Compact self-dual Hermitian surfaces

Abstract

Abstract. In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kähler.

Systems of symplectic forms on four-manifolds

“…Any subbundle E of 3 2 of rank three and maximal, in the sense that q |E is positive, determines a unique conformal structure on M such that E = 3 + [26]. The proof of this descends from the fact that q has signature (3,3) and that at any point x of M the set of maximal subspaces of 3 2 T x M is parametrised by…”

“…Conditions (i) and (ii) are equivalent on any compact, not necessarily Einstein, Hermitian complex surface [5,13]. Part (i) holds for compact self-dual Hermitian surfaces [3] as well.…”

Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

“…There has been many recent studies on the curvatures of special connections on Hermitian manifolds, see e.g. for the Chern connection ( [3], [4], [7], [16], [17], [20], [22]), the Levi-Civita connection ( [2], [14], [21]), the Strominger connection ( [8], [24], [27], [28]) and the Lichnerowicz connection ( [10], [13], [19]). There are also some recent work on the curvatures of general Gauduchon connections, see [1], [23], [26], [30] etc.…”

Holomorphic sectional curvature and Kähler-like metric

“…1985 年, Balas 和 Gauduchon [7] 证明了具有 非正的常值全纯截面曲率的紧致复 Hermite 面一定是 Kähler 流形. 1996 年, Apostolov 等 [8] 证明了 具有逐点常值的全纯截面曲率的紧致复面一定是 Kähler 流形. 一个自然的问题是, 具有常值全纯截 面曲率的高维紧致复流形, 是否为 Kähler 流形?…”