Classifying closed 2-orbifolds with Euler characteristics

A Complete Obstruction to the Existence of Nonvanishing Vector Fields on Almost-Complex, Closed, Cyclic Orbifolds

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 $H_{orb}^\ast (Q; \R)$.

The index of a vector field on an orbifold with boundary

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.

关于超Kähler Kummer 8轨形的分类

On Classifying HyperKähler Kummer 8-Orbifolds

本文构造并分类数百个基于E_8极大环面的Kummer型T^8/G超Kähler 8轨形，指出可消解情形对应已知两类流形，并推广至SU(4)与Spin(7)完整群新构造。

Almost complex structure, blowdowns and McKay correspondence in quasitoric orbifolds

We prove the existence of invariant almost complex structure on any positively omnioriented quasitoric orbifold. We construct blowdowns. We define Chen-Ruan cohomology ring for any omnioriented quasitoric orbifold. We prove that the Euler characteristic of this cohomology is preserved by a crepant blowdown. We prove that the Betti numbers are also preserved if dimension is less or equal to six. In particular, our work reveals a new form of McKay correspondence for orbifold toric varieties that are not Gorenstein. We illustrate with an example.

Characteristic Classes of Bad Orbifold Vector Bundles

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.

Scalar-flat Kähler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of $k$-dimensional quaternionic projective space by a $(k-1)$-torus.

A "Periodic Table" for Supersymmetric M-Theory Compactifications

We develop a systematic method for classifying supersymmetric orbifold compactifications of M-theory. By restricting our attention to abelian orbifolds with low order, in the special cases where elements do not include coordinate shifts, we construct a "periodic table" of such compactifications, organized according to the orbifolding group (order up to 12) and dimension (up to 7). An intriguing connection between supersymmetric orbifolds and G2-structures is explored.

可定义群胚的万有欧拉示性数与伯恩赛德群

The universal Euler characteristic and Burnside group for definable groupoids

本文为可定义群胚引入万有欧拉示性数，证明加性乘性不变量均可由环同态统一表示，并构造推广经典伯恩赛德环的伯恩赛德群。

关于曲面上复双曲圆盘丛的离散不变量之间的相互作用

On the interplay between discrete invariants of complex hyperbolic disc bundles over surfaces

本文研究复双曲圆盘轨道丛的三个离散不变量（欧拉示性数χ、欧拉数e和Toledo不变量τ）之间的关系，提出并部分证明了$-3|τ| = 2e+2χ$这一猜想。

具有$T^3$对称性的霍普夫猜想

The Hopf conjecture with $T^3$-symmetry

该研究针对具有$T^3$对称性的正截面曲率闭偶维流形及轨形，证明其具有正欧拉示性数，推进了霍普夫猜想相关研究。

Positively curved Killing foliations via deformations

We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact we prov…