Given any closed Kähler manifold we define, following an idea by Eugenio Calabi [8], a Riemannian metric on the space of Kähler metrics regarded as a infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space has positive constant sectional curvature and admits explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation.

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...

Abstract. In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics H β ; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson [29] to the case of Kähler manifolds with boundary; moreover we introduce a subspace H C of H β which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of C 1,1 β geodesics whose boundary values lie in H C . Moreover, we prove that such geodesic is the limit of a sequence of C 2,α β approximate geodesics under the C 1,1 β -norm. As a geometric application, we prove the metric space structure of H C .

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We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fiber bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.