On the Gauduchon curvature of Hermitian manifolds

Abstract

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of [Formula: see text]-Gauduchon Ricci-flat metrics on the suspension of a compact Sasaki–Einstein manifold, for all [Formula: see text]; in particular, for the Bismut, Minimal and Hermitian conformal connection. A monotonicity theorem is obtained for the Gauduchon holomorphic sectional curvature, illustrating a maximality property for the Chern connection and furnishing insight into known phenomena concerning hyperbolicity and the existence of rational curves. Moreover, we show a rigidity result for Hermitian metrics which have a pair of Gauduchon holomorphic sectional curvatures that are equal, elucidating a duality implicit in the recent work of Chen–Nie.

Levi-Civita Ricci-Flat Metrics on Non-Kähler Calabi-Yau Manifolds

“…(3) t < 1 and M is a suspension of a compact Sasaki-Einstein manifold [BS23], see also [LY17], [WY19], and [Cor23b]; (4) t = −1 and M is a total space of a principal T 2r -bundle over a Kähler-Einstein Fano manifold [GGP08,Gra11]. Motivated by Problem 1, from the construction provided in [GGP08], and the ideas introduced in [PT18], [Cor23a], we prove the following theorem.…”

“…As in the case of Riemannian manifolds, we would like to understand the differential geometry of each of these connections in terms of their curvature properties. The relations between the curvature tensors of ∇ Ch , ∇ SB , and ∇ LC have been extensively studied, see for instance [Gra76], [TV81], [Gau84], [AD99], [LY12,LY17], [YZ16,YZ18], [AOUV22], [WY19], [WYZ20], [HLY20], [BS23]. An important problem in this context is related to curvature conditions that generalize the concept of Calabi-Yau manifolds 1 to the non-Kähler context.…”