Space Of Bundle Research Articles

Stable vector bundles on the families of curves

We offer a new approach to proving the Chen–Donaldson–Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a family of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.

H-instanton bundles on three-dimensional polarized projective varieties

We propose a notion of instanton bundle (called H-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor H, that naturally generalizes the ones on P3 and on the flag threefold F(0,1,2). We briefly discuss the cases of Veronese and Fano threefolds. Then we deal with H-instanton bundles E on three-dimensional rational normal scrolls S(a0,a1,a2). We give a monadic description of H-instanton bundles and we prove the existence of μ-stable H-instanton bundles on S(a0,a1,a2) for any admissible charge k=c2(E)H. Then we deal in more detail with S(a,a,b) and S(a0,a1,a2) with a0+a1>a2 and even degree. Finally we describe a nice component of the moduli space of μ-stable bundles whose points represent H-instantons.

Shatz and Bialynicki-Birula stratifications of the moduli space of Higgs bundles

Let $X$ be a compact Riemann surface of genus $g\geq 2$. In this paper we study how two memorable stratifications of the moduli space of Higgs bundles of rank $4$ over $X$, Shatz stratification and Bialynicki-Birula stratification, relate to each other and provide dimensions for the Harder-Narasimhan type of a semistable rank 4 Higgs bundle over $X$. In particular, we prove that the two stratifications do not coincide. Thus, we extend to rank 4 the work of Hausel [7], who proved that both stratifications coincide in rank 2, and of Gothen and Zúñiga-Rojas [6], who proved that such a thing no longer occurred in rank 3.

Automorphism group of the moduli space of parabolic vector bundles with fixed degree

We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with fixed degree. We prove that every 3-birational map can be described as a composition of tensorization by a fixed line bundle, Hecke transformations, dualization, taking pullback by an isomorphism between the curves and the action of the group of automorphisms of the Jacobian variety of the curve which fix the r-torsion. In particular, we prove a Torelli type theorem, stating that the 3-birational class of the moduli space determines the isomorphism class of the curve.

Positive Ricci curvature on fiber bundles with compact structure group

Abstract This paper presents a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it generalizes and puts in a unified framework the results of Nash [12] and Poor [15]. With the intention of disseminating this result,we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.

Numerical Simulation of Flow-Induced Vibration of Three-Dimensional Elastic Heat Exchanger Tube Bundle Based on Fluid-Structure Coupling

The reliability of the heat exchanger tube bundle not only affects the economic efficiency of production but also relates to the normal development of production safety and health. To study the flow-induced vibration of tube bundles, a three-dimensional finite element model of heat exchange tubes and watersheds inside and outside the tubes was established to explore the flow-induced vibration characteristics of tube bundles and analyze the natural frequencies of single-span and multispan heat exchange tubes. Considering the randomness of the effective support between the tube bundle and the support plate of the heat exchanger, the natural frequency and vibration mode of the four-span tube with failure of the tube bundle support are analyzed. On this basis, the vibration caused by the two-way coupling flow between tube and tube outflow is calculated. Finally, the flow-induced vibration characteristics of the five-tube bundle with two different pitch-diameter ratios are analyzed. The calculation results show that the error between the calculated natural frequencies and the theoretical values is less than 3%, and within the allowable error range, the natural frequencies of the same order decrease with the increase of the number of support failures. The vibration frequencies of single-span and multispan tube bundles are consistent with the lift and drag frequencies, the vibration displacement curves show typical Strouhal modes, and the amplitude increases with the increase of fluid velocity. Vibration displacement curves of symmetrical spans of multispan tube bundles are similar in shape and amplitude. With the increase of tube bundle spacing, the vibration characteristics become more obvious.

Para-K\xe4hler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

Construction of the moduli space of Higgs bundles using analytic methods

Construction of the moduli space of Higgs bundles using analytic methods

Minimal rational curves on the moduli spaces of symplectic and orthogonal bundles

Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.

Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles

We study the existence of weighted extremal K\ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\ahler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal K\ahler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally K\ahler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For each admissible K\ahler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(-1,1)$, then there exists a weighted extremal K\ahler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no K\ahler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of K\ahler metrics of constant weighted scalar curvature in the rational admissible K\ahler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

Even and odd instanton bundles on Fano threefolds

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$ and $\mathrm{Pic}(X)$ is cyclic. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.

Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve

<abstract><p>We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $ 3 $ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.</p></abstract>

Quantizations of local surfaces and rebel instantons

We construct explicit deformation quantizations of the noncompact complex surfaces Z_k:=\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k)) and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.

On the completeness of total spaces of horizontally conformal submersions

Abstract In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

Equivariant fundamental classes in RO(C2)–graded cohomology with ℤ∕2–coefficients

Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this paper, we develop a theory of fundamental classes for equivariant submanifolds in $RO(C_2)$-graded cohomology in constant $\mathbb{Z}/2$ coefficients. We show the cohomology of any $C_2$-surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.

Automorphism group of the moduli space of parabolic bundles over a curve

We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking the dual, and Hecke transforms (using the filtrations given by the parabolic structure). A Torelli theorem for parabolic bundles with arbitrary rank and generic weights is also obtained. These results are extended to the classification of birational equivalences which are defined over “big” open subsets (3-birational maps, i.e. birational maps giving an isomorphism between open subsets with complement of codimension at least 3).Finally, an analysis of the stability chambers for the parabolic weights is performed in order to determine precisely when two moduli spaces of parabolic vector bundles with different parameters (curve, rank, determinant and weights) can be isomorphic.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

Monodromy of rank 2 parabolic Hitchin systems

We study the monodromy of the Hitchin fibration for moduli spaces of parabolic G-Higgs bundles in the cases when G=SL(2,R), GL(2,R) and PGL(2,R). A calculation of the orbits of the monodromy with Z2-coefficients provides an exact count of the components of the moduli spaces for these groups.

INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS

AbstractFor $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

An elementary discussion of the representation and geometric invariant theory of equioriented quivers of type D with an application to quiver bundles

We first review a constructive proof of Gabriel's theorem on the representation finiteness of Dynkin quivers for the case of an equioriented quiver of type D. Then, we proceed to the analysis of polystable representations of such a quiver, give a direct proof of the characterisationof its semistable representations, and explain how we may recover a theorem of Abeasis and Koike from this characterisation. Finally, we illustrate how these results may be applied to the study of moduli spaces of quiver bundles.