On the Chern–Yamabe problem

Abstract

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...

On the proportionality of Chern and Riemannian scalar curvatures

“…In fact, the authors believe that there do exist compact Kähler manifolds, satisfying these scalar curvature conditions, which admit no conformal non-Kähler sKlsc metrics. This is loosely reminiscent of the positive Gauduchon degree case for the Chern Yamabe problem [1].…”

“…In particular, note that Kähler metric are necessarily balanced. See [1] for an excellent reference regarding the conformal transformation of S C (g).…”

“…In particular, Hermitian manifolds, the Chern connection, its curvature, and the transformations of the Riemannian and Chern scalar curvatures under a conformal change are discussed. For a more detailed account of these topics see [15,3,17,1].…”

“…Much of the existing work in this direction focuses either on metrics which satisfy certain structural conditions that are a weakening of those found in Kähler geometry, or on Hermitian analogues of classical Riemannian questions concerning curvature. For example, see [2,11,19,6,1]. The work of this paper can be viewed as somewhere between these two ideas as we study the weakening of a relationship between scalar curvatures that characterizes the Kähler condition.…”

“…The motivation for studying Chern scalar curvature here, as opposed to a variety of other complex curvature scalars, comes mainly from the naturalness of the connection and the resulting complex geometry, as well as it's interplay with the underlying real geometry. It is worthwhile to note, though, that there has been much other recent interest in problems regarding Chern curvatures; for instance, in non-Kähler Calabi-Yau problems and the Chern Yamabe problem, see [21,1].…”