A Lefschetz hyperplane theorem for Mori dream spaces

Let X be a smooth Mori dream space of dimension ?? 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron?CSeveri spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking ??Mori dream hypersurfaces?? of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.     

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

We describe the equivariant Chow ring of the wonderful compactification X of a symmetric space of minimal rank, via restriction to the associated toric variety Y. Also, we show that the restrictions to Y of the tangent bundle T _X and its logarithmic analogue S _X decompose into a direct sum of line bundles. This yields closed formulas for the equivariant Chern classes of T _X and S _X , and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Toric varieties with huge Grothendieck group

In each dimension n?3, there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.     

Real parabolic vector bundles over a real curve

We define real parabolic structures on real vector bundles over a real curve. Let (X, σ _X ) be a real curve, and let S???X be a non-empty finite subset of X such that σ _X (S)?=?S. Let N?≥?2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y?∈?Y, one has $\sigma_Y(gy) = g^{-1}\sigma_Y(y)$ . We established an equivalence between the category of real parabolic vector bundles on (X, σ _X ) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σ _Y ).     

On module bundles and topological K-theory

Abstract  We define and study the group K(X) of a topological space X as the Grothendieck group of the category of suitable module bundles over X instead of the Grothendieck group of the category of vector bundles over X and prove some of its properties. Keywords Topological K-Theory, Module bundles, Waelbroeck algebra Mathematics Subject Classification (2000) 19 Lxx, 19 Axx, (Secondary) 46 Hxx     

Degenerations of the moduli spaces of vector bundles on curves I

LetY be a smooth projective curve degenerating to a reducible curveX with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on Ydegenerates to a moduli space onX which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles onY with fixed determinant and arbitrary rank.     

Vector bundles on a nearly degenerate Riemann surface

We show that on a Riemann surface X close to a maximally degenerate complex curve, any semistable vector bundle of degree 0 is associated with a linear representation of the Schottky group ?? _X uniformizing X. Further, we study the relationship between the representation space of ?? _X and the moduli space of semistable bundles on X as a higher rank version of Abel?CJacobi??s theorem.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C¹-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.     

Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?M_n of continuous functions from X to $\text{M}$_n can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.     

Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

Holonomy group scheme of an integral curve

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that is normal, being the algebraic closure of k. We define a notion of strong semistability for vector bundles on Y. We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case , we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G‐bundle , we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y.     

Integral approximation of the characteristic function of an interval and the Jackson inequality in C($$ \mathbb{T} $$)

We give an analog of D.O. Orlov’s theorem on semiorthogonal decompositions of the derived category of projective bundles for the case of equivariant derived categories. Under the condition that the action of a finite group on the projectivization X of a vector bundle E is compatible with the twisted action of the group on the bundle E, we construct a semiorthogonal decomposition of the derived category of equivariant coherent sheaves on X into subcategories equivalent to the derived categories of twisted sheaves on the base scheme.     

A note on compact Kähler manifolds with nef tangent bundles

In this paper, we study the compact Kähler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kähler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(?K _X )=1 if and only if there exists a finite étale coverY→X such thatY??¹×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].     

Projectivized Rank Two Toric Vector Bundles are Mori Dream Spaces

We prove that the Cox ring of the projectivization P(?) of a rank two toric vector bundle ?, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(?) is a Mori dream space if the toric variety X is projective and simplicial.     

Grothendieck Groups of Bundles on Varieties over Finite Fields

Let X be an irreducible, projective variety over a finite field, and let A be a sheaf of rings on X. In this paper, we study Grothendieck groups of categories of vector bundles over certain types of ringed spaces (X,A).