TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 RICCI CURVATURES ON HERMITIAN MANIFOLDS KEFENG LIU AND XIAOKUI YANG Abstract. In this paper, we introduce the first Aeppli-Chern class for com- plex 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 2n−1 1 explicit Levi-Civita Ricci-flat metrics on Hopf manifolds S × S . 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 Rie- mannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian man- ifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature. Contents 1. Introduction 1 2. Background materials 9 3. Geometry of the Levi-Civita Ricci curvature 12 4. Curvature relations on Hermitian manifolds 22 5. Special metrics on Hermitian manifolds 26 6. Levi-Civita Ricci-flat and constant negative scalar curvature metrics on Hopf manifolds 29 7. Appendix: the Riemannian Ricci curvature and ∗-Ricci curvature 35 References 37 1. Introduction Let (M, h) be a Hermitian manifold and g the background Riemannian metric. It is well-known that, when (M, h) is not K¨ahler,the complexification of the real curvature tensor R is extremely complicated. Moreover, on the Hermitian holo- morphic vector bundle (T 1,0M, h), there are two typical connections: the (induced) Received by the editors April 23, 2015 and, in revised form, May 21, 2016. 2010 Mathematics Subject Classification. Primary 53C55, 32Q25, 32Q20. The first author is supported in part by NSF Grant. The second author is partially supported by China’s Recruitment Program of Global Experts and National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. c XXXX American Mathematical Society 1 2 KEFENG LIU AND XIAOKUI YANG Levi-Civita connection and the Chern connection. The curvature tensors of them are denoted by R and Θ respectively. It is known that the complexified Riemannian curvature R, the Hermitian Levi-Civita curvature R and the Chern curvature Θ are mutually different. It is also obvious that R is closely related to the Riemann- ian geometry of M, and Θ can characterize many complex geometric properties of M whereas R can be viewed as a bridge between R and Θ, i.e. a bridge between Riemannian geometry and Hermitian geometry. i n Let {z }i=1 be the local holomorphic coordinates centered at a point p ∈ M. We can compare the curvature tensors of R, Θ when restricted on the space 1,1 ∗ 1,0 Γ(M, Λ T M ⊗ End(T M)). That is, we can find relations between Rijk` and Θijk`. We denoted by √ R(1) = −1R(1)dzi ∧ dzj with R(1) = hk`R , ij ij ijk` √ R(2) = −1R(2)dzi ∧ dzj with R(2) = hk`R , ij ij k`ij √ R(3) = −1R(3)dzi ∧ dzj with R(3) = hk`R , ij ij i`kj and √ R(4) = −1R(4)dzi ∧ dzj with R(4) = hk`R . ij ij kji` R(1) and R(2) are called the first Levi-Civita Ricci curvature and the second Levi- Civita Ricci curvature of (T 1,0M, h) respectively; R(3)and R(4) are the correspond- ing third and fourth Levi-Civita Ricci curvatures. Similarly, we can define the first Chern-Ricci curvature Θ(1), the second Chern-Ricci curvature Θ(2), the third and fourth Chern-Ricci curvatures Θ(3) and Θ(4) respectively. As shown in [41], R(2) and Θ(2) are closely related to the geometry of M, for example, we can use them to study the cohomology groups and plurigenera of compact Hermitian mani- folds. On the other hand, it is well-known that Θ(1) represents the first Chern class c (M) ∈ H1,1(M). However, in general, the first Levi-Civita Ricci form R(1) is not 1 ∂ d-closed, and so it can not represent a class in H1,1(M). ∂ We introduce two cohomology groups to study the geometry of compact com- plex (especially, non-K¨ahler) manifolds, the Bott-Chern cohomology and the Aeppli cohomology: Kerd ∩ Ωp,q(M) Ker∂∂ ∩ Ωp,q(M) Hp,q (M) := and Hp,q(M) := . BC Im∂∂ ∩ Ωp,q(M) A Im∂ ∩ Ωp,q(M) + Im∂ ∩ Ωp,q(M) Suppose α is a d-closed (p, q)-form. We denote by [α]BC and [α]A, the corresponding p,q p,q classes in HBC (M) and HA (M) respectively. Let Pic(M) be the set of holomorphic line bundles over M. As similar