Reflexive Sheaf Research Articles

Equivariant Chow cohomology of nonsimplicial toric varieties

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.

Vortex equation and reflexive sheaves

It is known that given a stable holomorphic pair $(E ,\phi)$, where $E$ is a holomorphic vector bundle on a compact K\ahler manifold $X$ and $\phi$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,\phi)$, where $\E$ is a reflexive sheaf on $X$.

Hyperholomorphic connections on coherent sheaves and stability

Abstract Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L 2-integrable. We show that such sheaves are polystable.

On possible Chern classes of stable bundles on Calabi–Yau threefolds

Supersymmetric heterotic string models, built from a Calabi–Yau threefold X endowed with a stable vector bundle V , usually lead to an anomaly mismatch between c 2 ( V ) and c 2 ( X ) ; this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In [M.R. Douglas, R. Reinbacher, S.-T. Yau, Branes, Bundles and Attractors: Bogomolov and Beyond, math.AG/0604597], a conjecture is stated which gives sufficient conditions on cohomology classes on X to be realized as the Chern classes of a stable reflexive sheaf V ; a weak version of this conjecture predicts the existence of such a V if c 2 ( V ) is of a certain form. In this note, we prove that on elliptically fibered X infinitely many cohomology classes c ∈ H 4 ( X , Z ) exist which are of this form and for each of them a stable S U ( n ) vector bundle with c = c 2 ( V ) exists.

Moduli Spaces of Reflexive Sheaves of Rank 2

AbstractLet ℱ be a coherent rank 2 sheaf on a scheme Y ⊂ ℙn of dimension at least two and let X ⊂ Y be the zero set of a section σ ∈ H0(ℱ). In this paper, we study the relationship between the functor that deforms the pair (ℱ, σ) and the two functors that deform F on Y, and X in Y, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme MY(P) of stable sheaves on a threefold Y at (ℱ), and the scheme structure at (X) of the Hilbert scheme of curves on Y become closely related. Using this relationship, we get criteria for the dimension and smoothness of MY(P) at (ℱ), without assuming Ext2(ℱ,ℱ) = 0. For reflexive sheaves on Y = ℙ3 whose deficiencymodule M = H*1 (ℱ) satisfies 0Ext2(M,M) = 0 (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H*0 (ℱ). Moreover, we show that every irreducible component of Mℙ3 (P) containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of Mℙ3 (P) that contains a reflexive stable sheaf with “small” deficiency module M.

Positivity of Chern classes for reflexive sheaves on P N

It is well known that the Chern classes c i of a rank n vector bundle on P N , generated by global sections, are non-negative if i ≤ 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 ≥ 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i ≤ 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 $${\mathcal {F}}$$ which is generated by $${H^0\mathcal {F}}$$ on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it.

Splitting criterion for reflexive sheaves

We give a criterion for a reflexive sheaf to split into a direct sum of line bundles.

Moduli of reflexive sheaves on smooth projective 3-folds

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.

An approximation theorem for sections of reflexive sheaves

We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map \(\) has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map \(\) has dense image, too.

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivisonof a region in k n (k algebraically closed) and r ∈ N, there is a reflexive sheaf K on P n , such that H 0 (K(d)) is essentially the space of piecewise polynomial functions on � , of degree at most d, which meet with order of smoothness r along common faces. In (9), Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle E on P n in terms of the top two Chern classes and the generic splitting type of E. We use a spectral sequence argument similar to that of (16) to characterize thosefor which K is actually a bundle (which is always the case for n = 2). In this situation we can obtain a formula for H 0 (K(d)) which involves only local data; the results of (9) cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P 2 and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.

The Degree of the Divisor of Jumping Rational Curves

For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $\P^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of the trivial bundle (for instance, if $E$ has balanced restriction on a generic line, or $r=2$ or $E$ is an exterior power of the tangent bundle). Assuming this, we give a formula for the 'virtual degree', interpreted enumeratively, of the locus of rational curves of degree $d$ on which the restriction of $E$ is not balanced, generalizing a classical formula due to Barth for the degree of the divisor of jumping lines of a semistable rank-2 bundle.

On curve sections of rank two reflexive sheaves

Let F be a normalized rank 2 reflexive sheaf on P3 with Chern classes c 1,c 2,c 3. Let α be the least integer such that 0≠H 0 F(α) and β be the smallest integer such that H 0 F(n) has sections whose zero scheme is a curve for all n≥ β . We show that if T 0 is the largest root of the cubic polynomial then β ≥ T 0-α-c 1-1. There are applications to the smallest degree of a surface containing a curves which are the zero schemes of sections of H 0 F(α).

