/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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Introduction to the Modern Theory of Dynamical Systems

Convex bodies: the brunn–minkowski theory

Demailly's Holomorphic Morse Inequalities.- Characterization of Moishezon Manifolds.- Holomorphic Morse Inequalities on Non-compact Manifolds.- Asymptotic Expansion of the Bergman Kernel.- Kodaira Map.- Bergman Kernel on Non-compact Manifolds.- Toeplitz Operators.- Bergman Kernels on Symplectic Manifolds.

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Holomorphic Morse Inequalities and Bergman Kernels

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

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The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1,1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a covering family, has non negative Kodaira dimension.

The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kahler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kahler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.

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Divisorial Zariski decompositions on compact complex manifolds

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+e-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kahler–Einstein volume forms with minimal singularities on varieties of general type.

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Monge-Ampère equations in big cohomology classes

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kahler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kahler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kahler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.

Sébastien Boucksom

Papers

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

Divisorial Zariski decompositions on compact complex manifolds

Monge-Ampère equations in big cohomology classes

A variational approach to complex Monge-Ampère equations