On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.
Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.
This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows...
We prove an extension theorem for Kähler currents with analytic singularities in a Kähler class on a complex submanifold of a compact Kähler manifold.
We prove that Kummer threefolds with algebraic dimension have Kodaira dimension 0.
Étant donnée une variété kählérienne compacte , on étudie dans l’espace vectoriel réel de cohomologie de Dolbeault le cône convexe des classes de Kähler ainsi que celui, plus grand, des classes de courants positifs fermés de type . Lorsque est projective, les traces de ces cônes sur l’espace de Néron–Severi engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.
In this appendix, we observe that Iitaka’s conjecture fits in the more general context of special manifolds, in which the relevant statements follow from the particular cases of projective and simple manifolds.
These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.
We study compact Kähler manifolds admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of , and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of . We extend Calabi’s theorem on the structure of compact Kähler...
Let f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we...
We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold have a pseudo-effective (instead of generically nef) determinant. A first consequence is that is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of is not covered by compact positive-dimensional analytic subsets,...