The Minimal Free Resolution of the Migliore–Peterson Rings in the Case That the Reflexive Sheaf Has Even Rank

LetRbe a commutative noetherian ring and ϕ:F→Gbe a homomorphism of freeR-modules where rankF=fand rankG=g. Fix an elementbg+1∈⋀g+1Fand a generator ωG*for ⋀gG*. The module action of ⋀·F* on ⋀·Fproduces the elementb1=[(⋀gϕ*)(ωG*)](bg+1) inF. LetJdenote the image ofb1:F*→R. Assume that gradeJ=f−g, which is the largest grade possible and is attained in the generic case. The idealJmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf−greflexive sheaf. It may also be interpreted as the order ideal of an element in a second syzygy module of rankf−g. Also,Jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two. Migliore and Peterson have studied the idealJunm, which is the unmixed part ofJ. Under geometric hypotheses, they have shown thatR/Junmis a Cohen–Macaulay ring and they have resolved this ring. Furthermore, iff−gis odd, thenJunmis a Gorenstein ideal and is not equal toJ. On the other hand, iff−gis even, thenJunm=J. In the present paper, we produce the resolution ofR/Jby freeR-modules in the case thatf−gis even and (f−g−2)! is a unit inR. Our resolution is minimal whenever the data are local or homogeneous. Our resolution is built from the differential graded algebra (⋀·F*〈X1,…,Xg〉,d), where the restriction ofdto ⋀·F* is the Koszul complex associated tob1:F*→Rand the degree two divided power variablesX1,…,Xghave been adjointed in order to kill the cycles ϕ*(G*)⊆⋀1F*. The acyclicity lemma is used to prove exactness. Ifg=1, then the idealJis equal to the Huneke–Ulrich almost complete intersection idealI1(yX), whereyis a 1×fmatrix andXis anf×falternating matrix. The resolution of this ideal is already known.

Rank 2 stable vector bundles on Fano 3-folds of index 2

Let X be a Fano 3-fold of the first kind with index 2. In this paper, we characterize the chern classes of rank 2 stable vector bundles on X and we find a bound for the least twist of a rank 2 reflexive sheaf on X which has a global section.

On the Second Section of a Rank 2 Reflexive Sheaf onP3

On the Second Section of a Rank 2 Reflexive Sheaf onP3

Some Vanishing Properties of the Intermediate Cohomology of a Reflexive Sheaf on Pn

Some Vanishing Properties of the Intermediate Cohomology of a Reflexive Sheaf on Pn

Chern classes and cohomology for rank 2 reflexive sheaves on P3

The paper shows that, if F is a nonsplit rank 2 reflexive sheaf on P 3, then the knowledge of the numbers dn = h2(F(n)) - h\F(n)) gives an explicit algorithm to compute the Chern classes C\, Cι, c?> and the dimensions h°(F(n)), for all n (in particular the first integer a such that the sheaf F(a) has some nonzero section). If the sheaf is a vector bundle it is also proved that the knowledge of the numerical sequence {hι(F(n))} together with the first Chern class gives all the information as above. In some special cases, i.e. when hx(F(n)) Φ 0 for at most three values of n, an algorithm is also produced to compute the first Chern class from the sequence {hι(F(n))} . Vector bundles with natural cohomology are also discussed. It must be remarked that, if one knows not only the dimensions hι (F(n)), for all n, but also the whole structure of the Rao-module Q)Hι(F(n)), then the first Chern class C is uniquely determined (as it is shown in a paper by P. Rao).

A bound for the vanishing of the first cohomology group of a rank 2 stable reflexive sheaf over p3

For a reflexive sheaf E, we obtain an upper bound, in terms of the Chern classes of E , for the vanishing of the groups H 1(P 3 E(k)).We also show that, for every c 2 ≥ 2, there exists a normalized rank 2 stable reflexive sheaf over P 3 for which the bounds given are the best possible.

Unstability order for an unstable surface of a rank 2 reflexive sheaf

The goal of this paper is to give a sharp bound for the order of unstability of an unstable surface X of a rank 2 stable reflexive sheaf E onP3 in terms of the Chern classes of E and the degree of X; and a sharp bound for the order of unstability of an unstable surface X of a rank 2 nonstable reflexive sheaf E onP3in terms of the Chern classes of E, the order of nonstability of E and the degree of X.

Unstable planes for rank 2 reflexive sheaves

SynopsisThe purpose of this paper is to give a sharp bound for the order of instability of an unstable hyperplane of a rank 2 stable reflexive sheaf E on ℙn, in terras of the Chern classes of E; and a sharp boundfor the order of instability of an unstable hyperplane of a rank 2 nonstable reflexive sheaf E on ℙn, in terms of the Chern classes of E and the order of nonstability of E.