Reflexive Sheaf Research Articles

Rational singularities and q-birational morphisms

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank [Formula: see text], link our notion of rational singularities with the notion of rational singularities in, [S. Kovacs, Irrational centers, Pure Appl. Math. Q. 7(4) (2011) 1495–1515], and prove generalizations of standard facts about rational singularities. Moreover, by using a definition of non-rational locus, we introduce the notion of [Formula: see text] as a dual notion of well-known Serre’s notion of [Formula: see text], and prove a theorem about [Formula: see text]-birational morphisms.

Toric sheaves, stability and fibrations

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.

Almost nef regular foliations and Fujita's decomposition of reflexive sheaves

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. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with nef anti-canonical bundles.

Canonical subsheaves of torsionfree semistable sheaves

Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety defined over an algebraically closed field. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ also admits a filtration of subbundles for which each successive quotient is a stable vector bundle whose slope is $\mu (F)$. 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 $\mu (F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $\phi \colon Y \longrightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $\phi _*\colon \pi ^{\textrm {et}}_{1}(Y) \longrightarrow \pi ^{\textrm {et}}_{1}(X)$ to be surjective. The criterion in question is expressed in terms of the above mentioned unique maximal locally free subsheaf associated to the direct image $\phi _*{\mathcal O}_Y$.

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.

Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties byQ\mathbb {Q}-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.

Analytic tangent cones of admissible Hermitian Yang–Mills connections

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.

Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.

Higher rank sheaves on threefolds and functional equations

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.

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

On the product elliptic threefold X=C×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 Db(X) induced by the classical Fourier-Mukai transform on Db(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 modification by the transform of a codimension 2 sheaf.

Algebraic integrability of foliations with numerically trivial canonical bundle

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

We prove an analogue of the Donaldson–Uhlenbeck–Yau theorem for asymptotically cylindrical (ACyl) Kähler manifolds: If is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a μ–stable holomorphic vector bundle, then it admits an asymptotically translation-invariant projectively Hermitian Yang–Mills metric (with curvature in across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau with the geometric regularization scheme introduced by Bando and Siu.

A Horrocks' theorem for 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 E on Pn is greater or equal to nm−m. We completely describe m-tail reflexive sheaves on Pn of minimal rank and we construct huge families of m-tail reflexive sheaves of higher rank.

The limit of the Hermitian–Yang–Mills flow on reflexive sheaves

In this paper, we study the asymptotic behavior of the Hermitian–Yang–Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder–Narasimhan–Seshadri filtration, this answers a question by Bando and Siu.

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

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 given.

Vector bundles whose restriction to a linear section is Ulrich

An Ulrich sheaf on an n-dimensional projective variety \(X \subseteq \mathbb {P}^{N}\) is an initialized 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 to \(\mathbb {P}^2\) is a semistable instanton bundle on \(\mathbb {P}^{2}\), we show that some high Veronese embedding of X admits a \(\delta \)-Ulrich sheaf with a global section.

$$T$$ 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 K\"ahler manifold, $\omega$-stability implies $T$-stabilty. As a consequence of this we obtain the $T$-semistability of any reflexive Higgs sheaf with an admissible Hermitian-Yang-Mills metric. Finally, we prove that $T$-stability implies $\omega$-stability if, as in the classical case, some additional requirements on the base manifold are assumed. In that case, we obtain the existence of admissible Hermitian-Yang-Mills metrics on any $T$-stable reflexive sheaf.

An effective bound for reflexive sheaves on canonically trivial 3-folds

We give effective bounds for the third Chern class of a semistable rank 2 reflexive sheaf on a canonically trivial threefold.

Reflexive and Spanned Sheaves on $${\mathbb{P}^3}$$ P 3

We investigate the reflexive sheaves on $${\mathbb{P}^3}$$ 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 $${\mathbb{P}^3}$$ spanned in codimension 2 with c 1 = 2 is spanned.

Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit

Abstract In the following article we study the limiting properties of the Yang–Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A 0 defining the holomorphic structure, then the Yang–Mills flow with initial condition A 0 converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E ∞, which is isomorphic to the associated graded object of the Harder–Narasimhan–Seshadri filtration of (E,A 0). Moreover, E ∞ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of one and two complex dimensions and proves the general case of a conjecture of Bando and Siu.