We prove that the Gromov-Hausdorff compactification of the moduli space of Kähler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kähler-Einstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

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...

In this article we prove the existence of Kähler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings. P N by sections of K −λ X . A weak Kähler-Einstein metric on X is a Kähler current in 2πc 1 (X) with locally continuous potential, and that is a smooth Kähler-Einstein metric on the smooth part X reg of X. Note there are different definitions in the literature, but they are all equivalent in this context [23]. A Q-Fano variety X is called Q-Gorenstein smoothable 1 if there is a flat family π : X → ∆, over a disc ∆ in C so that X ∼ = X 0 , X t is smooth for t = 0, and X admits a relatively Q-Cartier anti-canonical divisor −K X /∆ (in this case π : X → ∆ is called a Q-Gorenstein smoothing of X 0 ). It is well-known that, by possibly shrinking ∆, we may assume that X t is a Fano manifold for t = 0, and that there exists an integer λ > 0 such that K −λ Xt are very ample line bundles with vanishing higher cohomology for all t ∈ ∆. Moreover, the dimension, denoted by N (λ), of the corresponding linear systems | − λK Xt | is constant in t. Thus, when needed, we may assume that the family X is relatively very ample, i.e. there is a smooth embedding i :Xt . The main theorem of this paper is the following result, which extends the results of [12,13,14,15] to Q-Gorenstein smoothable Q-Fano varieties and simultaneusly gives some understanding on the way such singular metric spaces are approached by smooth KE metrics on the (analytically) nearby Fano manifolds.Theorem 1.1. Let π : X → ∆ be a Q-Gorenstein smoothing of a Q-Fano variety X 0 . If X 0 is K-polystable then X 0 admits a weak Kähler-Einstein metric ω 0 .Moreover, assuming that the automorphism group Aut(X 0 ) is discrete, X t admit smooth Kähler-Einstein metrics ω t for all |t| sufficiently small and (X 0 , ω 0 ) is the limit in the Gromov-Hausdorff topology of (X t , ω t ), in the sense of [21].Few remarks are in place. First, by the generalized Bando-Mabuchi uniqueness theorem [6] the above weak Kähler-Einstein metric ω 0 is unique up to Aut 0 (X 0 ), the identity component of Aut(X 0 ), thus can be viewed as a canonical metric on X 0 . Second, the above theorem does not just state that there is a sequence of nearby KE Fano manifolds which converges, in the Gromov-Hausdorff sense, to the weak KE metric, but that all the nearby KE Fano manifolds actually converge to the unique singular limit (X 0 , ω 0 ). Thus this property provides a good topological correspondence between complex analytic deformations (alias flat-families) and the notion of Gromov-Hausdorff convergence. For example, by the Bando-Mabuchi theorem we have the following immediate corollary: Corollary 1.2. Let π : X → ∆ and π ′ : X ′ → ∆ be two Q-Gorenstein smoothings of Q-Fano varieties X 0 and X ′ 0 . Suppose X t and X ′ t are bi-holomorphic for all t = 0, and X 0 and X ′ 0 are both K-polystable with discrete automorphism group, then X 0 and X ′ 0 are isomorphic varieties.Notice t...

We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of K-stable Fano varieties.