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

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 $\mathbb{C}$ 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

Extension and approximation of mm-subharmonic functions

Let $\Omega\subset \mathbb C^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial \Omega$. The aim of this paper is to find a characterization of the functions $f$ which can be extended to the inside to a $m$-subharmonic function under suitable assumptions on $\Omega$. We shall do so by using a function algebraic approach with focus on $m$-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of $m$-subharmonic functions

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

Regularity of Kobayashi metric

We review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.Comment: 14 pages, 8 figures - The previously announced main result had a gap. In this new version the corrected statement is given. To appear on the volume "Geometric Complex Analysis - Proceedings of KSCV 12 Symposium

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

Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.Comment: 40 pages, the constants in Theorems 1.1-1.8 have been modified by a multiplicative constant 1/2 ; v.2 is a final updat

Big Line Bundles over Arithmetic Varieties

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution

Intersecting Solitons, Amoeba and Tropical Geometry

We study generic intersection (or web) of vortices with instantons inside, which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1 supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1} \times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (Nf=Nc=1), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Ampere measure with respect to a plurisubharmonic function on (C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin function. The general form of the Kahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.Comment: 39 pages, 11 figure