ON THE VOLUME OF A LINE BUNDLE

By S. Boucksom. 2002

190 citations recorded by the source corpus.

Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic Morse inequalities in order to express the volume of a line bundle as the maximum of the mean curvatures of all the singular Hermitian metrics on it, with a way to pick an element at which the maximum is reached and satisfying a singular Monge–Ampere equation. This enables us to introduce the volume of any (1,1)-class on a compact Kahler manifold, and Fujita's theorem is then extended to this context.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dc759f3a323036bc1abeb49f1847c48069a6a55d

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, methodology): “It is a pleasure to thank Jean-Pierre Demailly and Sébastien Boucksom for several illuminating discussions.”
The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension (2004, background): “family ωe ; then T is a closed positive current, and the arguments in [ Bou02a ] show that its absolutely continuous part satisfies”
Convex Bodies Associated to Linear Series (2008, background): “When X is a compact complex manifold, Boucksom [6], [7] has defined and studied the volume (and other invariants) of an arbitrary pseudo-effective (1, 1)-class α on X.”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “Then one can check easily from the construction that the total mass of MA( ϕ ) is less than or equal to the volume vol( α ) of the class α (cf [ Bou02 ]): A particular class of θ -psh functions that appears naturally is the one for which the last inequality is an equality.”
Growth of balls of holomorphic sections and energy at equilibrium (2008, result): “The following result is proved in [25] using [24].”
Divisorial Zariski decompositions on compact complex manifolds (2004, result): “We have shown in [ 2 ] that the equality v(p )= pdim X is always true when p is a nef class, so the contended equality holds on a surface.”
Monge–Ampère equations in big cohomology classes (2008, background): “In particular we show that ∫ X 〈T〉6vol(α), where the right-hand side denotes the volume of the class α:={T} [10], which implies in particular that 〈T〉 is non-trivial only if the cohomology class α is big.”