Ricci curvatures on Hermitian manifolds. (English) Zbl 1365.53065

Liu, Kefeng; Yang, Xiaokui

Trans. Am. Math. Soc. 369, No. 7, 5157-5196 (2017).

In this paper, the authors introduce two cohomology groups to study the geometry of compact, complex (especially, non-Kähler) manifolds, namely the Bott-Chern cohomology
\[ H^{p,q}_{BC}(M):= \frac{\text{Ker } d \cap \Omega^{p,q}(M)}{\text{Im } \partial\overline{\partial} \cap \Omega^{p,q}(M)} \]
and the Aeppli cohomology
\[ H^{p,q}_{A}(M):=\frac{\text{Ker } \partial\overline{\partial} \cap \Omega^{p,q}(M)}{\text{Im } \partial \cap \Omega^{p,q}(M)+ \text{Im }\overline{\partial} \cap \Omega^{p,q}(M)}. \] Let \(\mathrm{Pic}(M)\) be the set of holomorphic line bundles over \(M\). Similar to the first Chern class map \(c_1:\mathrm{Pic}(M)\rightarrow H^{1,1}_{\overline{\partial}}(M)\), there is a first Aeppli-Chern class map \(c_1^{{AC}}:\mathrm{Pic}(M)\rightarrow H^{1,1}_{{A}}(M)\), which can be described as follows. Let \(L\to M\) be a holomorphic line bundle over \(M\). The first Aeppli-Chern class is defined as \(c^{AC}_1(L)=\left[-\sqrt{-1}\partial\overline{\partial}\log h\right]_A \in H^{1,1}_{A}(M)\) where \(h\) is an arbitrary smooth Hermitian metric on \(L\). The authors 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.
The authors systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, they study non-Kähler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, they construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds \(\mathbb S^{2n-1}\times \mathbb S^1\). The authors also construct a smooth family of Gauduchon metrics on a compact Hermitian manifolds 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 \(-\infty\) 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.

Reviewer: Nicolai K. Smolentsev (Kemerovo)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
32Q25	Calabi-Yau theory (complex-analytic aspects)
32Q20	Kähler-Einstein manifolds

Keywords:

Bott-Chern cohomology; Aeppli cohomology; holomorphic line bundles; first Aeppli-Chern class; non-Kähler Calabi-Yau manifolds; Gauduchon metrics; Hermitian manifolds; almost Hermitian manifolds; compact complex non-Kähler manifolds

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