Orbifolds and Finite Group Representations

Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groups

The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the authors defined a notion of the Grothendieck ring $K_0^{\rm fGr}(Var)$ of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from $K_0^{\rm fGr}(V…

Kobayashi-Ochiai关于一般型orbifold对的有限性定理

Kobayashi-Ochiai's finiteness theorem for orbifold pairs of general type

本文将Kobayashi-Ochiai关于固定簇到一般型簇支配映射集合的有限性定理，自然地推广至Campana的orbifold对情形。

The Chevalley-Weil Formula for Orbifold Curves

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.

轨形Hirzebruch-Riemann-Roch定理

Orbifold Hirzebruch-Riemann-Roch

该研究针对商Deligne-Mumford栈的轨形HRR定理展开，分析其与有限群表示论的联系，借助轨形Mukai配对得到了新的轨形HRR公式，丰富了代数几何领域的相关理论。

Orbifold Kähler Groups related to Mapping Class groups

We construct certain orbifold compactifications of the moduli stack of pointed stable curves over $\mathbb C$ and study their fundamental groups by means of their quantum representations. This enables to construct interesting Kähler groups and to settle most of the candidates for a counter-example to the Shafarevich conjecture on holomorphic convexity proposed in 1998 by Bogomolov and Katzarkov, using TQFT representations of the mapping class groups.

LG/CY Correspondence for Elliptic Orbifold Curves via Modularity

We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.

论通用与广义轨形欧拉示性数

On the universal and generalized orbifold Euler characteristics

该研究探讨通用轨形欧拉示性数与对应有限生成群$A$的广义$A$-欧拉示性数，证明形如$A'\ imes Z$（$Z$为整数群、$A'$有限）的所有$A$-欧拉示性数可确定通用轨形欧拉示性数。

Orbifold Milnor lattice and orbifold intersection form

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.

Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface over it, called a Dolgachev surface. Using the theory of Higgs bundles on these Dolgachev surfaces, we explicitly compute all values taken on by the orbifold Toledo invariant.

Localization and Gluing of Orbifold Amplitudes: The Gromov-Witten Orbifold Vertex

We define a formalism for computing open orbifold GW invariants of [C^3/G] where G is any finite abelian group. We prove that this formalism and a suitable gluing algorithm can be used to compute GW invariants in all genera of any toric CY orbifold of dimension 3. We conjecture a correspondence with the DT orbifold vertex of Bryan-Cadman-Young.

尼金利恩型 orbifold 的自同构

Automorphisms of Nikulin-type orbifolds

研究尼金利恩型 orbifold 的 monodromy 群，并通过其对第二整数上同调群的作用分类有限阶的 syplectic 自同构。

Logarithmic orbifold Euler numbers of surfaces with applications

We introduce orbifold Euler numbers for normal surfaces with Q-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov-Miyaoka-Yau type inequality. As a corollary we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type. Finally, we give some applications to singularities of plane curves.

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_θ/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebras $Λ$. As an application we finish the computation of $K_0$-groups of the crossed product algebras describing the above orbifolds initiated by Kumjian, Walters and Buck. Also, we describe a torsion pair in the category of $Λ$-modules, such that th…

Mok张量和无球因子的有界对称域的环面商

Mok tensors and Orbifold Quotients of Bounded symmetric domains without ball factors

本文研究了无球因子有界对称域的环面商，通过 Mok 曲率张量和 ample 分割条件，确定了有限光滑覆盖的存在性。

Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.

Orbifold conformal blocks and the stack of pointed G-covers

Starting with a vertex algebra $V$, a finite group $G$ of automorphisms of $V$, and a suitable collection of twisted $V$--modules, we construct (twisted) $D$--modules on the stack of pointed $G$--covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a $G$--equivariant version of the Virasoro uniformization theorem.