Carla Farsi
In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.
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In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.
Dirac Operators on Orbifold Resolutions: Uniform Elliptic Theory
本文通过将轨形视为锥纤维奇异空间并用渐近锥纤维化进行消解，发展了一致椭圆理论，从而得到轨形消解上狄拉克算子的Fredholm性质、一致有界性及指标分解公式。
We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.
We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed point formula for orbifolds. These cohomological formulae for the localized indices give rise to a class of refined topological invariants…
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics on 4-manifolds, and related results for metrics which are critical points of other natural Riemannian functionals on the space of metrics.
Bounding Shortest Closed Geodesics with Diameter on compact 2-dimensional Orbifolds Homeomorphic to $S^2$
将长度有界扫除方法推广到紧致二维轨形，建立了最短闭轨形测地线长度与直径之间的不等式。
Resolution of compact Einstein orbifolds in general dimensions
研究高维紧致爱因斯坦流形非塌缩序列的Gromov-Hausdorff极限为紧致爱因斯坦orbifold，证明其存在有限奇点，并给出超几何orbifold的负障碍物，简化了Ozuch四维工作中的技术难点。
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.
We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
Closed geodesics on compact orbifolds and on noncompact manifolds
该研究证明奇数维紧黎曼orbifold及特定非紧流形上存在非平凡闭测地线，并得出不存在该测地线的紧黎曼orbifold的相关限制。
We study the analytic torsion of odd-dimensional hyperbolic orbifolds $Γ\backslash \mathbb{H}^{2n+1}$, depending on a representation of $Γ$. Our main goal is to understand the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.
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…
Orbifold Bogomolov-Gieseker inequalities on compact Kähler varieties
本文研究了具klt奇点的紧Kähler簇上稳定反演层的Orbifold Bogomolov-Gieseker不等式，并给出了当等式成立时该层的特征刻画。
On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that n-connected n-orbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions we prove slightly better bounds. We obtain sharp results up to dimension 5.
For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal comp…
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As …
We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.
We study isometric circle actions on 7 dimensional positively curved Eschenburg spaces which are almost free, thus giving rise to orbifold fibrations of these spaces. This shows in particular that every known example of compact manifolds with positive sectional curvature is the total space of an orbifold fibration. We then study in detail the geometry of the singular locus and how to minimize it. As a consequence we obtain, for example, a six dimensional compact positively curved orbifold with only two singular points with orbifold groups Z_2.
We describe a class of compact $G_2$ orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can be mapped to F-theory on a Calabi-Yau fourfold, and we find that they are dual to an example studied by Donagi, Grassi and Witten. Finally, we give two different descriptions of our main example and the associative submanifolds as a twisted connected sum.
We prove every oriented compact cyclic $3$-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compact cyclic 3-orbifolds. We show every contact structure in an oriented compact cyclic $3$-orbifold contactified by our method is homotopic to an overtwisted structure with the overtwisted disc intersecting the singular locus of the orbifold. We pose Eliashberg's like characterizat…
We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.
In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensional orbifolds with zero-dimensional singular locus and orbifold fundamental group infinite torsion and of odd exponent.
We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.
We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian $\RR P^2$ in $T^*\RR P^2$ is weakly contractible.
On full asymptotics of analytic torsions for compact locally symmetric orbifolds
本文研究紧局部对称轨形上平坦向量丛序列的渐近 Ray-Singer 解析挠率，利用 Selberg 迹公式并借助几何局部化公式将椭圆轨道积分化为中心化子的恒等轨道积分。
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.
We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds. We find that different hypersurfaces within the same family of Calabi-Yau 4-orbifolds may result in different Spin(7)-manifolds.
We prove that Demailly's holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric criterion for a compact connected orbifold to be a Moishezon orbifolds, thus generalizing Siu's and Demailly's answers to the Grauert-Riemenschneider conjecture to the orbifold case.
Topology of singular foliations of closed 1-forms on orbifolds
研究闭合1形式诱导的奇异叶状结构的拓扑性质，确定叶的紧致性条件及共存机制，扩展Calabi在orbifold上的经典结果。
We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order to prove it, we also solve the orbifold Yamabe problem under a certain condition. We use these results to give some exact computations of the Yamabe invariant of compact orbifolds.
For a compact, smooth C^r orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for finite r \ge 1 and a Frechet manifold if r=infty. In each case, the local model is the separable Banach (Frechet) space of C^r (C^infty, resp.) orbisections of the tangent orbibundle.
We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on $p$-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean $\R^3$ and are shown to be isospectral on functions using dimension formulas for the eigensp…
We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.
Remarks on Kähler orbifolds of non-negative Ricci curvature
本文证明了凯勒轨道折面的非负曲率版本的 Kobayashi 定理，证明了非负曲率的紧凯勒轨道折面在满足一定条件时是单连通的。
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.
The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-Einstein geometry. We generalize the expansion for compact Kähler orbifolds with only finite isolated singularities and analyze the behavior of the expansion near the singularities.
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
On the Euler Characteristic of Odd-Dimensional Orbifolds and Their Boundaries
本文证明了紧致奇维orbifold的Euler示性数等于其边界Euler示性数的一半。
In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.
We classify representations of compact connected Lie groups whose induced action on the unit sphere has an orbit space isometric to a Riemannian orbifold.
A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $ω$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a one-to-one correspondence between symplectic toric orbifolds and convex rational simple polytopes with positive integers attached to each facet.
On Counting Flat Connections over $G_2$-Orbifolds
本文研究无挠 $G_2$-轨道空间上平坦丛的 $G_2$-瞬子模空间，通过引入和乐扰动证明其紧致与光滑性，定义了在 $C^0$-形变下不变的整数值不变量，并计算了 Joyce 构造中特定轨道空间的该不变量。
We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.
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.
We construct compact $G_2$-orbifolds with ADE-singularities that carry exactly one parallel spinor. Our examples are related to certain quotients of $\mathbb{C}^2\times T^3$ that have been investigated in arXiv:hep-th/9812205. We shortly discuss the physical applications of our examples.
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