Ricci Curvatures on Hermitian manifolds

Liu, Kefeng; Yang, Xibei

doi:10.48550/arxiv.1404.2481

Cited by 12 publications

(25 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We should remark that this goes in different direction with respect to both the classical Yamabe problem and the Yamabe problem for almost Hermitian manifolds studied by H. del Rio and S. Simanca in [3] (compare Remark 2.2). One motivation for considering exactly such "complex curvature scalar", between other natural ones [12,13], comes from the importance of the Chern Ricci curvature in non-Kähler Calabi-Yau problems (compare, for example, [21]). We also stress that it would be very interesting, especially in view of having possibly "more canonical" metrics on complex manifolds, to study the problem of existence of metrics satisfying both cohomological and curvature conditions (e.g., Gauduchon metrics with constant Chern scalar curvature).…”

Section: Introductionmentioning

confidence: 99%

On the Chern–Yamabe problem

Angella

Calamai

Spotti

2017

Mathematical Research Letters

View full text Add to dashboard Cite

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when the expected constant Chern scalar curvature is non-positive. In particular, this includes the case when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.2 DANIELE ANGELLA, SIMONE CALAMAI, AND CRISTIANO SPOTTI also called Gauduchon, metric ω, that is, satisfying ∂∂ ω n−1 = 0. On the other hand, one can look instead at metrics with special "curvature" properties, related to the underlying complex structure. The focus of the present note is exactly on this second direction. In particular on the property of having constant Chern scalar curvature in fixed conformal classes (hence neglecting cohomological conditions, for the moment). We should remark that this goes in different direction with respect to both the classical Yamabe problem and the Yamabe problem for almost Hermitian manifolds studied by H. del Rio and S. Simanca in [3] (compare Remark 2.2). One motivation for considering exactly such "complex curvature scalar", between other natural ones [12,13], comes from the importance of the Chern Ricci curvature in non-Kähler Calabi-Yau problems (compare, for example, [21]). We also stress that it would be very interesting, especially in view of having possibly "more canonical" metrics on complex manifolds, to study the problem of existence of metrics satisfying both cohomological and curvature conditions (e.g., Gauduchon metrics with constant Chern scalar curvature).We now describe the problem in more details. Let X be a compact complex manifold of complex dimension n endowed with a Hermitian metric ω. Consider the Chern connection, that is, the unique connection on T 1,0 X preserving the Hermitian structure and whose part of type (0, 1) coincides with the Cauchy-Riemann operator associated to the holomorphic structure. The Chern scalar curvature can be succinctly expressed aswhere ω n denotes the volume element.Denote by C H X the space of Hermitian conformal structures on X. On a fixed conformal class, there is an obvious action of the following "gauge group":where HConf(X, {ω}) is the group of biholomorphic automorphisms of X preserving the conformal structure {ω} and R + the scalings. It is then natural to study the moduli spaceIn analogy with the classical Yamabe problem, it is tempting to ask whether in each conformal class there always exists at least one metric having constant Chern scalar curvature. That is, one can ask whether the following Chern-Yamabe conjecture holds.Conjecture 2.1 (Chern-Yamabe conjecture). Let X be a compact complex manifold of complex dimension n, and let {ω} ∈ C H X be a Hermitian conformal structure on...

show abstract

Section: Introductionmentioning

confidence: 99%

On the Chern–Yamabe problem

Angella

Calamai

Spotti

2017

Mathematical Research Letters

View full text Add to dashboard Cite

show abstract

“…Remark 3.5. As shown in [31], the Hopf surface H a,b (and every diagonal Hopf manifold [23]) has a Hermitian metric with semi-positive holomorphic bisectional curvature. Since b 2 (H a,b ) = b 2 (S 1 × S 3 ) = 0, we see c 1 (H a,b ) = 0 and so H a,b is a non-Kähler Calabi-Yau manifold with semi-positive holomorphic sectional curvature.…”

Section: Hermitian Manifolds With Semi-positive Holomorphic Sectional...mentioning

confidence: 92%

“…It is also well-known that, on Hermitian manifolds, there are many curvature notations and the curvature relations are more complicated than the relations in the Kähler case because of the non-vanishing of the torsion tensor (e.g. [21,23]).…”

Section: Hermitian Manifolds With Semi-positive Holomorphic Sectional...mentioning

confidence: 99%

“…Note also that all diagonal Hopf manifolds are non-Kähler Calabi-Yau manifolds with semi-positive holomorphic sectional curvature( [23,31]). For reader's convenience, we present some recent progress about the relationship between the positivity of holomorphic sectional curvature and the algebraic positivity of the anti-canonical line bundle, which is inspired by the following conjecture of Yau: Conjecture 1.5.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hermitian manifolds with semi-positive holomorphic sectional curvature

Yang

2016

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

show abstract

“…Hence, an equivalent statement of ( 5) is n • ρ (3) = ρ (1) . The first and third Ricci forms satisfy the following general relation (see [13,17])…”

Section: Negative Projectively Flat Metricsmentioning

confidence: 99%

Positive projectively flat manifolds are locally conformally flat-Kähler Hopf manifolds

Calamai¹

2017

Preprint

View full text Add to dashboard Cite

We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler metrics on Hopf manifolds, explicitly characterized by Vaisman [23]. Finally, we review the known characterization and properties of zero projectively flat metrics. As applications, we make sharp a list of possible projectively flat metrics by Li, Yau, and Zheng [16, Theorem 1]; moreover we prove that projectively flat astheno-Kähler metrics are in fact Kähler and globally conformally flat.

show abstract

Ricci Curvatures on Hermitian manifolds

Cited by 12 publications

References 39 publications

On the Chern–Yamabe problem

On the Chern–Yamabe problem

Hermitian manifolds with semi-positive holomorphic sectional curvature

Positive projectively flat manifolds are locally conformally flat-Kähler Hopf manifolds

Contact Info

Product

Resources

About