We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a boundedω-plurisubharmonic solution of an elliptic complex Monge-Ampère equation is smooth under an assumption on the background Hermitian metricω. This generalizes a result of Székelyhidi and Tosatti on Kähler manifolds.

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-Kähler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.a variety of Ricci curvatures on a Hermitian manifold. Among other results, they derive explicit relations between all kinds of Ricci curvatures on general Hermitian manifolds. In the locally conformal Kähler case, we can get a simpler formula (see Proposition 3.2). Then we are able to reduce the theorem to the conformally Kähler case. Actually we prove the following result under the more general pointwise constant condition.Theorem 1.2. Let (M, ω) be a compact locally conformal Kähler manifold with pointwise nonpositive constant holomorphic sectional curvature. Then (M, ω) is globally conformal Kähler.We remark that Vaisman [33] has proved that a locally conformal Kähler metric with pointwise constant (Chern) sectional curvature is either globally conformal Kähler or has vanishing first Chern class. The constancy of sectional curvature is of course stronger than the constancy of holomorphic sectional curvature. For example, the sectional curvatures of CP n and B n (n ≥ 2) are not pointwise constant.An important class of locally conformal Kähler manifolds is called Vaisman manifolds, whose Lee form is parallel with respect to the Levi-Civita connection. It is shown [27] that a Vaisman metric on a compact manifold must be Gauduchon. Then we obtain Corollary 1.3. A compact Vaisman manifold with pointwise nonpositive constant holomorphic sectional curvature is Kähler.Considering the constancy of holomorphic sectional curvature, a natural question is: does the pointwise constancy of H imply the global constancy?When ω is Kähler and n ≥ 2, it is always true by the Schur's Lemma, as ω is Kähler-Einstein and H is constant multiple of the scalar curvature (see Proposition 3.4). If ω is non-Kähler, we construct counterexamples showing that the Schur type result does not hold in general (example 3.8).Proposition 1.4. There exist non-Kähler, conformally flat metrics on C n (n ≥ 2) with pointwise negative constant (or pointwise positive constant) but not globally constant holomorphic sectional curvature. In the negative case, the metric is complete.Remark 1.5. If the holomorphic sectional curvature is defined using the Levi-Civita connection, it is Gray-Vanheche [13] who first discovered that the Schur type result does not hold in non-Kähler setting. Since the Levi-Civita connection coincides with the Chern connection if and only if the metric is Kähler, our result are obviously different from theirs. We refer to [13] [28] [29] and the references therein for more results and development in that direction. Also see [22] for some recent resul...

No abstract

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds of dimension n under a mild assumption.

No abstract