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Shengda Hu

Publications and source records attributed to Shengda Hu.

18 recordsLinked to original sources
arXiv2022

Differential calculus for generalized geometry and geometric Lax flows

Employing a class of generalized connections, we describe certain differential complices $\left(\tilde \Omega^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic properties, where $\mathbb{T} M = T M \oplus T^*M$ is the generalized tangent bundle on $M$. A number of classical geometric notions are extended to $\mathbb{T} M$, such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when $\mathbb{T} M$ is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenb\"ock identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest.

math.DG↗
arXiv2019

Commuting Pairs, Generalized para-K\"ahler Geometry and Born Geometry

In this paper, we study the geometries given by commuting pairs of generalized endomorphisms ${\cal A} \in \text{End}(T\oplus T^*)$ with the property that their product defines a generalized metric. There are four types of such commuting pairs: generalized K\"ahler (GK), generalized para-K\"ahler (GpK), generalized chiral and generalized anti-K\"ahler geometries. We show that GpK geometry is equivalent to a pair of para-Hermitian structures and we derive the integrability conditions in terms of these. From the physics point of view, this is the geometry of $2D$ $(2,2)$ twisted supersymmetric sigma models. The generalized chiral structures are equivalent to a pair of tangent bundle product structures that also appear in physics applications of $2D$ sigma models. We show that the case when the two product structures anti-commute corresponds to Born geometry. Lastly, the generalized anti-K\"ahler structures are equivalent to a pair of anti-Hermitian structures (sometimes called Hermitian with Norden metric). The generalized chiral and anti-K\"ahler geometries do not have isotropic eigenbundles and therefore do not admit the usual description of integrability in terms of the Dorfman bracket. We therefore use an alternative definition of integrability in terms of the generalized Bismut connection of the corresponding metric, which for GK and GpK commuting pairs recovers the usual integrability conditions and can also be used to define the integrability of generalized chiral and anti-K\"ahler structures. In addition, it allows for a weakening of the integrability condition, which has various applications in physics.

hep-th↗
arXiv2018

The moduli space of real vector bundles of rank two over a real hyperelliptic curve

The Desale-Ramanan Theorem is an isomorphism between the moduli space of rank two vector bundles over complex hyperelliptic curve and the variety of linear subspaces in an intersection of two quadrics. We prove a real version of this theorem for the moduli space of real vector bundles over a real hyperelliptic curve. We then apply this result to study the topology of the moduli space, proving that it is relatively spin and identifying the diffeomorphism type for genus two curves. Our results lay the groundwork for future study of the quantum homology of these moduli spaces.

math.AG↗
arXiv2017

Suslov problem with the Klebsh-Tisserand potential

In this paper, we study a nonholonomic mechanical system, namely the Suslov problem with the Klebsh-Tisserand potential. We analyze the topology of the level sets defined by the integrals in two ways: using an explicit construction and as a consequence of the Poincar\'e-Hopf theorem. We describe the flow on such manifolds.

math-ph↗
arXiv2015

On generalized Kähler geometry on compact Lie groups

We present some fundamental facts about a class of generalized Kähler structures defined by invariant complex structures on compact Lie groups. The main computational tool is the BH-to-GK spectral sequences that relate the bi-Hermitian data to generalized geometry data. The relationship between generalized Hodge decomposition and generalized canonical bundles for generalized Kähler manifolds is also clarified.

math.DG↗
arXiv2014

A Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic bundles

In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that $α$-stability of a vector bundle, in this sense, can depend on the parameter $α$.

math.DG↗
arXiv2014

An example concerning Hamiltonian groups of self product, II

We describe the natural identification of $FH_*(X \times X, \triangle; ω\oplus -ω)$ with $FH_*(X, ω)$. Under this identification, we show that the extra elements in $Ham(X \times X, ω\oplus -ω)$ found in (Part I), for $X = (S^2 \times S^2, ω_0 \oplus λω_0)$ for $λ> 1$, do not define new invertible elements in $FH_*(X, ω)$.

math.SG↗
arXiv2014

An example concerning Hamiltonian groups of self product, I

We show that $(S^2\times S^2, ω_0 \oplus λω_0)$, with $λ> 1$, is an example of symplectic manifold $(X, ω)$ such that the $π_1 Ham(X \times X, ω\oplus -ω)$ contains extra elements than those from $π_1 Ham(X, ω) \times π_1 Ham(X, -ω)$.

math.SG↗
arXiv2012

Regularization of the Kepler problem on the Sphere

In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.

math.DS↗
arXiv2010

Homological Lagrangian monodromy

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.

math.SG↗
arXiv2009

A relative Seidel morphism and the Albers map

In this note, we introduce a relative (or Lagrangian) version of the Seidel homomorphism that assigns to each homotopy class of paths in Ham(M), starting at the identity and ending on the subgroup that preserves a given Lagrangian submanifold L, an element in the Floer homology of L. We show that these elements are related to the absolute Seidel elements by the Albers map. We also study for later use, the effect of reversing the signs of the symplectic structure as well as the orientations of the generators and of the operations on the Floer homologies.

math.SG↗
arXiv2008

Chern-Weil homomorphism in twisted equivariant cohomology

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of $G$. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite dimensional approximations of the Borel construction $M\times_G EG_k$, whenever the generalized complex manifold $M$ possesses a Hamiltonian $G$ action.

math.DG↗
arXiv2007

Extended manifolds and extended equivariant cohomology

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, à la Atiyah-Bott.

math.DG↗
arXiv2006

A deRham model for Chen-Ruan cohomology ring of abelian orbifolds

We present a deRham model for Chen-Ruan cohomology ring of abelian orbifolds. We introduce the notion of \emph{twist factors} so that formally the stringy cohomology ring can be defined without going through pseudo-holomorphic orbifold curves. Thus our model can be viewed as the classical description of Chen-Ruan cohomology for abelian orbifolds. The model simplifies computation of Chen-Ruan cohomology ring. Using our model, we give a version of wall crossing formula.

math.SG↗
arXiv2006

Hamiltonian symmetries and reduction in generalized geometry

A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian reduction in the category of generalized complex geometry is then constructed. The definitions and constructions are natural extensions of the corresponding ones in the symplectic geometry. We describe cutting in generalized complex geometry to show that it's a general phenomenon in generalized geometry that topology change is often accompanied by twisting (class) change.

math.DG↗
arXiv2006

Reduction and duality in generalized geometry

Extending our reduction construction in \cite{Hu} to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) $T$-duality with $H$-fluxes for principle bundles naturally arise as reductions of factorizable Poisson Lie group actions. In particular, the group may be non-abelian.

math.DG↗
arXiv2001

Semi-Stable Degeneration of Toric Varieties and Their Hypersurfaces

We provide a construction of examples of semistable degeneration via toric geometry. The applications include a higher dimensional generalization of classical degeneration of K3 surface into 4 rational components, an algebraic geometric version of decomposing K3 as the fiber sum of two E(1)'s as well as it's higher dimensional generalizations, and many other new examples.

math.AG↗