as the first Chern class map c1 : Pic(M) → H1,1(M), there is a first Aeppli-Chern class map ∂ AC 1,1 (1.1) c1 : Pic(M) → HA (M), which can be described as follows. Definition 1.1. Let L → M be a holomorphic line bundle over M. The first Aeppli-Chern class is defined as √ AC 1,1 (1.2) c1 (L) = − −1∂∂ log h A ∈ HA (M) RICCI CURVATURES ON HERMITIAN MANIFOLDS 3 √ where h is an arbitrary smooth Hermitian metric on L. Note that − −1∂∂ log h is the (local) curvature form Θh of the√ Hermitian line bundle (L, h). If we choose 0 h a different metric h , then Θh0 − Θh = −1∂∂ log h0 is globally ∂∂-exact. Hence AC 1,1 c1 (L) is well-defined in HA (M) and it is independent of the metric h. For a AC AC −1 −1 complex manifold M, c1 (M) is defined to be c1 (KM ) where KM is the anti- canonical line bundle ∧nT 1,0M. BC AC Note that, for a Hermitian line bundle (L, h), the√ classes c1(L), c1 (L) and c1 (L) have the same (1, 1)-form representative Θh = − −1∂∂ log h (in different classes). On K¨ahler manifolds or Hermitian manifolds the first Chern classes and first Bott-Chern classes are well-studied in the literatures by using the Chern connec- tion. In particular, the related Monge-Amp`ere type equations are extensively inves- tigated since the celebrated work of Yau. However, the geometry of the Levi-Civita connection is not well understood in complex geometry although it has rich real Riemannian geometry structures. 1.1. The complex geometry of the Levi-Civita connection. In the first part of this paper, we study (non-K¨ahler)Hermitian manifolds by using Levi-Civita Ricci AC forms and Aeppli-Chern class c1 (M). At first, we establish the following result which is analogous to the classical result that the (first) Chern-Ricci curvature Θ(1) represents the first Chern class c1(M). Theorem 1.2. On a compact Hermitian manifold (M, ω), the first Levi-Civita (1) AC 1,1 Ricci form R represents the first Aeppli-Chern class c1 (M) in HA (M). More precisely, 1 ∗ (1.3) R(1) = Θ(1) − (∂∂∗ω + ∂∂ ω). 2 In particular, we obtain (1) R(1) = Θ(1) if and only if d∗ω = 0, i.e. (M, ω) is a balanced manifold; ∗ (1) (2) if ∂∂ ω = 0, then R represents the real first Chern class c1(M) ∈ 2 AC 2 HdR(M), i.e. c1(M) = c1 (M) in HdR(M). We also show that, on complex manifolds supporting ∂∂-lemma (e.g. manifolds in Fujiki class C and in particular, Moishezon manifolds), the converse statement of (2) holds. There is an important class of manifolds, so called Calabi-Yau manifolds which are extensively studied by mathematicians and also physicians. In this paper, a Calabi-Yau manifold is a complex manifold with c1(M) = 0 and we will focus on non-K¨ahler Calabi-Yau manifolds. There are many fundamental results on non- K¨ahler Hermitian manifolds with vanishing first Bott-Chern classes. They are al- ways characterized by using the first Chern-Ricci curvature Θ(1) and the related Monge-Amp`eretype equations (e.g.[19, 26, 52, 53, 54, 55, 56, 8]). For more details on this subject, we refer the reader to the nice survey paper [51]. Next, we make the following observation: Corollary 1.3. Let M be a complex manifold. Then BC AC c1 (M) = 0 =⇒ c1(M) = 0 =⇒ c1 (M) = 0. 4 KEFENG LIU AND XIAOKUI YANG Moreover, on a complex manifold satisfying the ∂∂-lemma, BC AC c1 (M) = 0 ⇐⇒ c1(M) = 0 ⇐⇒ c1 (M) = 0. That means, it is very natural to study non-K¨ahler Calabi-Yau manifolds by using AC (1) the first Aeppli-Chern class c1 and the first Levi-Civita Ricci curvature R . Definition 1.4. A Hermitian metric ω on M is called Levi-Civita Ricci-flat if R(1)(ω) = 0. It is known that Hopf manifolds M = S2n−1 × S1 are all non-K¨ahler Calabi-Yau manifolds (n ≥ 2), i.e. c1(M) = 0. However, there is no Chern Ricci-flat Hermitian metrics on M, i.e. there does not exist a Hermitian metric ω such that Θ(1)(ω) = 0 BC since c1 (M) 6= 0 (see Remark 6.1 for more details). On the contrary, we can construct explicit Levi-Civita Ricci-flat metrics on them.