Rank 2 arithmetically Cohen‐Macaulay bundles on a general quintic surface

Rank 2 arithmetically Cohen‐Macaulay vector bundles on a general quintic hypersurface of the three‐dimensional projective space are classified (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stable vector bundles over cuspidal cubics

We give a complete classification of stable vector bundles over a cuspidal cubic and calculate their cohomologies. The technique of matrix problems is used, similar to [2, 3].     

On higher order geometry on anchored vector bundles

Some geometric objects of higher order concerning extensions, semi-sprays, connections and Lagrange metrics are constructed using an anchored vector bundle. The paper was partially supported by a 2005 MEC-CNCSIS grant.     

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.     

Rank 2 totally arithmetically Cohen–Macaulay vector bundles on Hirzebruch surfaces

Here we study the totally arithmetically Cohen–Macaulay rank 2 vector bundles on any Hirzebruch surface F _e . E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).     

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T ^*(J ^(r,s,q)(Y, R ^1,1)₀)^*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T ^*(J ^(r,s) (Y, R)₀)^*→R for any Y as above.     

ACM vector bundles on scrolls

Here we show that certain low rank ACM vector bundles on scrolls over smooth curves are iterated extensions of line bundles. Partially supported by MIUR and GNSAGA of INDAM (Italy)     

Globally generated vector bundles on the Segre threefold with Picard number two

We classify globally generated vector bundles on with small first Chern class, i.e. , . Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne–Serre correspondence.     

Rank two vector bundles with canonical determinant

Denote by B^k_2,K the locus of vector bundles of rank two that have canonical determinant and at least k sections. We show that for a generic curve of genus g,B^k_2,K is non‐empty and has a component of the expected dimension if g is sufficiently large. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with _c1≤3$c_1\le 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q$Q$ with _c1=3$c_1=3$ is an odd instanton up to twist.     

On the classification of complex vector bundles of stable rank

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle. Since deceased.     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Holomorphic vector bundles on non-algebraic tori of dimension 2

We describe the Chern classes of holomorphic vector bundles on non-algebraic complex torus of dimension 2.     

Holomorphic vector bundles on a neighborhood of non-Stein compact subsets of Stein spaces

LetK be a compact subset of a complex spaceX. Here we give conditions onX andK assuring the existence of a fundamental systemU of open neighborhoods, ofK such that for everyU∈U there is a holomorphic vector bundleE onU which is not holomorphically trivial.

---

Sunto SiaX uno spazio complesso eK∩X un compatto. In questo lavoro diamo condizioni suX eK che garantiscono l'esistenza di un sistema fondamentale di intorni apertiU diK inX tali che per ogniU∈U esiste un fibrato olomorfo non-triviale suU.

     

K-theory of blow-ups and vector bundles on the cone over a surface

TheK-theory of the derived categories as well as someK-theoretic invariants associated to the resolution of singularities are applied in order to compute theK ₀-groups of a variety with finitely many singular points. Explicit computations are given in order to determine theK ₀ and the relative CH* groups of the affine cone over a nonsingular surface of degreed in ^d+1 not contained in any hyperplane.     

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.     

P-Geometries of Rank 3

Under minor extra assumptions, we classify P-geometries of rank 3. Previously, a classification was known only for flag-transitive P-geometries.