaves have a projective coarse moduli scheme; it is smooth and even symplectic if X is an abelian or K3 surface and A is Azumaya. We explain a relation to the classical theory of the Brandt groupoid.

Moduli spaces of SL(r)-bundles on singular irreducible curves

For a stable irreducible curve $X$ and a torsion free sheaf $L$ on $X$ of rank one and degree $d$, D.S. Nagaraj and C.S. Seshadri ([NS]) defined a closed subset $\Cal U_X(r,L)$ in the moduli space of semistable torsion free sheaves of rank $r$ and degree $d$ on $X$. We prove that $\Cal U_X(r,L)$ is irreducible, when a smooth curve $Y$ specializes to $X$ and a line bundle $\Cal L$ on $Y$ specializes to $L$, the specialization of moduli space of semistable rank $r$ vector bundles on $Y$ with fixed determinant $\Cal L$ has underlying set $\Cal U_X(r,L)$. For rank 2 and 3, we show that there is a …

三维流形上秩二无挠层的点奇点与局部第三陈类

Point Singularities and Local Third Chern Classes for Rank-Two Torsion-free Sheaves on Threefolds

本文研究复三维流形上秩二无挠层孤立点奇点携带的局部第三陈类贡献，证明其可由全族数据计算且形变不变，并给出相对K-理论解释。

奇异埃尔米特度量与Catanese、Fujita和Kawamata分解定理

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

本文证明带半正曲率奇异埃尔米特度量且满足最小延拓性质的无挠层可分解为埃尔米特平坦丛与泛型丰沛层的直和，并将其应用于相对多重典范丛直接像，推广了Fujita、Catanese–Kawamata和Iwai的结果。

多稳定局部阿贝尔抛物Higgs丛的Kollár工具包

Kollár's package for polystable locally abelian parabolic Higgs bundles

本文将Kollár工具包推广到带乘子理想扭转的多稳定局部阿贝尔抛物Higgs丛，统一复几何中的相关定理，并推出高正像层的弱正性与一般消灭性质。

带状结构上层的模

Moduli of sheaves on ribbons

本文研究了带状结构上无挠层模栈的几何性质，通过完全类型引入分层并分析其性质与闭包关系，揭示了Fano、Calabi-Yau和典范极化情形间的分叉以描述不可约分支，并计算了模栈在给定层处的切空间。

The generalized Lipman-Zariski problem

We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x \in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$, the sheaf $Ω_X^{[p]}$ of reflexive differential $p$-forms is free. Does this imply that $(x \in X)$ is smooth? We give an example showing that the answer is no even for $p = 2$ and $X$ a terminal threefold. However, we prove that if $p = n - 1$, then there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. As an application, we show that if $X$ is a projecti…

离散赋环 adic 空间的驯顺真基变换

Tame proper base change for discretely ringed adic spaces

本文在离散赋环 adic 空间中，针对真态射与局部闭浸入证明了驯顺拓扑下关于 Abel $p$-挠层导出前推的真基变换定理。

Wall-crossing for punctual Quot-schemes

We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their tautological integrals. In the fourfold case, we first construct these classes under additional conditions. We use novel methods relying on the wall-crossing of Joyce. Our results include -the dependence of the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface, -relations to the previous results in the literature, which addressed the …

关于光滑射影曲线上某些点状 Quot 概形的一个评注

A remark on some punctual Quot schemes on smooth projective curves

本文研究光滑射影曲线上局部自由层的点状 Quot 概形，证明其除具有正规性和典范 Gorenstein 奇点外，还是 Picard 数为一的 $\mathbb{Q}$-factorial Fano 簇。

拟平凡层的模空间

Moduli spaces of quasi-trivial sheaves

研究拟平凡层的Gieseker--Maruyama模空间$\mathcal{N}_X(r,n)$，通过点的商概形分析其性质，证明特定条件下的空性、同构性，构造$r<n$时的不可约分支并确定$X=\mathbb{P}^3$且$n\le10$时的唯一性。

形式推前的有限性

Finiteness of formal pushforwards

在温和假设下，研究形式概形上挠自由凝聚层的推前，证明有限性但非凝聚性，并关联到形式函数的粘合问题。

高 genus 减少的 Gromov-Witten 不变量 via sheaves 的 desingularizations

Higher genus reduced Gromov--Witten invariants via desingularizations of sheaves

本文通过 sheaves 的 desingularizations 构造了 stacks，用于定义大规模 GIT 商在所有 genus 中的减少 Gromov-Witten 不变量。