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 $\Omega_F=\{(z,w)\in \mathbb{C}^{n+1}:{\rm Im}w-|F(z)|^2>0\},$ where $F=(f_1,...,f_m)$ is a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^m$, in terms of the complex singularity exponent of $F$.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 $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. 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 $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.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 $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$