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We give sufficient conditions for the existence of Kahler-Einstein and constant scalar curvature Kahler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kahler classes and the branching divisor. This result generalizes previous work on Kahler-Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kahler-Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927-973], and extends Chen-Cheng's existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kahler metrics (II)-Existence results, J. Amer. Math. Soc. 34 2021, 4, 937-1009].

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kähler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful paper by Ding and Tian [16] which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday.

Given a compact constant scalar curvature Kähler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kähler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kähler desingularization. We also give complete proofs of a number of analytic results which have been used in this context by various authors. A series of explicit examples is discussed

Toeplitz operators on quantized compact symplectic manifolds were introduced by Guillemin and Boutet de Monvel, who studied their spectral asymptotics in analogy with the theory developed by Duistermaat, Guillemin, and H\"{o}rmander for pseudodifferential operators. In this survey, we review some recent results concerning eigenfunction asymptotics in this context, largely based on the microlocal description of Szeg\"{o} kernels by Boutet de Monvel and Sj\"{o}strand, and its revisitation and generalization to the almost complex symplectic category by Shiffman and Zelditch. For simplicity, the exposition is restricted to the complex projective setting.

We investigate how to obtain various flows of Kãhler metrics on a fixed manifold as variations of Kãhler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchiâs metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the Kãhler-Ricci flow. In the latter case we rederive the V-soliton equation of La Nave-Tian.

In this paper, we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the Kähler cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities are sufficient to check K-semistability. A similar improvement on Donaldson's lower bound for the Calabi energy is given. © The Author(s) 2011

We consider the problem of existence of constant scalar curvature Kähler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kähler-Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.

La responsabilità sociale d'impresa si propone come un nuovo modello manageriale capace di coniugare sostenibilità e coinvolgimento degli stakeholder, in particolare i consumatori che in questo approccio possono svolgere il ruolo di guardiani del mercato e stimolatori di innovazione

In this paper we study the first eigenvalue of the Laplacian on a compact Kaehler manifold using stable bundles and balanced bases.

We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too.

We consider Fano manifolds M that admit a collection of finite automorphism, groups G(1),..., G(k), such that the quotients M/G(i) are smooth Fano manifolds possessing a Kahler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kahler-Einstein metric too