337 research outputs found
On the cohomology of pseudoeffective line bundles
The goal of this survey is to present various results concerning the
cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and
related properties of their multiplier ideal sheaves. In case the curvature is
strictly positive, the prototype is the well known Nadel vanishing theorem,
which is itself a generalized analytic version of the fundamental
Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested
here in the case where the curvature is merely semipositive in the sense of
currents, and the base manifold is not necessarily projective. In this
situation, one can still obtain interesting information on cohomology, e.g. a
Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a
surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his
PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing
theorem that depends on the concept of numerical dimension of a given
pseudoeffective line bundle. The proof of these results depends in a crucial
way on a general approximation result for closed (1,1)-currents, based on the
use of Bergman kernels, and the related intersection theory of currents.
Another important ingredient is the recent proof by Guan and Zhou of the strong
openness conjecture. As an application, we discuss a structure theorem for
compact K{\"a}hler threefolds without nontrivial subvarieties, following a
joint work with F.Campana and M.Verbitsky. We hope that these notes will serve
as a useful guide to the more detailed and more technical papers in the
literature; in some cases, we provide here substantially simplified proofs and
unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the
Abel Symposium, Trondheim, July 201
Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties
The goal of this survey is to describe some recent results concerning the L 2
extension of holomorphic sections or cohomology classes with values in vector
bundles satisfying weak semi-positivity properties. The results presented here
are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow
many techniques from the long series of papers by T. Ohsawa. The recent
achievement that we want to point out is that the surjectivity property holds
true for restriction morphisms to non necessarily reduced subvarieties,
provided these are defined as zero varieties of multiplier ideal sheaves. The
new idea involved to approach the existence problem is to make use of L 2
approximation in the Bochner-Kodaira technique. The extension results hold
under curvature conditions that look pretty optimal. However, a major unsolved
problem is to obtain natural (and hopefully best possible) L 2 estimates for
the extension in the case of non reduced subvarieties -- the case when Y has
singularities or several irreducible components is also a substantial issue.Comment: arXiv admin note: text overlap with arXiv:1703.00292,
arXiv:1510.0523
Numerically flat Higgs vector bundles
After providing a suitable definition of numerical effectiveness for Higgs
bundles, and a related notion of numerical flatness, in this paper we prove,
together with some side results, that all Chern classes of a Higgs-numerically
flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if
and only if it is has a filtration whose quotients are flat stable Higgs
bundles. We also study the relation between these numerical properties of Higgs
bundles and (semi)stability.Comment: 11 page
Green Currents for Meromorphic Maps of Compact K\"ahler Manifolds
We consider the dynamics of meromorphic maps of compact K\"ahler manifolds.
In this work, our goal is to locate the non-nef locus of invariant classes and
provide necessary and sufficient conditions for existence of Green currents in
codimension one.Comment: Statement of Theorem 1.5 is slightly improved. Proposition 5.2 and
Theorem 5.3 are adde
Del Pezzo surfaces and local inequalities
I prove new local inequality for divisors on smooth surfaces, describe its
applications, and compare it to a similar local inequality that is already
known by experts.Comment: 13 pages; to appear in the proceedings of the conference "Groups of
Automorphisms in Birational and Affine Geometry", Levico Terme (Trento), 201
Towards the Green-Griffiths-Lang conjecture
The Green-Griffiths-Lang conjecture stipulates that for every projective
variety X of general type over C, there exists a proper algebraic subvariety of
X containing all non constant entire curves f : C X. Using the
formalism of directed varieties, we prove here that this assertion holds true
in case X satisfies a strong general type condition that is related to a
certain jet-semistability property of the tangent bundle TX . We then give a
sufficient criterion for the Kobayashi hyperbolicity of an arbitrary directed
variety (X,V). This work is dedicated to the memory of Professor Salah
Baouendi.Comment: version 2 has been expanded and improved (15 pages
Local syzygies of multiplier ideals
In recent years, multiplier ideals have found many applications in local and
global algebraic geometry. Because of their importance, there has been some
interest in the question of which ideals on a smooth complex variety can be
realized as multiplier ideals. Other than integral closure no local
obstructions have been known up to now, and in dimension two it was established
by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is
locally a multiplier ideal. We prove the somewhat unexpected result that
multiplier ideals in fact satisfy some rather strong algebraic properties
involving higher syzygies. It follows that in dimensions three and higher,
multiplier ideals are very special among all integrally closed ideals.Comment: 8 page
Interpolation in non-positively curved K\"ahler manifolds
We extend to any simply connected K\"ahler manifold with non-positive
sectional curvature some conditions for interpolation in and in
the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a
comparison theorem for the Hessian in K\"ahler geometry due to Greene, Wu and
Siu, Yau.Comment: 9 pages, Late
Effective algebraic degeneracy
We prove that any nonconstant entire holomorphic curve from the complex line
C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary
dimension n (at least 2) must be algebraically degenerate provided X is generic
if its degree d = deg(X) satisfies the effective lower bound: d larger than or
equal to n^{{(n+1)}^{n+5}}
Extension and approximation of -subharmonic functions
Let be a bounded domain, and let be a
real-valued function defined on the whole topological boundary . The aim of this paper is to find a characterization of the functions
which can be extended to the inside to a -subharmonic function under
suitable assumptions on . We shall do so by using a function algebraic
approach with focus on -subharmonic functions defined on compact sets. We
end this note with some remarks on approximation of -subharmonic functions
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