Torsion-free Sheaf Research Articles

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.

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.

Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O} _S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld-Mukai bundles to prove that $A$ is cut out by the global sections of a rank $1$ torsion-free sheaf $\mathcal{G} $ on $S$. Furthermore, we show that $c_1(\mathcal{G} )$ with one exception is adapted to $\mathcal{O} _S(C)$ and satisfies $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case.In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G} )_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O} _S(C)$.

Framed symplectic sheaves on surfaces

A framed symplectic sheaf on a smooth projective surface [Formula: see text] is a torsion-free sheaf [Formula: see text] together with a trivialization on a divisor [Formula: see text] and a morphism [Formula: see text] satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for [Formula: see text]. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

An algorithm for construction of commuting ordinary differential operators by geometric data

We give a description and program realisation of an algorithm which produces explicit examples of commuting ordinary differential operators with rational coefficients by geometric data consisting of a rational curve and a torsion free sheaf with some trivialisation.

Holomorphic bundles for higher dimensional gauge theory

Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the $\rm G_2-$manifolds obtained from asymptotically cylindrical Calabi-Yau $3-$folds studied by Kovalev and by Corti-Haskins-Nordstr\"om-Pacini et al. The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties $X$ with $\operatorname{Pic}{X}\simeq\mathbb{Z}^l$, a result which may be of independent interest. Finally, we apply monads to produce a prototypical model of the curvature blow-up phenomenon along a sequence of asymptotically stable bundles degenerating into a torsion-free sheaf.

Brill–Noether theory for curves on generic abelian surfaces

We completely describe the Brill-Noether theory for curves in the primitive linear system on generic abelian surfaces, in the following sense: given integers $d$ and $r$, consider the variety $V^r_d(|H|)$ parametrizing curves $C$ in the primitive linear system $|H|$ together with a torsion-free sheaf on $C$ of degree $d$ and $r+1$ global sections. We give a necessary and sufficient condition for this variety to be non-empty, and show that it is either a disjoint union of Grassmannians, or irreducible. Moreover, we show that, when non-empty, it is of expected dimension. This completes prior results by Knutsen, Lelli-Chiesa and Mongardi.

On the splitting of Lazarsfeld–Mukai bundles on K3 surfaces

Let X be a K3 surface, let C be a smooth curve on X, and let Z be a base point free pencil on C. Then, the Lazarsfeld–Mukai bundle EC,Z of rank 2 associated with C and Z is given by an extension of the torsion free sheaf JZ⊗OX(C) by OX, where JZ is the ideal sheaf of Z in X. We can see that if C is very ample as a divisor on X, EC,Z is an ACM bundle with respect to OX(C). In this paper, by using this fact, we will characterize a necessary condition for EC,Z to be given by an extension of two line bundles on X, by ACM line bundles with respect to OX(C).

Framed sheaves on projective stacks

Given a normal projective irreducible stack X over an algebraically closed field of characteristic zero we consider framed sheaves on X, i.e., pairs (E,ϕE), where E is a coherent sheaf on X and ϕE is a morphism from E to a fixed coherent sheaf F. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If X is a projective irreducible orbifold of dimension two and F a locally free sheaf on a smooth divisor D⊂X satisfying certain conditions, we consider (D,F)-framed sheaves, i.e., framed sheaves (E,ϕE) with E a torsion-free sheaf which is locally free in a neighbourhood of D, and ϕE|D an isomorphism. These pairs are μ-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of X. This implies the existence of a fine moduli space parameterizing isomorphism classes of (D,F)-framed sheaves on X with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on “stacky” compactifications of them. In particular, in a subsequent paper [20] these results are used to study gauge theories on ALE spaces of type Ak.

Gonality of non-Gorenstein curves of genus five

We establish sufficient conditions for some curves to be trigonal and derive from them that most of non-Gorenstein curves of genus five are so. Afterwards, we show that the gonality of such a curve ranges from 2 to 5. Gonality is understood within a broader context, i.e., the gd1 may possibly admit a base point and correspond to a torsion free sheaf of rank one instead of a line bundle. This study comes along with a thorough description of possible canonical models and kinds of singularities.

Stability of tautological bundles on the Hilbert scheme of two points on a surface

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

Stability of tautological bundles on the Hilbert scheme of two points on a surface

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

A GIT interpretation 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.

