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 \cite{No1}. 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 this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.
In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.
Étale Nori finite vector bundles are those bundles defined by representations of a finite étale group scheme in the usual way. In this note we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and of a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case a result of Pink-Roessler.
For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the si…
Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $ϕ_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{°ϕ_1...\degϕ_m}=0$.
We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $π_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve.
This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[π_1, π_1]$ of the étale fundamental group $π_1$ is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1 $ is abelian without $p$-power quotient. This answers positively a conject…
v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ of the étale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.
For a regular pair $(X,Y)$ of schemes of pure codimension 1 on which 2 is invertible, we consider quadric bundles on $X$ which are nondegenerate on $X-Y$, but are minimally degenerate on $Y$. We give a formula for the behaviour of the cohomological invariants (characteristic classes) of the nondegenerate quadric bundle on $X-Y$ under the Gysin boundary map to the etale cohomology of $Y$ with mod 2 coefficients. The results here are the algebro-geometric analogs of topological results for complex bundles proved earlier by Holla and Nitsure, continuing further the algebraization program which wa…
In this article we deduce criteria for the splitting and the triviality of vector bundles, by restricting them to partially ample divisors. This allows to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with a number of examples. For products of minuscule homogeneous varieties, our results allow to test the splitting of vector bundles by restricting them to products of Schubert 2-planes. The triviality criteria obtained inhere are particularly suited to Frobenius split varieties, whose splitting is defined by a section in the anti-canonical li…
Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $π^{alg}_{1}(X)$ is large, we show that the Seshadri constant $ε(p^{*}L)$ can be made arbitrarily large by passing to a finite étale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when $π_{1}(X)$ is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle $L$ on $X$ and a positive number $N…
We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical …
For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $\mathbb Q_p$, we construct a $p$-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous $K$-linear representations of the Tate module and a certain subcategory of the Higgs bundles on $A$. To do so, our central object of study is the category of vector bundles for the $v$-topology on the diamond associated to $A$. We prove that any pro-finite-étale $v$-vector bundle can be built from pro-finite-étale $v$-line bundles and unipotent $v$-bundles. To describe the latte…
We prove that the triviality of the Galois action on the suitably twisted odd-dimensional étale cohomogy group of a smooth projective varietiy with finite coefficients implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serre's Lectures on the Mordell--Weil theorem.
We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods include the use of complex integral forms and the recently developed theory of superholonomy.
We extend our previous theory of etale parallel transport to a larger class of slope zero vector bundles on p-adic curves. The new class is stable under pullback by ramified coverings. We also construct p-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.
Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f: C -> M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite etale quotient of an abelian variety answering a conjecture of I. Biswas.
A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation. A field theory satisfies spin-statistics if it is both spin and super, and $360^\circ$-rotation of the spin structure agrees with the operation of flipping the signs of all fermions. We set up a framework in which these two notions are precisely analogous. In this framework, field theories are defined over $\mathrm{Vect}_{\mathbb R}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of …
Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the étale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of étale …
Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.
We use Scholze's framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb{C}_p$ this enables us to prove a category equivalence between $\mathbb{C}_p$-local systems with integral…
Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^*E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.
In Comm. Math. Physics 118 (1988), 651-701, A. Beilinson and V. Schechtman define on the total space of a smooth family of curves a so-called trace complex associated to a vector bundle, the 0-th relative cohomology of which is the Atiyah algebra of the determinant bundle. Their proof reduces the general case to the acyclic one. In particular, one needs a comparison of the image of the trace complex for a bundle, and its twist by an étale multisection. We analyse this and correct a point in the original proof.
On the number of Frobenius periodic vector bundles on elliptic curves
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（同济大学智能计算与应用重点实验室数学科学学院）
Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $…
We conjecture the equality of the numerical and Kodaira dimensions $ν_1^*(X)$ and $κ_1^*(X)$ for the cotangent bundle of compact Kähler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $ν_1^*(X)=κ_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite étale cover of $X$. The proof rests on the Beauville-Bog…
We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental group approximates the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statement about vector bundles which are true in characteristic zero are proved. Some of these are positive characteristic avatar of notions which appear in the second aut…
For conic bundles on a smooth variety (over a field of characteristic $\ne 2$) which degenerate into pairs of distinct lines over geometric points of a smooth divisor, we prove a theorem which relates the Brauer class of the non-degenerate conic on the complement of the divisor to the covering class (Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate conic, via the Gysin homomorphism in etale cohomology. This theorem is the algebro-geometric analogue of a topological result proved earlier.
