257 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
Semistability vs. nefness for (Higgs) vector bundles
According to Miyaoka, a vector bundle E on a smooth projective curve is
semistable if and only if a certain numerical class in the projectivized bundle
PE is nef. We establish a similar criterion for the semistability of Higgs
bundles: namely, such a bundle is semistable if and only if for every integer s
between 0 and the rank of E, a suitable numerical class in the scheme
parametrizing the rank s locally-free Higgs quotients of E is nef. We also
extend this result to higher-dimensional complex projective varieties by
showing that the nefness of the above mentioned classes is equivalent to the
semistability of the Higgs bundle E together with the vanishing of the
discriminant of E.Comment: Comments: 20 pages, Latex2e, no figures. v2 includes a generalization
to complex projective manifolds of any dimension. To appear in Diff. Geom.
App
Unbalance Responses of Rotor/Stator Systems with Nonlinear Bearings by the Time Finite Element Method
Since the early 1970s, major works in rotordynamics were oriented toward the calculation of critical speeds and unbalance responses. The current trend is to take into account many kinds of non-linearities in order to obtain more realistic predictions. The use of algorithms based on nonlinear methods is therefore needed. This article first describes the time finite element method. The method is then applied to nonlinear rotor/stator systems where bearings present a radial clearance. Keywords Bearing clearance, Hertz contact, Nonlinear dynamics, Stability, Time finite element Many investigations have taken place concerning the calculation of critical speeds and unbalance responses in rotor dynamics and research now tends to get more realistic predictions Engineers now have to take into account non-linearities in their models. In the aircraft engine domain, those non-linearities come from components, such as bearings or squeeze film dampers Thus, frequential methods like the incremental harmonic balance In this article, the time finite element method is described. This time-based method enables us to get steady-state solutions and to assess their stability. To prove its reliability in rotordynamics problems, two examples are addressed. Both consist of rotor/stator systems with radial clearance in the bearings. THE TIME FINITE ELEMENT METHOD Description of the Method The aim of the time finite element method is to find out the periodic solutions of forced systems. It is based on Hamilton's Law of Varying Action i.e., The principle of this method is to interpolate the displacement of all spatial degrees of freedom between given instants t i and t i+1 by polynomial
The openness conjecture and complex Brunn-Minkowski inequalities
We discuss recent versions of the Brunn-Minkowski inequality in the complex
setting, and use it to prove the openness conjecture of Demailly and Koll\'ar.Comment: This is an account of the results in arXiv:1305.5781 together with
some background material. It is based on a lecture given at the Abel
symposium in Trondheim, June 2013. 13 page
Bergman kernel and complex singularity exponent
We give a precise estimate of the Bergman kernel for the model domain defined
by where
is a holomorphic map from to ,
in terms of the complex singularity exponent of .Comment: to appear in Science in China, a special issue dedicated to Professor
Zhong Tongde's 80th birthda
Holomorphic symplectic geometry: a problem list
A list of open problems on holomorphic symplectic, contact and Poisson
manifolds
Novel Branches of (0,2) Theories
We show that recently proposed linear sigma models with torsion can be
obtained from unconventional branches of conventional gauge theories. This
observation puts models with log interactions on firm footing. If non-anomalous
multiplets are integrated out, the resulting low-energy theory involves log
interactions of neutral fields. For these cases, we find a sigma model geometry
which is both non-toric and includes brane sources. These are heterotic sigma
models with branes. Surprisingly, there are massive models with compact complex
non-Kahler target spaces, which include brane/anti-brane sources. The simplest
conformal models describe wrapped heterotic NS5-branes. We present examples of
both types.Comment: 36 pages, LaTeX, 2 figures; typo in Appendix fixed; references added
and additional minor change
Exceptional del Pezzo hypersurfaces
We compute global log canonical thresholds of a large class of quasismooth
well-formed del Pezzo weighted hypersurfaces in
. As a corollary we obtain the existence
of orbifold K\"ahler--Einstein metrics on many of them, and classify
exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted
hypersurfaces in .Comment: 149 pages, one reference adde
Weakly--exceptional quotient singularities
A singularity is said to be weakly--exceptional if it has a unique purely log
terminal blow up. In dimension , V. Shokurov proved that weakly--exceptional
quotient singularities are exactly those of types , , ,
. This paper classifies the weakly--exceptional quotient singularities
in dimensions and
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