We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $Γ$ an orbifold called the space of $Γ$-sectors of $Q$. The obstruction occurs as the Euler-Satake characteristics of the $Γ$-sectors for an appropriate choice of $Γ$; in the case that $Q$ is oriented, this obstruction is expressed as a cohomology class, the $Γ$-Euler-Satake class. We also acquire a complete obstruction in the case that $Q$ is compact with boundary and in the case that $Q$ is an open suborbifold of a closed orbifold.
We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlying topological spaces of $Q$ and its sectors, and in terms of the orbifold Euler class $e_{orb}(Q)$ in Chen-Ruan orbifold cohomology $H_{orb}^\ast (Q; \R)$.
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.
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function whe…
A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces.
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
In this paper, we establish an equality between the analytic torsion introduced by Dar\cite{MR876230} and the orbifold analytic torsion defined by Ma \cite{MR2140438} on a compact manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an honest vector bundle, although the metric on the flat bundle may not be flat.
We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for $1|1$-dimensional EFTs from the data of a gerbe with connection.
We present the theory of pseudodifferential operators acting on a vector orbibundle over an orbifold, construct the zeta function of an elliptic pseudodifferential operator and show the existence of a meromorphic extension to the complex plane with at most simple poles. We give formulas for generalized densities on the orbifold whose integrals compute the residues of the zeta function.
We prove residual formulas for vector fields defined on compact complex orbifolds with isolated singularities and give some applications of these on weighted projective spaces.
The goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem and the index of the vector field in the case of the Poincaré-Hopf Theorem) is related to Satake's orbifold Euler characteristic, a rational number which depends on the orbifold structure. For the second pair of generalizations, we use the Chen-Ruan orbifold cohomology to express the local data in a way…
We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $λ$ on $L$. Much of…
In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.
Let $Γ$ be a finite group acting linearly on a vector space $V$. We compute the Lie algebra cohomology of the Lie algebra of $Γ$-invariant formal vector fields on $V$. We use this computation to define characteristic classes for foliations on orbifolds.
SU(3)-structures on quotients of 3-Sasakian orbifolds
Institut Camille Jordan, Université Claude Bernard Lyon 1（里昂第一大学卡米耶·乔丹研究所）；Institut de Physique des Deux Infinis de Lyon, Université de Lyon, UCBL, UMR 5822, CNRS/IN2P3（里昂两无穷物理研究所，里昂大学，里昂第一大学）
We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle S^1 (here termed weighted hyperprojective spaces) are homotopy equivalent to weighted projective spaces, they are not S^1-representation homotopic. Indeed, we show that their Chen-Ruan cohomology rings (over the rationals) are distinct.
We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of certain orbifold fundamental groups.
This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general overview are advised to start there. We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of Spivak's 'derived manifolds' (s…
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a fam…
This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with bound…
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie a…