Instability of truncated symmetric powers of sheaves

Let X be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p>0. Let FX:X→X be the absolute Frobenius morphism, and E a torsion free sheaf on X. We give an upper bound of instability of truncated symmetric powers Tl(E)(0⩽l⩽rk(E)(p−1)) in terms of Lmax(ΩX1), I(ΩX1) and I(E) (Theorem 3.5). As an application, we obtain an upper bound of the instability of Frobenius direct image FX⁎(E) and some sufficient conditions of FX⁎(E) being slope semi-stable. In addition, we study the slope (semi)-stability of sheaves of locally exact (resp. closed) forms BXi (resp. ZXi).

A local ampleness criterion of torsion free sheaves

We give an ampleness criterion of a torsion free coherent sheaf at a given point in terms of a curvature positivity of a singular Hermitian metric.

SPLITTING TYPE, GLOBAL SECTIONS AND CHERN CLASSES FOR TORSION FREE SHEAVES ON PN

In this paper we compare a torsion free sheaf F on <TEX>$P^N$</TEX> and the free vector bundle <TEX>$\oplus^n_{i=1}O_{P^N}(b_i)$</TEX> 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 F. 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 <TEX>$c_i$</TEX>(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on <TEX>$P^N$</TEX>, whose splitting type has no gap (i.e., <TEX>$b_i{\geq}b_{i+1}{\geq}b_i-1$</TEX> 1 for every i = 1,<TEX>$\ldots$</TEX>,n-1), the following formula for the discriminant: <TEX>$$\Delta(F):=2_{nc_2}-(n-1)c^2_1\geq-\frac{1}{12}n^2(n^2-1)$$</TEX>. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes <TEX>$c_3$</TEX>(F(t)),<TEX>$\ldots$</TEX>,<TEX>$c_n$</TEX>(F(t)) for the dimension of the cohomology modules <TEX>$H^iF(t)$</TEX> and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on <TEX>$c_1(F)$</TEX>, <TEX>$c_2(F)$</TEX>, the splitting type of F and t.

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

Let K be a number field and OK the ring of integers of K. Let (E, h) be a hermitian finitely generated flat OK-module. For an OK-submodule F of E, let us denote by hF↪→E the submetric of F induced by h. It is well known that the set of all saturated OK-submodules F with deg(F, hF↪→E) ≥ c is finite for any real numbers c (for details, see [4, the proof of Proposition 3.5]). In this note, we would like to give its generalization on a projective arithmetic variety. Let X be a normal and projective arithmetic variety. Here we assume that X is an arithmetic surface to avoid several complicated technical definitions on a higher dimensional arithmetic variety. Let us fix a nef and big C∞-hermitian invertible sheaf H on X as a polarization of X. Then we have the following finiteness of saturated subsheaves with bounded arithmetic degree, which is also a generalization of a partial result [5, Corollary 2.2].

Theta-characteristics on singular curves

On a smooth curve a theta–characteristic is a line bundle L with square that is the canonical line bundle ω. The equivalent conditionHom(L, ω) ∼= L generalizes well to singular curves, as applications show. More precisely, a theta–characteristic is a torsion–free sheaf F of rank 1 with Hom(F , ω) ∼= F . If the curve has non ADE–singularities then there are infinitely many theta–characteristics. Therefore, theta–characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta–characteristics (i.e. F with h(C,F) ≡ 0 resp. h(C,F) ≡ 1 modulo 2) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local type. Mathematics Subject Classification (2000). 14H60, 14H42, 14H40. A theta–characteristic on a smooth projective curve C over C is a line bundle L whose square is the canonical bundle K. Let Θ be the set of all theta–characteristics. If g is the genus of the curve, then there are 2 theta– characteristics on C. Naturally, one would like to know the dimension of the linear systems |L|, L ∈ Θ. However, these dimensions depend not only on the genus alone, but also on the complex structure of the curve. Nevertheless, using the his theory of theta–functions Riemann proved that the dimension modulo 2 depends only on the genus. If Θ = {L ∈ Θ | h(L) is even.} Θ− = {L ∈ Θ | h(L) is odd.} are the even resp. odd theta–characteristics, then there are 2g−1(2g + 1) even and 2g−1(2g − 1) odd theta–characteristics. Atiyah gave another analytic proof of this, and Mumford the first algebraic one [A, M]. Mumford’s ideas were refined and extended by Harris to include singular curves into the theory — at least Gorenstein curves, for example plane curves [H]. On a singular curve a line bundle is defined to be a locally free sheaf of rank 1. Harris showed that the number of even and odd theta–characteristics can be computed in terms of the genus of the curve and certain discrete invariants of the singularaties. He also remarked that it would be desirable to be able to treat torsion–free sheaves of rank 1 alongside with the line bundles. This is what we want to do in this article.