M. Nori proved that on a projective smooth variety, a bundle is finite, (that is the ring it generates has dimension 0), if and only if it trivializes on a finite cover. In this note, we consider bundles of degree 0 on an elliptic curve. We prove that the dimension of the ring generated by such a bundle is the same as the dimension of a torsor trivializing the bundle (and is 1).
Given a variety $X$, and a normal crossings divisor $D\subset X$, we relate, in the case of abelian monodromy, the following two: 1. existence of a $G$-torsor with prescribed ramification, and 2. existence of essentially finite parabolic vector bundles with prescribed weights.
We show some of the conjectures of Pappas and Rapoport concerning the moduli stack of $\mathcal{G}$-torsors on a curve C, where $\mathcal{G}$ is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of the stack of torsors in case $\mathcal{G}$ is simply connected.
The object of this paper is the notion of r-spin structure: a line bundle whose r-th power is isomorphic to the canonical bundle. Over the moduli functor M_g of smooth genus-$g$ curves, $r$-spin structures form a finite torsor under the group of r-torsion line bundles. Over the moduli functor Mbar_g of stable curves, r-spin structures form an 'etale stack, but the finiteness and the torsor structure are lost. In the present work, we show how this bad picture can be definitely improved simply by placing the problem in the category of Abramovich and Vistoli's twisted curves. First, we find that …
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 …
For curves of genus bigger than one we prove that Buium's first arithmetic jet spaces (p-jet spaces) admit the structure of a torsor under some line bundle. This result lifts a known constructions in characteristic p where the first $p$-jet space modulo p is a sheaf under the Frobenius tangent sheaf (parametrizing Frobenius linear derivations). In particular we show there is a natural family of lifts of the Frobenius tangent bundle so that the first $p$-jet space (and hence higher order lifts of the Frobenius) form torsor a under this bundle. The Cech cohomology classes associated to this tors…
Let ${\mathcal B}_g(r)$ be the moduli space of triples of the form $(X,\, K^{1/2}_X,\, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{\mathcal B}_g(r)$--torsor ${\mathcal H}_g(r)$ over ${\mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundle…
In the first part of our paper, we show that there exist non-isomorphic derived equivalent genus $1$ curves, and correspondingly there exist non-isomorphic moduli spaces of stable vector bundles on genus $1$ curves in general. Neither occurs over an algebraically closed field. We give necessary and sufficient conditions for two genus $1$ curves to be derived equivalent, and we go on to study when two principal homogeneous spaces for an abelian variety have equivalent derived categories. We apply our results to study twisted derived equivalences of the form $D^b(J,α)\simeq D^b(J,β)$, when $J$ i…
We study how forcing algebras give rise to ${\mathbb A}^1$-bundles and ${\mathbb A}^1$-torsors and how they are related to ${\mathbb A}^1$-patches. In particular we discuss the affineness of torsors and how algebraic properties of ${\mathbb A}^1$-patches can be deduced from this property. This is in part a survey article.
We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.
Let $W$ be a finite group and $T$ be an abelian group. Consider an extension $0 \ra T \ra N \ra W \ra 0$. For a smooth projective curve $X$, we give a precise description of the fiber of the quotient by $T$ map $q_T: \cM_X(N) \ra \cM_X(W)$ as a torsor over an abelian variety. We also prove a result on Mumford groups.
Let 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.
Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the base scheme Y to be affine and consider non-constant groups schemes G. There is a fundamental new feature in this case - in that the torsor may now be ramified over Y. The natural framework for handling the case of a non-constant group scheme over the affine base is provided by the quadratic theory of Hopf-algebras.
We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are -- in favorable cases -- again group schemes, called Néron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $\mathfrak g$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under Néron blowups of $G$.
We establish a relative version of the abstract "affine representability" theorem in ${\mathbb A}^1$--homotopy theory from Part I of this paper. We then prove some ${\mathbb A}^1$--invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass--Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in ${\mathbb A}^1$--homotopy theory.
Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single finite torsor. Let $G$ be any finite $k$-group scheme and $Y$ any $G$--torsor over $X$ pointed in $y\,\in\, Y(k)$; we define over $Y$, which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois $k$-group schemes $π^S(Y,\,y)$ and $π^N(Y,\,y)$, …
Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $λ:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $λ:\hat{X}_Y\to Y$ 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 finit…
Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $Ω_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (α^*L)$, where $α$ is the addition map on $A\times A$, and $p_1$ is the projection of $A\times A$ to the first factor. In \cite{BHR}, it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.
The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a no…
We prove an extension of the nonabelian Hodge theorem in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beilinson and Bernstein; on the other side, we endow them with appropriately defined twisted Higgs data. The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we…
We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with ir…
We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.
We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem-Noether theorem for certain algebra objects in the derived category, which allows us to give an explicit description of the boundary points in our compactified moduli problem.
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$ is generically trivial. We prove that in this case $\mathcal G$ is Zariski locally trivial. This result confirms the conjecture.
We establish comparison results between the Hasse-Witt invariants w_t(E) of a symmetric bundle E over a scheme and the invariants of one of its twists E_α. For general twists we describe the difference between w_t(E) and w_t(E_α) up to terms of degree 3. Next we consider a special kind of twist, which has been studied by A. Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. We show how to explicitly describe the twist for representations arising from very general tame actions. This involves the ``square root of the inverse different'' which Serre, Esna…
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…
Raynaud and Gruson showed that there is a reasonable algebro-geometric notion of family of discrete (infinite-dimensional) vector spaces. The author introduces a notion of family of Tate spaces ("Tate" means "locally linearly compact") and claims that it is local. The definition takes in account that the K_{-1} of a ring is not necessarily zero. However, we prove that K_{-1} always vanishes after Nisnevich sheafification. As a discrete counterpart of families of Tate spaces, we introduce the notion of almost projective module. We discuss the notions of dimension torsor and determinant gerbe of…
We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an $\mathbb{A}^1$-homotopy equivalent model, built from $\mathbb{A}^1$-fiber bundles not just algebraic line bundles, we construct an "algebraic uniformization": X is a quotient of affine space by a solvable group action. This provides a clean dictionary, using a single coordinate system, between the algebra and geometry of hypersurfaces: effective divisors are characterized via toric and invariant-theoretic techniques. In particular, the Cox ring is an …
We investigate the elliptic analogs of multi-indexed polylogarithms that appear in the theory of the fundamental group of the projective line minus three points as sections of a universal nilpotent bundle with regular singular connection. We use an analytic uniformisation to derive the fundamental nilpotent De Rham torsor of a single elliptic curve in terms of a double Jacobi form introduced by Kronecker. We then extend this result to any smooth family, relatively to the base, i.e., to the moduli stack $M_{1,2}$ over $M_{1,1}$. Everything relies on explicit formulas that turn out to be algebra…
Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes
CNRS, Université de Poitiers, LMA（法国国家科学研究中心，普瓦捷大学，数学实验室）；Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学，法国国家科学研究中心，IMB UMR 5584）；Department of Mathematics “F. Enriques”, University of Milan（米兰大学，“F.恩里克斯”数学系）
