The vortex equation on affine manifolds

Abstract

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, φ), consisting of a flat vector bundle E over M and a flat nonzero section φ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on Kähler manifolds to the situation of flat pairs on affine manifolds.

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

“…Research into the DUY theorem gained momentum in the 1980s, driven by numerous eminent mathematicians, as documented in works such as [1][2][3][4][5]. Over the past two decades, this theorem has continuously piqued the interest of numerous researchers, as evidenced by various publications ( [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references within). On 9 September 2021, Mochizuki was awarded the Breakthrough Prize in Mathematics for his remarkable contributions to the field of twistor D-modules.…”

The Limit of the Yang-Mills-Higgs Flow for twisted Higgs pairs

“…It was originally proved by Narasimhan-Seshadri ( [34]), Donaldson ([15,16]) and Uhlenbeck-Yau ( [42]) for holomorphic bundles. There are also many interesting and important generalized Donaldson-Uhlenbeck-Yau theorems (see [4,5,6,9,25,31,32,33,37,44,45] and references therein).…”

Analytically stable Higgs bundles on some non-Kähler manifolds

“…A Higgs bundle (E, ∂ E , θ) over M is a holomorphic bundle (E, ∂ E ) coupled with a Higgs field θ ∈ Ω 1,0 X (End(E)) such that ∂ E θ = 0 and θ ∧ θ = 0. Higgs bundles first emerged thirty years ago in Hitchin's ( [13]) reduction of self-dual equation on R 4 to Riemann surface and in Simpson's ( [29]) work on nonabelian Hodge theory, they have rich structures and play an important role in many different areas including gauge theory, Kähler and hyperkähler geometry, group representations and nonabelian Hodge theory. Letting H be a Hermitian metric on the bundle E, we consider the Hitchin-Simpson connection: D H,θ = D H + θ + θ * H , where θ * H is the adjoint of θ with respect to the metric H. The curvature of this connection is…”

“…If M is compact, it has been proved by Gauduchon ( [12]) that there exists a Gauduchon metric in the conformal class of every Hermitian metric ω. When the base Hermitian manifold is compact and Gauduchon, the Donaldson-Uhlenbeck-Yau theorem is also valid (see [3,4,8,18,20,21]). Inspired by Mochizuki's result ( [25]), we consider the case that the base manifold (M, ω) is non-compact Gauduchon and satisfies the following assumption.…”