1,345 research outputs found
Quasiplurisubharmonic Green functions
Given a compact K\"ahler manifold , a quasiplurisubharmonic function is
called a Green function with pole at if its Monge-Amp\`ere measure is
supported at . We study in this paper the existence and properties of such
functions, in connection to their singularity at . A full characterization
is obtained in concrete cases, such as (multi)projective spaces.Comment: 23 pages; to appear in Journal de Mathematiques Pures et Appliquee
Valuative analysis of planar plurisubharmonic functions
We show that valuations on the ring R of holomorphic germs in dimension 2 may
be naturally evaluated on plurisubharmonic functions, giving rise to
generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic
function thus defines a real-valued function on the set V of valuations on R
and--by way of a natural Laplace operator defined in terms of the tree
structure on V--a positive measure on V. This measure contains a great deal of
information on the singularity at the origin. Under mild regularity
assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two
plurisubharmonic functions. As a consequence, any generalized Lelong number can
be interpreted as an average of valuations. Using our machinery we also show
that the singularity of any positive closed (1,1) current T can be attenuated
in the following sense: there exists a finite composition of blowups such that
the pull-back of T decomposes into two parts, the first associated to a divisor
with normal crossing support, the second having small Lelong numbers.Comment: Final version. To appear in Inventiones Math. 37 pages, 5 figure
A converse to the Andreotti-Grauert theorem
The goal of this paper is to show that there are strong relations between
certain Monge-Amp\`ere integrals appearing in holomorphic Morse inequalities,
and asymptotic cohomology estimates for tensor powers of holomorphic line
bundles. Especially, we prove that these relations hold without restriction for
projective surfaces, and in the special case of the volume, i.e. of asymptotic
0-cohomology, for all projective manifolds. These results can be seen as a
partial converse to the Andreotti-Grauert vanishing theorem.Comment: 12 page
Regularity of plurisubharmonic upper envelopes in big cohomology classes
The goal of this work is to prove the regularity of certain
quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way
in the construction of hermitian metrics with minimal singularities on a big
line bundle over a compact complex manifold. We prove that the complex Hessian
forms of these envelopes are locally bounded outside an analytic set of
singularities. It is furthermore shown that a parametrized version of this
result yields a priori inequalities for the solution of the Dirichlet problem
for a degenerate Monge-Ampere operator; applications to geodesics in the space
of Kahler metrics are discussed. A similar technique provides a logarithmic
modulus of continuity for Tsuji's "supercanonical" metrics, which generalize a
well-known construction of Narasimhan-Simha.Comment: 27 pages, no figure
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