Intermediate Cohomology Research Articles

Rank two bundles on Pn$ {\mathbf {P}}^{n}$ with isolated cohomology

AbstractThe purpose of this paper is to study minimal monads associated to a rank two vector bundle on . In particular, we study situations where has for , except for one pair of values . We show that on , if , then must be decomposable. More generally, we show that for , there is no indecomposable bundle for which all intermediate cohomology modules except for are zero.

ACM line bundles on polarized K3 surfaces

An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this paper, we give a necessary and sufficient condition for a non-trivial line bundle $\mathcal{O}_X(D)$ on $X$ with $|D|=\emptyset$ and $D^2\geq L^2-6$ to be an ACM and initialized line bundle with respect to $L$, for a given K3 surface $X$ and a very ample line bundle $L$ on $X$.

Purely Log Terminal Threefolds with Non-Normal Centres in Characteristic Two

We show that many classical results of the minimal model programme do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is not normal, a three dimensional klt singularity which is not rational nor Cohen-Macaulay, and a klt Fano threefold with non-trivial intermediate cohomology.

Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section

In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding P2×P2⊆P8 and its general hyperplane sections.Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3.

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝ d - or ℤ d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ) d /Γ and are a step in the direction of the “top-degree cohomology” part.

ACM bundles on cubic surfaces

In this paper we prove that, for every r ≥ 2 , the moduli space M^s_X (r; c_1,c_2) of rank r stable vector bundles with Chern classes c_1 = rH and c_2 = \frac 1 2 (3r^2 − r) on a nonsingular cubic surface X ⊂ ℙ^3 contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on X .

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.

Vector bundles with no intermediate cohomology on Fano threefolds of type V22

We classify rank-2 vector bundles with no intermediate cohomology on the general prime Fano threefold of index 1 and genus 12. The structure of their moduli spaces is given by means of a monad-theoretic resolution in terms of exceptional bundles.

Rank 4 vector bundles on the quintic threefold

Abstract By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P 4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext 1 (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.

A few splitting theorems for rank two vector bundles on P^4

The paper investigates vanishing conditions on the intermediate cohomology of a normalized rank 2 vector bundle on which force to split or, at least, to be a non-stable bundle (with few possible exceptions). The results are applied to see when subcanonical surfaces in are forced to be complete intersections of two hypersurfaces, since subcanonical surfaces are zero loci of non-zero sections of rank vector bundles.

Liaison classes of modules

We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being perfect, Cohen–Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module liaison classes. Furthermore, we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one.

Monads on projective spaces

Let ℰ be a vector bundle on Pn. There is a strong relationship between ℰ and its intermediate cohomology modules. In the case where ℰ has low rank, we exploit this relationship to provide various splitting criteria for ℰ. In particular, we give a splitting criterion for ℰ in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P4 with a Buchsbaum second cohomology module.

Constructing schemes with prescribed cohomology in arbitrary codimension

In this paper we show how to construct varieties of arbitrary dimension in Pn with prescribed intermediate cohomology modules (up to shift). If the codimension is not too small then we use Kleiman's Bertini theorems to guarantee that the varieties are integral or even smooth.

Vector bundles on fano 3-folds without intermediate cohomology

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology.

Rank-two vector bundles on general quartic hypersurfaces in P4

In this paper all non-splitting rank-two vector bundles E without intermediate cohomology on a general quartic hypersurface X in P4 are classified. In particular, the existence of some curves on a general quartic hypersurface is proved.

Vector Bundles onG(1, 4) without Intermediate Cohomology

It is a famous result due to G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] that line bundles on a projective space are the only indecomposable vector bundles without intermediate cohomology. This fact generalizes to quadric and grassmannians if we add cohomological conditions. In this paper the case of G(1, 4) is studied completely, and a characterization-classification of vector bundles on it without intermediate cohomology is obtained.

Submodules of the Deficiency Modules and an Extension of Dubreil's Theorem

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S , when does IJ = I ∩ J ? More precisely, our setting will be in a polynomial ring k [ x 0 , …, x n ], and the ideals I and J will define subschemes of the projective space ℙ n k over k . In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J . By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space. Our main technique in this paper is a study of the deficiency modules of a subscheme V of ℙ n . These modules are important algebraic invariants of V , and reflect many of the properties of V , both geometric and algebraic. For instance, when V is equidimensional and dim V [ges ]1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [ 14, 11, 15, 7 ], although they do not in general completely determine the even liaison class, except in the case of curves in ℙ 3 [ 14 ]. On the algebraic side, at least for curves in ℙ 3 , the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [ 12 ]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space ℙ n . We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y , respectively, of ℙ n , we relate the quotient module ( I ∩ J )/ IJ to the cohomology of V , at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product. In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in ℙ n , and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V . The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product. Finally, in Section 3, we give an extension of a surprising result of Amasaki [ 1 ] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in ℙ 3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of ℙ n [ 2 ]). Easier proofs were subsequently given by Geramita and Migliore in [ 6 ], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of ℙ n whose intermediate cohomology vanishes, we are able to extend these considerations.

The intermediate cohomology modules of a rank two vector bundle on p3 are not split

It is shown that the intermediate cohomology modules of a rank two vector bundle on P3 cannot be written as the direct sum of two graded submodules with disjoint supports. As a consequence the same property holds for the Rao module of a subcanonical curve in P3.

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

Nonstable reflexive sheaves on 𝑃³

The spectrum is defined for nonstable rank two reflexive sheaves on P 3 {{\mathbf {P}}^3} and is used to establish vanishing theorems for intermediate cohomology in terms of the Chern classes and the order of nonstability. These results are shown to be best possible and the extremal cases are classified. Some applications to Cohen-Macaulay generically local complete intersection curves in P 3 {{\mathbf {P}}^3} are given.