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 that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with "as many" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily A^1-contractible.
The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group $\mathbb{G}_{m}\simeq\mathbb{C}^{\text{*}}$, using the language of polyhedral divisors developed by Altmann and Hausen as generalization of $\mathbb{Q}$-divisors.
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.
We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine A1-contractible 3-folds. Moreover, we give examples of stably A1-contractible smooth affine 4-folds containing a Brieskorn-Pham surface, and a family of smooth affine 3-folds with a higher dimensional A1-contractible total space.
We construct a smoothing of a particular normal crossing surface, which dual complex is a duncehat, thus obtaining an example of a surface degeneration with non-collapsible, contractible dual complex. The general fiber of this family has $h^{1,0} = h^{2,0} = 0$ and $h^{1,1}=9$, we conjecture that it is in the deformation class of the Barlow surface. This text is a complete rewrite of author's previous work [14].
In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$. More generally, we even prove that this contraction is possible without blowing up an assigned cluster of points on S. Using the theory of peeling, we are also able to give some information in the case D is not connected.
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）
We investigate the topological properties of Berkovich analytifications over hybrid fields, that is a field equipped with the maximum of its native norm and the trivial norm. We prove that the analytification of the affine line or of a smooth projective curve over a countable Archimedean hybrid field is contractible, and show that it can be non-contractible when the field is uncountable. Further, we prove that the analytification of affine space over a non-Archimedean hybrid field or over a discrete valuation ring is contractible. As an application, we show that the Berkovich affine line over …
Let $E_G$ be a $Γ$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $Γ$, where $G$ and $Γ$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $Γ$--space with a dense orbit, and $x_0 \in X$ is a $Γ$--fixed point. We show that if $Γ$ is reductive, then $E_G$ admits a $Γ$--equivariant isomorphism with the product principal $G$--bundle $X \times_ρ E_G(x_0)$, where $ρ\,:\, Γ\, \longrightarrow\, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over…
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
Let $X \subset \mathbb P^3$ be a smooth hypersurface of degree $6$ over complex numbers. In this paper, we give a characterization of initialized and ACM bundles of rank $1$ on $X$ with respect to the line bundle given by a smooth hyperplane section of $X$.
Given a vector bundle $E$ on a tree of smooth rational curves $C$, we give necessary and sufficient conditions for a vector bundle $E'$ on $\mathbb{P}^1$ to specialize to $E$ on $C$, generalizing the rank 2 case, due to Coskun.
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.
Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.
In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.
Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at gene…
Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.
We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^1$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^1$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^1$-contractible smooth schemes.
Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this article, we give criterion for existence of a logarithmic connection on $E_G$ singular along $D$.
In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.
Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\geq 2$. We prove that the Brauer group ${\rm Br}(\mathcal{M}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$. We also prove that ${\rm Br}(\mathcal{M}_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN…
In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. I…
