
THE ABEL SYMPOSIUM 2013 

AbstractsYou can find a pdf file with the program and abstracts here.Eric BedfordAutomorphisms of blowups of projective spaceAbstract: We will discuss the existence of automorphisms and pseudoautomorphisms of complex manifolds which have positive entropy. Bo BerndtssonThe openness problem and complex BrunnMinkowski inequalitiesAbstract: The openness conjecture of DemaillyKollár says that if $u$ is a plurisubharmonic function, then the set of all real numbers $t$ such that $e^{tu}$ is locally integrable near a certain point, is open. I will give a proof of this and discuss the relation to complex BrunnMinkowski inequlities. Zbigniew BłockiHörmander's $\bar\partial$estimate, some generalizations and new applicationsAbstract: We will present some new applications of the classical Hörmander's $L^2$ estimate for the $\bar\partial$ equation. Among them the OhsawaTakegoshi extension theorem with optimal constant, onedimansional Suita conjecture, as well as Nazarov's approach to the BourgainMilman inequality in convex geometry. JeanPierre DemaillyOn the cohomology of pseudoeffective line bundlesAbstract: The lecture will present various results concerning the cohomology of pseudoeffective line bundles on compact Kähler manifolds, twisted with corresponding multiplier ideal sheaves. In case the curvature is strictly positive in the sense of currents, the prototype is the well known Nadel vanishing theorem. We are interested here in the case where the curvature is merely semipositive. Various results and applications will be discussed, including a recent vanishing theorem due to Junyan Cao (forthcoming PhD thesis in Grenoble), and a study of simple compact Kähler 3folds (joint work with F. Campana and M. Verbitsky from April 2013). TienCuong DinhPositive closed (p,p)currents and applications in complex dynamicsAbstract: I will present some recent progress in the study of positive closed currents of arbitrary bidegree: regularization, superpotential, density and intersection. Several applications to dynamics will be given: entropy estimates, properties of dynamical degrees and equidistribution. The talk is based on joint works with Nessim Sibony. Peter EbenfeltPartial rigidity of degenerate CR embeddings into spheresAbstract:
We shall consider degenerate CR embeddings $f$ of a strictly
pseudoconvex hypersurface $M\subset {\mathbb C}^{n+1}$ into a
sphere $ {\mathbb S}$ in a higher dimensional complex space $
{\mathbb C}^{N+1}$. The degeneracy of the mapping $f$ will be
characterized in terms of the ranks of the CR second fundamental
form and its covariant derivatives. In 2004, the speaker, together
with X.
Huang and D. Zaitsev, established a rigidity result for CR embeddings
$f$ into spheres in low codimensions. A key step in the proof of
this result was to show that degenerate mappings are necessarily
contained in a complex plane section of the target sphere (partial
rigidity). In the 2004 paper, it was shown that if the total rank
$d$ of the second fundamental form and all of its covariant
derivatives is $ Franc ForstneričComplex analysis and the CalabiYau problemAbstract: I shall describe how methods of complex analysis can be used to give new results on the conformal CalabiYau problem. I will show that every bordered Riemann surface admits a proper complete holomorphic immersion into the ball of $\mathbb{C}^2$, and a proper complete embedding as a holomorphic null curve in the ball of $\mathbb{C}^3$. Since the real and the imaginary parts of a null curve in $\mathbb{C}^3$ are conformally immersed minimal surfaces in $\mathbb{R}^3$, this gives a bounded complete conformal minimal immersion of any bordered Riemann surface into $\mathbb{R}^3$. The main advantage of our methods when compared to the existing ones is that we do not need to change the conformal type of the Riemann surface. (Joint work with A. Alarcon, University of Granada.) Samuel GrushevskyMeromorphic differentials with real periods, and the geometry of the moduli spaces of Riemann surfacesXiaojun HuangAnalyticity of the local hull of holomorphy for a codimension two realsubmanifold in $C^n$Abstract: We discuss the formal and analytic flattening for a real submanifold of codimension two in a complex Euclidean space near a CR singular point. This has an immediate application for obtaining the regularity result for the local hull of holomorphy or finding a Leviflat hypersurface with a prescribed realanalytic boundary. Joseph KohnWeakly Pseudoconvex CR