Holonomy and the Ricci curvature of complex Hermitian manifolds (Q6656083)

The aim of this paper is to prove two results on geometric consequences of the representation of restricted holonomy group of a hermitian connection. The first result concerns when such a hermitian manifold is Kähler in terms of the torsion and the irreducibility of the holonomy action. As a consequence the author obtains a criterion of when a hermitian manifold and connection is a generalized Calabi-Yau in the sense that the first Ricci vanishes or equivalently that the restricted holonomy is inside \(SU(m)\). The second result of the paper concerns the projectivity of a compact Kähler manifold, namely when a compact Kähler manifold can be embedded into some complex projective space of a higher dimension. It concerns when a compact Kähler manifold with a generic restricted holonomy group is projective, under some nonnegativity assumptions in terms of the Ricci and other curvatures. This paper is organized as follows: Section 1 is an introduction to the subject. Some preliminaries are given in Section 2. Section 3 is devoted to the proof of the following result. For a complex hermitian manifold \((M^m, g)\), and a hermitian connection \(\nabla\), assume that the restricted holonomy group \(G\) action on \(T^{1,0}(M)\) is irreducible. Assume further that \(\mbox{Ric}^1\neq0\) and the torsion of \(\nabla\) is parallel. Then \((M^m, g)\) must be Kähler. Equivalently, for any non-Kähler hermitian manifold whose restricted holonomy with respect to a hermitian connection with a parallel torsion is irreducible, it must be a generalized Calabi-Yau in the sense that its the first Ricci vanishes. One may find it interesting to compare the result with the Hopf manifold example in the appendix of [\textit{L. Ni} and \textit{F. Zheng}, ``A classification of locally Chern homogeneous Hermitian manifolds'', Preprint, \url{arXiv:2301.00579}]. Section 4 deals with a Lie algebraic structure on \(T_\mathbb{C}M\). Here the author proves that for any hermitian manifold \((M, g)\), if its Bismut torsion is parallel, then the torsion of the Chern connection endows \(T_\mathbb{C}M\) with a complex Lie algebra structure. Section 5 is devoted to the proof of the following result. Let \((M^m, g)\) be a compact Kähler manifold with the restricted holonomy being \(U(m)\) (or \(SU(m)\)). Assume any one of the following : (i) The sum of the smallest two eigenvalues of Ric is nonnegative; (ii) The mixed curvature \(\mathcal{C}_{\alpha,\beta}(X)\geq0\), for \(\alpha>0\), \(3\alpha+2\beta>0\); (iii) The second scalar curvature \(S_2\geq0\). In a recent arXiv preprint [\textit{J. Chu} et al., ``On Kähler manifolds with non-negative mixed curvature'', Preprint, \url{arXiv:2408.14043}], a result close to part (ii) appeared. The argument of [loc. cit.] uses a conformal change, while the proof here follows closely the argument of [\textit{K. Tang}, ``Quasi-positive curvature and vanishing theorems'', Preprint, \url{arXiv:2405.03895}]. The paper is supported by an appendix (Section 6) on a computation using standard formulae concerning a lemma used in the previous section.