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Bohui Chen

Publications and source records attributed to Bohui Chen.

At least 19 recordsLinked to original sources
arXiv2026

Equivariant basic cohomology of Lie groupoids

This paper develops equivariant basic cohomology for Lie groupoids equipped with weak actions of Lie groups. The weak action is encoded by a Kan fibration over the classifying groupoid, and the basic complex of the fiber is shown to carry the structure needed for Weil and Cartan models. The construction is compared with Bott--Shulman--Stasheff cohomology, where the equivariant theory is obtained from the quotient groupoid. For orbifolds, basic forms are interpreted as orbifold differential forms, and the resulting equivariant basic cohomology is used to formulate differential-geometric constructions such as equivariant integration and localization. The paper also studies the induced weak action on the inertia groupoid and uses it to define an equivariant refinement of the Chen--Ruan cohomology ring. In this framework the sectorwise equivariant cohomology, obstruction bundle, equivariant Euler class, Gysin maps and three-point functions are assembled into an equivariant Chen--Ruan product whenever the corresponding pairing is nondegenerate.

math.DG↗
arXiv2026

Automorphisms of Lie groupoids and symplectic reduction on orbifolds

In this paper, the 2-group BAut(X) of automorphisms of a Lie groupoid X is constructed. Considering the 2-group G action on X, we explain the equivalence between 2-group homomorphisms from G to BAut(X) with Kan fibrations over G with fiber X. This justifies the notion of Kan fibration for 2-group actions on Lie groupoids. As an application, we formulate Hamiltonian actions of \'etale Lie 2-groups on orbifolds in terms of Kan fibrations and study the symplectic reductions. We show that, in general, the reduction is in fact a symplectic Lie 2-groupoid, and under certain isotropic free condition, the reduction is still an orbifold. Also the slice theorem of a group G action on Lie groupoids is proved.

math.DG↗
arXiv2025

Generalization Bounds for Equivariant Networks on Markov Data

Equivariant neural networks play a pivotal role in analyzing datasets with symmetry properties, particularly in complex data structures. However, integrating equivariance with Markov properties presents notable challenges due to the inherent dependencies within such data. Previous research has primarily concentrated on establishing generalization bounds under the assumption of independently and identically distributed data, frequently neglecting the influence of Markov dependencies. In this study, we investigate the impact of Markov properties on generalization performance alongside the role of equivariance within this context. We begin by applying a new McDiarmid's inequality to derive a generalization bound for neural networks trained on Markov datasets, using Rademacher complexity as a central measure of model capacity. Subsequently, we utilize group theory to compute the covering number under equivariant constraints, enabling us to obtain an upper bound on the Rademacher complexity based on this covering number. This bound provides practical insights into selecting low-dimensional irreducible representations, enhancing generalization performance for fixed-width equivariant neural networks.

stat.ML↗
arXiv2023

Sum of Hamiltonian manifolds

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,ω^+,μ^+)$ and $(X^-,ω^-,μ^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$.

math.SG↗
arXiv2020

Orbifold Gromov--Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid $\sf S$ of a compact symplectic orbifold groupoid $(\mathsf X,ω)$. Let $\mathsf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$, and $\mathsf D_{\mathfrak a}=\mathsf{PN}_{\mathfrak a}$ be the exceptional divisor, where $\sf N$ is the normal bundle of $\sf S$ in $\sf X$. In this paper we show that the absolute orbifold Gromov--Witten theory of $\mathsf X_{\mathfrak a}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\sf X$, $\sf S$ and $\mathsf D_{\mathfrak a}$, the natural restriction homomorphism $H^*_{\text{CR}}({\sf X})\rightarrow H^*_{\text{CR}}({\sf S})$ and the first Chern class of the tautological line bundle over $\mathsf D_{\mathfrak a}$. To achieve this we first prove similar results for the relative orbifold Gromov--Witten theories of $(\mathsf X_{\mathfrak a}|\mathsf D_{\mathfrak a})$ and $(\mathsf N_{\mathfrak a}|\mathsf D_{\mathfrak a})$. As applications of these results, we prove an orbifold version of a conjecture of Maulik--Pandharipande on the Gromov--Witten theory of blowups along complete intersections, a conjecture on the Gromov--Witten theory of root constructions and a conjecture on Leray--Hirsch result for orbifold Gromov--Witten theory of Tseng--You.

math.SG↗
arXiv2020

Double ramification cycles with orbifold targets

In this paper, we consider double ramification cycles with orbifold targets. An explicit formula for double ramification cycles with orbifold targets, which is parallel to and generalizes the one known for the smooth case, is provided. Some applications for orbifold Gromov--Witten theory are also included.

