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In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

In this paper we study compact Sasaki manifolds in view of transverse K¨ahler geometry and extend some results in K¨ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse K¨ahler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse K¨ahler metric of constant scalar curvature. We prove the existence of transverse K¨ahler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the U(1)-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

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We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson’s theorem: the existence of an orbifold K¨ahler metric of constant scalar curvature implies K-semistability. By extending the notion of slope stability to orbifolds, we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold K¨ahler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, Ghigi-Koll´ar, Rollin-Singer, the AdSCFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-Yau) fit into this framework.