For any complex vector bundle $E^k$ of rank $k$ over a manifold $M^m$ with Chern classes $c_i \in H^{2i}(M^m,\Z)$ and any non-negative integers $l_1, >..., l_k$ we show the existence of a positive number $N(k,m)$ and the existence of a complex vector bundle $\hat E^k$ over $M^m$ whose Chern classes are $ N(k,m) \cdot l_i\cdot c_i\in H^{2i} (M^m,\Z)$. We also discuss a version of this statement for holomorphic vector bundles over projective algebraic manifolds.
Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to $L_\infty$-algebroids to show that two homotopy equivalent bundles have the same $L_\infty$-algebroid structures and explore some consequence on the theory of shifted Poisson structures.
Let K be a a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.
A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (200…
The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally accosiated with differential operators, are found. These geometrical structures are used to solve the problems of local as well as global equivalency of differential operators.
We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen-Ruan orbifold cohomology of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern-Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. This paper proves the existence of suitably equivariant transition functions for such groupoids, generalising consequently the classification of principal bundles by means of their transition functions, to extensions of princip…
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Chern polynomial of the tensor product. In the second formula we express the total Chern class of the tensor product as resultant of two explicit polynomials with coefficient involving Chern classes of each vector bundles. This approach leads to determinantal formulas for the total …
This paper exhibits equivalences of 2-stacks between certain models of $\mathbb{S}^1$-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to $\mathbb{S}^1$-bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting o…
We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kähler manifolds. We provide a simple and independent proof of Reznikov's result and show that our characteristic classes of Higgs bundles and the characteristic classes of flat vector bundles are compatible via the nonabelian Hodge theorem.
We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden-Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupo…
We show that the set of the equivalence classes of multifoliate structures is in one-to-one correspondence with the set of equivalence classes of finite complete projective systems of vector space epimorphisms. After that we give the complete description of all product preserving bundle functors on the categories of multifibered and multifoliate manifolds.
Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that a…
We continue the study of the anti-Hermitian structures of general natural lift type on the tangent bundles. We get the conditions under which these structures are in the eight classes obtained by Ganchev and Borisov. We complete the characterization of the general natural anti-Kahlerian structures on the tangent bundles with necessary and sufficient conditions, then we present some results concerning the remaining classes.
Let $G$ be a Lie group and $G\to\Aut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[G\to\Aut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e.\ between principal $[G\to\Aut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)\to 1]$-group bundles over differentiable stacks, wher…
We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we …
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.
We prove that for a suitable class of metric measure spaces, the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of $L^2$-sections of the `Gromov-Hausdorff tangent bundle'. The class of spaces $({\rm X},{\sf d},{\mathfrak m})$ we consider are PI spaces that for every $\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\mathfrak m}$-a.e.\ ${\rm X}$ and corresponding $(1+\varepsilon)$-biLipschitz maps $φ_i:U_i\to{\mathbb R}^{k_i}$ such that $(φ_i)_*{\mathfrak m}\lower3pt\hbox{$|_{U_i}$}\ll\mathcal L^{k_i}$.…
A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we construct explicitly the parallel transport of a connection on a principal 2-bundle. Parallel transport along a path is a Morita equivalence between the fibres over the end points, and parallel transport along a surface is an intertwiner between Morita equivalences. We prove that our …
We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor called regression, and complements a similar equivalence for bundles and gerbes equipped with connections, derived previously in Part II of this series of papers. The two equivalences provide a complete loop space formulation of the geometry of gerbes; functorial, monoidal, natura…
We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group and the automorphisms group. The action of the elements in the connected component of the identity of the group generalizes known results in the literature. The action of the elements not connected with the identity is shown to be determined by a characteristic class by using di…
In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle $\mathcal{D}_{A}^{\textrm{tr}}$ instead of $Q_A$, which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, $Q_A$ is not really an object associated to the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).
We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canonical frames for these structures can be given in the two prolongation steps: the first step, based on our previous works, gives the canonical bundle of moving frames for the fixed submanifolds of flags on each fiber and the second step consists of the prolongation of the bundle ob…
The characteristic forms in the bundle of connections of a principal bundle P over M determine the characteristic classes of P for degree less or equal to the dimension of M, and differential forms on the space of connections for higher degree. The equivariant characteristic classes provide canonical equivariant extensions of this forms, and so cohomology classes in the quotient space of connections modulo gauge transformations. More generally, given a closed differential form in M and a characteristic form, we obtain a cohomology class in the space of connections modulo gauge transformations,…
We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles whose structure group is the group of based loops, $ΩG$. In particular, we give a formula for characteristic classes in odd dimensions for such bundles which are associated to characteristic classes for $G$-bundles in the same way that the string class is related to the first Po…
The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.
We review the caloron correspondence between $G$-bundles on $M \times S^1$ and $ΩG$-bundles on $M$, where $ΩG$ is the space of smooth loops in the compact Lie group $G$. We use the caloron correspondence to define characteristic classes for $ΩG$-bundles, called string classes, by transgression of characteristic classes of $G$-bundles. These generalise the string class of Killingback to higher dimensional cohomology.
A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any principal bundle yielding canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally the de Rham differential of the integrands…
We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,π_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $ε=ε(n,G)$ such that the tangent bundle of any manifold in the class $M_{…
Faculty of Physics, University of Warsaw（华沙大学物理系）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Institute of Mathematics, University of Warsaw（华沙大学数学研究所）