math.AG↗
arXiv2020

On fibrations of Lie groupoids

As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber $\sf A$, i.e., \[ 1\rightarrow {\sf A} \rightarrow {\sf G} \rightarrow {\sf K}\rightarrow 1 \] where $\sf A,\sf G$ and $\sf K$ are Lie groupoids. Similar to the theory of group extensions, we show that the existence of locally topological product Lie groupoid fibrations with fiber $\sf A$ over $\sf K$ is obstructed by a groupoid cohomology of $H^3_{\bar Λ}({\sf K},Z_{\sf A})$, and these locally topological product Lie groupoid fibrations are classified by $H^2_{\bar Λ}({\sf K},Z_{\sf A})$ once exists. Here $Z_{\sf A}$ is the center of $\sf A$. This generalizes the theory of group extensions, of gerbes over manifolds/groupoids and etc.

math.DG↗
arXiv2019

The groupoid structure of groupoid morphisms

In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are equivalent. Then we study the automorphism groupoid of a groupoid.

math.CT↗
arXiv2019

Weighted blowup correspondence of orbifold Gromov--Witten invariants and applications

Let $\sf X$ be a symplectic orbifold groupoid with $\sf S$ being a symplectic sub-orbifold groupoid, and $\sf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$ with $\sf Z$ being the corresponding exceptional divisor. We show that there is a weighted blowup correspondence between some certain absolute orbifold Gromov--Witten invariants of $\sf X$ relative to $\sf S$ and some certain relative orbifold Gromov--Witten invariants of the pair $(\sf X_{\mathfrak a}|Z)$. As an application, we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weighted blowup invariant.

math.SG↗
arXiv2018

Equivariant commutative stringy cohomology rings on almost complex manifolds

In this paper, motivated by Chen--Ruan's stringy orbifold theory on almost complex orbifolds, we construct a new cohomology ring $\mathscr H^\ast_{G,cs}(X)$ for an equivariant almost complex pair $(X,G)$, where $X$ is a compact connected almost complex manifold, $G$ is a connected compact Lie group which acts on $X$ and preserves the almost complex structure.

math.SG↗
arXiv2017

The Asymptotic Behavior of Finite Energy Symplectic Vortices with Admissible Metrics

Assume $(X, ω)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $μ$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics grow up at least cylindrically fast, without assuming the group action on the level set $μ^{-1}(0)$ is free. It generalizes the corresponding results by Ziltener [23, 24] under the free action assumption. The result of this paper is the first step in setting up the quotient morphism moduli space induced by the authors in [6]. Necessary preparations in understanding the structure of such moduli spaces are also introduced here. The quotient morphism constructed in [6] is a part of the project on the quantum Kirwan morphism by the authors (see [3, 4, 5]).

math.SG↗
arXiv2016

Prescribed Scaler Curvatures for Homogeneous Toric Bundles

In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

math.DG↗
arXiv2015

Extremal metrics on toric surfaces

In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.

math.DG↗
arXiv2015

Gluing principle for orbifold stratified spaces

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth structure on the stratified space. As an application, we provide an orbifold structure on the coarse moduli space $\bar{M}_{g, n}$ of stable genus $g$ curves with $n$-marked points. Using the gluing theory for $\bar{M}_{g, n} $ associated to horocycle structures, there is a natural orbifold gluing atlas on $\bar{M}_{g, n} $. We show this gluing atlas can be refined to provide a good orbifold gluing structure and hence a smooth orbifold structure on $\bar{M}_{g,n}$. This general gluing principle will be very useful in the study of the gluing theory for the compactified moduli spaces of stable pseudo-holomorphic curves in a symplectic manifold.

math.GT↗
arXiv2014

Virtual neighborhood technique for pseudo-holomorphic spheres

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, ω, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \C)$-orbit of a pseudo-holomorphic map representing a point in the moduli space $\cM_{0, 0}(X, A)$. In Sections 2 and 3 of this paper, we explain an integration theory on virtual orbifolds using proper étale groupoids and establish the virtual neighborhood technique for a general orbifold Fredholm system. When the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, ω, J)$ is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, ω, J)$ and show that the genus zero Gromov-Witten invariant is well-defined.

math.GT↗
arXiv2014

$L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends

Let $(X,ω)$ be a compact symplectic manifold with a Hamiltonian action of a compact Lie group $G$ and $μ: X\to \mathfrak g$ be its moment map. In this paper, we study the $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends. We studied a circle-valued action functional whose gradient flow equation corresponds to the symplectic vortex equations on a cylinder $S^1\times \mathbb R$. Assume that $0$ is a regular value of the moment map $μ$, we show that the functional is of Bott-Morse type and its critical points of the functional form twisted sectors of the symplectic reduction (the symplecitc orbifold $[μ^{-1}(0)/G]$). We show that any gradient flow lines approaches its limit point exponentially fast. Fredholm theory and compactness property are then established for the $L^2$-Moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends.

math.SG↗