We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over $\mathbb{Q}$.
We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.
A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets. The conjecture is equivalent to the claim that the relevant admissible m…
Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups $f_* : π_1^{\rm et}(X)\rightarrowπ_1^{\rm et}(Y)$ is surjective. Fix a polarization on $Y$ and equip $X$ with the pullback, by $f$, of this polarization on $Y$. Given a stable vector bundle $E$ on $X$, we prove that there is a vector bundle $W$ on $Y$ with $f^*W$ isomorphic to $E$ if and only if the direct image $f_*E$ contains a stable vector bundle $F$ such that $$ \…
Let $G$ be a reductive algebraic group over a perfect field $k$ and $\cG$ a $G$-bundle over a scheme $X/k$. The main aim of this article is to study the motive associated with $\cG$, inside the Veovodsky Motivic categories. We consider the case that $\charakt k=0$ (resp. $\charakt k\geq 0$), the motive associated to $X$ is geometrically mixed Tate (resp. geometrically cellular) and $\cG$ is locally trivial for the Zariski (resp. étale) topology on $X$ and show that the motive of $\cG$ is geometrically mixed Tate. Moreover for a general $X$ we construct a nested filtration on the motive associa…
Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined étale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in $P^n$.
Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double étale …
Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is isotropic. We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H^1(A\times_k R,G)-> H^1(A\times_ k K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local ring of several points on a k-smooth scheme, then it suffices to require that k is infinite and keep the same assumption concerning G.
In this note we define Chern-Simons classes of a superconnection $D+L$ on a complex supervector bundle $E$ such that $D$ is flat and preserves the grading, and $L$ is an odd endomorphism of $E$ on a supermanifold. As an application we obtain a definition of Chern-Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. We extend Reznikov's theorem on triviality of these classes when the manifold is a compact Kähler manifold or a smooth complex quasi--projective variety, in degrees > 1.
The Venereau polynomials v-n:=y+x^n(xz+y(yu+z^2)), n>= 1, on A4 have all fibers isomorphic to the affine space A3. Moreover, for all n>= 1 the map (v-n, x) : A4 -> A2 yields a flat family of affine planes over A2. In the present note we show that over the punctured plane A2\0, this family is a fiber bundle. This bundle is trivial if and only if v-n is a variable of the ring C[x][y,z,u] over C[x]. It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Venereau established that v-n is indeed a variable of C[x][y,z,u] over C[x] for n>= 3. In this note we give…
We show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding …
The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibil…
We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,φ)$, where $C$ is a compact connected Riemann surface and $φ: C\to X$ a holomorphic map, such that the pull back $φ^*TX$ is not semistable. 2. The variety $X$ admits an étale covering by an abelian variety. 3. The dimension $\dim X \leq 1$. We conjecture that all complex projective manifolds are of the above type, and prove that the following classes are among those that are of the above type. i) All $X$ with a finite fundamental group. ii) All $X$ such that t…
Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975)), Gieseker conjectured that every stratified bundle (i.e. every O-coherent D-module) on X is trivial, if and only if the étale fundamental group of X is trivial. This was proven by Esnault-Mehta in "Simply connected projective manifolds in characteristic p> 0 have no nontrivial stratified bundles" (Invent. Math. 181 (2010)). Building on the classical situa…
We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the non-affine case this is false: we show…
This work establishes a structure theorem for compact Kähler manifolds with semipositive anticanonical bundle. Up to finite étale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the non existence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient for this is the characterization of unirul…
Given an etale quotient q:X->Y of smooth projective varieties we relate the simple Seshadri constant of a line bundle M on Y with the multiple Seshadri constant of q*M in the points of the fiber. We apply this method to compute the Seshadri constant of polarized abelian surfaces in the points of a finite subgroup.
