Search results for "Vector bundle"
showing 10 items of 29 documents
Some families of big and stable bundles on $K3$ surfaces and on their Hilbert schemes of points
2021
Here we investigate meaningful families of vector bundles on a very general polarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for any integer $k \geqslant 2$. In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers $n$ such that the twist of the tangent bundle of $X$ by the line bundle $nH$ is big and stable on~$X$; we then prove a similar result for a natural twist of the tangent bundle of $X^{[k]}$. Next, we prove global generation, bigness and stability results for tautological bundles on $X^{[k]}$ arising either from line bundles…
On the K-stability of complete intersections in polarized manifolds
2011
We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kaehler-Einstein metric and a strong evidence of K-stability of complete intersections on Grassmannians.
On a question of Mehta and Pauly
2013
In this short note we provide explicit examples in characteristic $p$ on certain smooth projective curves where for a given semistable vector bundle $\mathcal{E}$ the length of the Harder-Narasimhan filtration of $F^\ast \mathcal{E}$ is longer than $p$. This answers a question of Mehta and Pauly raised in arXiv:math/0607565.
Quillen superconnections and connections on supermanifolds
2013
Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;\Omega (M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.
The X-Ray Transform for Connections in Negative Curvature
2016
We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connect…
On the stability of flat complex vector bundles over parallelizable manifolds
2017
We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / \Gamma$, where $G$ is a complex connected Lie group and $\Gamma$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_\rho$ associated to any irreducible representation $\rho : \Gamma \rightarrow \text{GL}(r,{\mathbb C})$. More precisely, we prove that $E_{\rho}$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector bundle. All the rational Chern classes of $E$ vanish, in particular, its degree is zero. We deduce a stability result for flat holomorphic vector bundles $E_{\r…
Nonlinear Evolution Equations, Quasi-Solitons and their Experimental Manifestation
1990
We review the typical experimental facts which characterize quasisolitons in one-dimensional real systems, in connection with their modeling by nonlinear partial differential equations.We consider these nonlinear waves or excitations in two different domains of the real world : the macroworld and the microworld. In the macroworld we examine typical one-dimensional devices : the electrical networks, the Josephson transmission lines and the optical fibers, where the localized waves or pulses can be simply and coherently created, easily observed and manipulated on a macroscopic scale. In the microworld, we consider the magnetic chains and polymers, where the indirect experimental signatures of…
On the universal bundle for gravity
1991
Abstract We construct a supergravity type theory based on a superspace whose odd directions consist of a vector, together with a scalar representing a topological BRST shift symmetry. As such, the resulting theory is a theory of topological gravity. The gravitino is interpreted as a ghost field for this shift symmetry and plays the usual role of gauge field for local supersymmetry. Our construction is within the bundle of frames approach to superspace where covariant torsion constraints are analyzed, and we find that the resulting theory contains additional fields which are not present in existing theories of topological gravity. In particular, a minimal solution exists which contains a BRS…
Unirationality of Hurwitz spaces of coverings of degree <= 5
2011
Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.
$$O_2(\mathbb {C})$$O2(C)-Vector Bundles and Equivariant Real Circle Actions
2020
The main goal of this article is to give an expository overview of some new results on real circle actions on affine four-space and their relation to previous results on \(O_2(\mathbb {C})\)-equivariant vector bundles. In Moser-Jauslin (Infinite families of inequivalent real circle actions on affine four-space, 2019, [13]), we described infinite families of equivariant real circle actions on affine four-space. In the present note, we will describe how these examples were constructed, and some consequences of these results.