The Hitchin–Kobayashi Correspondence for Quiver Bundles over Generalized Kähler Manifolds

Abstract

In this paper, we establish the Hitchin-Kobayashi correspondence for the I±-holomorphic quiver bundle E = (E, φ) over a compact generalized Kähler manifold (X, I+, I−, g, b) such that g is Gauduchon with respect to both I+ and I−, namely E is (α, σ, τ )-polystable if and only if E admits an (α, σ, τ )-Hermitian-Einstein metric.where ·, · denotes the natural inner product on T X ⊕ T * X. Definition 2.2. ([12]) A manifold X is called a generalized Kähler manifold if it carries the data (I + , I − , g, b), where• I ± are two complex structures on X,• g is a Riemannian metric on X, • b is a two-form on X, • I ± are parallel with respect to the connections ∇ ± = ∇ ± 1 2 g −1 H, respectively, where ∇ is the Levi-Civita connection of g and H = db.The generalized Calabi-Yau manifold is an important kind of generalized Kähler manifold.Definition 2.3. ([13])A generalized Calabi-Yau manifold is a generalized Kähler manifold (X, J 1 , J 2 ) such that both nowhere vanishing pure spinors ψ 1 , ψ 2 corresponding to J 1 , J 2 , respectively satisfy the following conditions

Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold

“…Leveraging the elliptic regularity, we can deduce that there exists a subsequence H v (t) converging to H v,∞ in the C ∞ loc -topology. Applying (24), we recognize that H v,∞ is the desired metric that satisfies the boundary condition. Finally, the uniqueness is guaranteed by the maximum principle and Proposition 3.…”

“…Later, Zhang [23] generalized their results to the compact almost-Hermitian manifold. Recently, the DUY theorem for quiver bundles has been extensively studied by Hu and Huang [24] to the compact generalized Kähler manifold. In summary, they showed that the stability of the holomorphic quiver bundle and the existence of Hermite-Yang-Mills metric over some compact Hermitian manifolds are equivalent.…”