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.
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.
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…
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 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.
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.
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.
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.
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 …
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.
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…
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…
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.
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…
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…
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…
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.
A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications. This generalizes a result of Ewald on pseudo-symmetric nonsingular toric Fano varieties.
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 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 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.
A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
Université Côte d’Azur（蔚蓝海岸大学）；CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université（法国国家科学研究中心，朱西厄-巴黎左岸数学研究所，索邦大学）；Tohoku University（东北大学）
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…
In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective and weakly positive, by using singular Hermitian metrics. As an applications, we obtain a generalization of Mori's result. We also give a characterization of the augmented base locus by using singular Hermitian metrics on vector bundles and the Lelong numbers.
We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use $L^{2}$-harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the res…
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for K"ahler morphisms and singular hermitian metrics with non-algebraic…
An Ulrich sheaf on an n-dimensional projective variety X, embedded in a projective space, is a normalized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby-Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves delta-Ulrich. In the case n=2, where delta-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection is a semistable instanton bundle, we show t…
By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundl…
The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which caracterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism of coherent sheaves, which generalizes the concept of stable isomorphism. An N-type resolution for a family of curves $C$ over the local ring $A$, defined by an ideal $J$, is an exact sequence $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on $P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. We prove the two following results, when the residual field of $A$ is inf…
We study constructions of stable holomorphic vector bundles on Calabi-Yau threefolds, especially those with exact anomaly cancellation which we call extremal. By going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds. We then introduce a generalized version of Hartshorne-Serre construction and use it to yield extremal bundles of general ranks and study their geometry. In light of this probing the geometry of the space of stable vector bundles, we revisit the DRY conjecture on stable r…