ManifoldsAbstract: Let $\Omega\subset\mathbb C^n$ be a domain with a smooth pseudoconvex boundary $b\Omega$ which is weakly pseudoconvex at $P\in b\Omega$ (i.e. the Levi form is not positive definite at $P$). Then the regularity of solutions of $\bar\partial\varphi=\alpha$ near $P$, where $\varphi$ is a $(0,q)$form and $\alpha$ a $(0,q+1)$form, depends on the behavior of the germs of complex analytic varieties of dimension $q$ through $P$. If $U$ is a neighborhood of $P$ regularity regularity on $U\cap\bar\Omega$ is studied by means of subelliptic estimates of the ``energy'' form $$ Q(\varphi,\varphi)=\\bar\partial\varphi\^2+\\bar\partial^*\varphi\^2, $$ defined on $(q,0)$forms in $C^\infty(\bar\Omega)$ which are supported in $U\cap\bar\Omega$ and belong to the domain of $\bar\partial^*$. The subelliptic estimate holds if there exist constants $\varepsilon$ and $C$ such that $\\varphi\^2_\varepsilon\le CQ(\varphi,\varphi)$, for all such forms, where the left hand side denotes the $\varepsilon$ Sobolev norm. The D'Angelo $q$type of $P$ is the maximum order of contact that a complex $q$dimensional variety through $P$ can have with $b\Omega$. The main theorem, due to Catlin, is that a subelliptic estimate holds at $P$ if and only if the D'Angelo type of $P$ is finite. In the case of when $b\Omega$ is real analytic in a neighborhood of $P$ there is an equivalent necessary and sufficient condition for subellipticity, namely that the ideal type of $P$ is finite. This condition is expressed in terms of ideals of subelliptic multipliers, these consist of germs of funtions at P. When the ideal qtype is infinite the above subelliptic estimate does not hold. This follows from the fact that infinite ideal qtype of $P$ is equivalent to the existence of a qdimensional complex analytic variety through P that is contained in $b\Omega$. Here we present an explicit construction of such a variety which gives insight into the relation between the D'Angelo type and the ideal type. The study of these ideals have a direct application to the study of singularities of complex analytic varieties. Consider the variety $V\subset\mathbb C^{n1}$ given by $h_1(z_1\dots z_{n1})= \dots =h_m(z_1\dots z_{n1})= 0$, where the $h_j$ are holomorphic functions that vanish at the origin. Let $\Omega\subset\mathbb C^n$ be a pseudoconvex domain defined by $z\in\Omega\mid r(z)<0$ where in a neighborhood of the origin we have $$ r(z_1\dots z_n)= Re(z_n)+\sumh_j(z_1\dots z_{n1})^2. $$ The multiplier ideals at the origin are generated by ideals of germs of holomorphic functions which are invariants of V. Curtis McMullenEntropy and dynamics on complex surfacesAbstract: The log of Lehmer's number – a degree 10 algebraic integer, approximately 1.17628 – provides a lower bound on the entropy for all automorphisms of compact complex surfaces. We will discuss explicit constructions of automorphisms with minimal entropy and, more generally, the synthesis of *projective* K3 surfaces from small Salem numbers, using algebraic number theory, glue groups, integer linear programming and the Torelli theorem. Joël MerkerSiuYeung holomorphic sections of ${\sf Sym}^m T_X^*$Abstract: An $n$dimensional complete intersection complex projective algebraic $X^n \subset \mathbb P^{n+c}$ of codimension $c \geqslant n$ larger than its dimension is known to have a wealth of high order symmetric differentials that are everywhere holomorphic (no poles!), although such a (limited) cohomological knowledge happens to be ineffective. Drawing a guided inspiration from Siu and Yeung's seminal 1996 hyperbolicity paper, I will present a construction in which Geometry, Algebra and Combinatorics share their strengths. Ngaiming MokOn the Zariski closure of an infinite number of totally geodesic subvarieties of $\Omega/\Gamma$Abstract: Let $\Omega$ be a bounded symmetric domain, $\Gamma \subset \text{Aut}(\Omega)$ be a torsionfree lattice, $X := \Omega/\Gamma$. Let $Z \subset X$ be an irreducible quasiprojective variety such that $Z$ is the Zariski closure of the union of an infinite family of totallygeodesic complex subvarieties $S_\alpha \subset Z, \, \alpha \in A$. Under a nondegeneracy condition one expects $Z$ to be also totally geodesic, so that $Z$ is again uniformized by a bounded symmetric domain. This setup is related to a wellknown problem on Shimura varieties $X = \Omega/\Gamma$ in which one tries to characterize the Zariski closure of an infinite number of `special' subvarieties $S_\alpha$. Here special