Finite Vector Bundles Research Articles

Essentially finite G-torsors

Let X be a smooth projective curve of genus g, defined over an algebraically closed field k, and let G be a connected reductive group over k. We say that a G-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups G. We give a Tannakian interpretation of such torsors, and we prove that all essentially finite G-torsors have torsion degree, and that the degree is 0 if X is an elliptic curve. We then study the density of the set of k-points of essentially finite G-torsors of degree 0, denoted MGef,0, inside MGss,0, the k-points of all semistable degree 0 G-torsors. We show that when g=1, MGef⊂MGss,0 is dense. When g>1 and when char(k)=0, we show that for any reductive group of semisimple rank 1, MGef,0⊂MGss,0 is not dense.

Étale triviality of finite vector bundles over compact complex manifolds

A vector bundle E over a projective variety M is called finite if it satisfies a nontrivial polynomial equation with nonnegative integral coefficients. Introducing finite bundles, Nori proved that E is finite if and only if the pullback of E to some finite étale covering of M is trivializable [14]. The definition of finite bundles extends naturally to holomorphic vector bundles over compact complex manifolds. We prove that a holomorphic vector bundle over a compact complex manifold M is finite if and only if the pullback of E to some finite étale covering of M is holomorphically trivializable. Therefore, E is finite if and only if it admits a flat holomorphic connection with finite monodromy. In [4] this result was proved under the extra assumption that the compact complex manifold M admits a Gauduchon astheno-Kähler metric.

Essentially Finite Vector Bundles on Normal Pseudo-Proper Algebraic Stacks

Let X be a normal, connected and projective variety over an algebraically closed field k . In [ I. Biswas and J. P. P. dos Santos , J. Inst. Math. Jussieu 10, No. 2, 225–234 (2011; Zbl 1214.14037)] and [ M. Antei and V. B. Mehta , Arch. Math. 97, No. 6, 523–527 (2011; Zbl 1236.14041)] it is proved that a vector bundle V on X is essentially finite if and only if it is trivialized by a proper surjective morphism f:Y\longrightarrow X . In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field k .

A note on certain Tannakian group schemes

In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.

A characterization of finite vector bundles on Gauduchon astheno-Kahler manifolds

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.

Essentially finite vector bundles on varieties with trivial tangent bundle

Let X X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle T X TX is trivial. Let F X : X ⟶ X F_X\, :\, X\,\longrightarrow \, X be the absolute Frobenius morphism of X X . We prove that for any n ≥ 1 n\, \geq \,1 , the n n –fold composition F X n F^n_X is a torsor over X X for a finite group–scheme that depends on n n . For any vector bundle E ⟶ X E\,\longrightarrow \, X , we show that the direct image ( F X n ) ∗ E (F^n_X)_*E is essentially finite (respectively, F F –trivial) if and only if E E is essentially finite (respectively, F F –trivial).

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 H 0 ( Y , O Y ) = k ; this Galois closure is a torsor p : X ˆ Y → X dominating f by an X -morphism λ : X ˆ Y → Y and universal for this property. Moreover, we show that λ : X ˆ Y → Y 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 H 0 ( Y , O Y ) = k , Y has a fundamental group scheme π 1 ( Y , y ) fitting in a short exact sequence with π 1 ( X , x ) .

Vector bundles trivialized by proper morphisms and the fundamental group scheme

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

A criterion for virtual global generation

Let X be a smooth projective curve defined over an algebraically closed field k, and let FX denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually glob- ally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F m X ) ∗ E ∼ Ea ⊕ E f for some m, where Ea is some ample vector bundle and E f is some finite vector bundle over X (either of Ea and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

Hodge Cohomology of Étale Nori Finite Vector Bundles

Etale Nori finite vector bundles are the bundles defined by representations of a finite etale group scheme in the usual way. In this article, we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case the result of Pink-Roessler [13, Proposition 3.5].

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle E E over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f f and g g , with nonnegative integral coefficients, such that the vector bundle f ( E ) f(E) is isomorphic to g ( E ) g(E) . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.