subvarieties are defined by arithmetic conditions, but it is known that they are always totally geodesic. While the case where $S_\alpha$ are 0dimensional, for which the problem is called the AndréOort Conjecture, cannot be dealt with directly using methods of complex geometry, the case where $S_\alpha$ are of positive dimension is naturally a problem in complex geometry. From our complexanalytic perspective, no arithmeticity assumption is placed on $\Gamma \subset \text{Aut}(\Omega)$, and the `distinguished' subvarieties are simply the totallygeodesic subvarieties of $X$. Using methods of Kähler geometry, we solve the aforementioned problem in the rank1 case. A generalization of the argument to bounded symmetric domains $\Omega$ leads to the study of holomorphic isometries from complex unit balls $B^m$ to $\Omega$. We explain a method along this line of thoughts for solving the general problem, and illustrate how the problem is solved when $Z$ is a complex surface and $S_\alpha \subset Z$ are totallygeodesic holomorphic curves using our recent results on holomorphic isometries with respect to the Bergman metric. Stefan NemirovskiTopology and Several Complex VariablesAbstract: The talk will be a survey of what is known about the topology of holomorphically, rationally, and polynomially convex domains. Takeo OhsawaLevi flats in Hopf surfacesAbstract: A compact Levi flat hypersurface in a complex manifold is said to be of qconcave type if it admits a neighborhood system consisting of qconcave manifolds in the sense of AndreottiGrauert. The real analytic Levi flat hypersurfaces of 1concave type in Hopf surfaces are classified. Nessim SibonyDynamics of foliations by Riemann surfacesAbstract: I will discuss some basic facts for dynamics of foliations (with singularities) by Riemann surfaces. I will emphasis ergodicty results à la Birkhoff and their relation with Nevanlinna theory. The lecture is based on joint works with J.E Fornaess, T.C Dinh and V.A Nguyen. Berit StensønesReal analytic domains and plurisubharmonic functionsAbstract: We study pseudoconvex domains in $\mathbb C^3$ with real analytic boundary. The focus will be on demonstrating how one can use the underlying structure to bump the domain out to type in different directions. Tetsuo UedaSemiparabolic fixed points and their bifurcations in complex dimension 2Abstract: I will talk about bifurcations of semiparabolic fixed points in dimension 2. The intrinsic structure of a semiattracting fixed point is investigated and this is used to explain the discontinuity of (filled) Julia sets for the Family of Hénon mappings. This talk is based on a joint work with Eric Bedford and John Smillie. Sidney WebsterLocal and global invariants of a complex cotangent line fieldAbstract: Such a field on a complex nmanifold is locally spanned by a smooth, nonzero, (1,0)form, satisfying a (1,0)integrability condition. Attached to the field is an invariant complex GodbillonVey cohomology class, provided a certain primary obstruction vanishes. Under an additional Levitype nondegeneracy condition, the field has a complete system of local biholomorphic invariants, which are derived from a connection on the principal bundle of adapted coframes. ShingTung YauPeriod Integrals, Counting Curves and Mirror SymmetryStephen YauNonconstant CR morphisms between compact strongly pseudoconvex CR manifolds and etale covering between resolutions of isolated singularitiesAbstract: Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities. We prove that any nonconstant CR morphism between two (2n1)dimensional strongly pseudoconvex CR manifolds lying in a ndimensional Stein variety with isolated singularities are necessarily a CR biholomorphism. As a corollary, we prove that any nonconstant self map of (2n1)dimensional strongly pseudoconvex CR manifold is a CR automorphism. We also prove that a finite etale covering map between two resolutions of isolated normal singularities must be an isomorphism. This is a joint work with YUCHAO TU and HUAIQING ZUO SaiKee YeungComplex hyperbolicity on the moduli of some higher dimensional manifolds.Abstract: The purpose of the talk is to explain some joint work with WingKeung To on the problem of complex hyperbolicity on families of some higher dimensional manifolds. XiangYu ZhouSome results on $L^2$ extension problem with optimal estimateAbstract: In this talk, we'll present some recent results about $L^2$ extension problem with optimal estimate, and give some applications.
