Speakers & Abstracts

Title: A residue formula for meromorphic connections and applications to stable sets of foliations

Abstract: We define the residues of a holomorphic connection of a holomorphic line bundle along a simple normal crossing divisor in a complex manifold by only assuming that its curvature extends across the divisor. We then prove a residue formula that localizes the first Chern class to the singular locus of the given holomorphic connection. As applications, we discuss proofs for a nonexistence theorem of Levi flat hypersurfaces with transversely affine Levi foliation in compact Kaehler surfaces and Brunella's conjecture about exceptional minimal sets of codimension one holomorphic foliations with ample normal bundle.
This is joint work with S. Biard and J. Brinkschulte.

Title: Harmonic embeddings of open Riemann surfaces into R^4

Abstract: We shall prove that every open Riemann surface admits a proper harmonic embedding into R^4. This reduces by one the previously known embedding dimension in this framework, due to Greene and Wu and dating back to 1975. This is a joint work with Francisco J. López.

Title: Tame sets in homogeneous spaces

Abstract: The notion of a tame set has been introduced by Rosay and Rudin in 1988 for closed discrete sets in complex-Euclidean space. Recently, the notion of a tame set has been generalized to complex manifolds. We prove the existence of tame sets in affine algebraic homogeneous spaces of complex linear algebraic Lie groups. Joint work with R. Ugolini.

Title: Gromov hyperbolicity methods in holomorphic iteration

Abstract: Karlsson proved in 2001 that the classical Denjoy-Wolff theorem about iteration of holomorphic functions in the unit disc can be generalized to the setting of nonexpanding maps of Gromov hyperbolic metric spaces. In this talk we will consider the same setting, and we will discuss the interplay between the horofunction compactification and the Gromov compactification. Applying our results to the case of strongly pseudoconvex domains and of convex domains of D’Angelo finite type we will obtain the existence of horospheres, a Julia Lemma, and an analog of the Denjoy-Wolff theorem for backward iteration. This is based on joint works with Fiacchi-Gontard-Guerini and Fiacchi-Guerini-Karlsson.

Title: On Picard Theorems over the Quaternions and the Octonions

Abstract: The classical theorem of Picard states that a non-constant holomorphic function $f: \mathbb{C} to \mathbb{C}$ can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable $f : \mathbb{H} \to \mathbb{H}$, or, of an octonionic variable, $f : \mathbb{O} \to \mathbb{O}$ may avoid.

Title: New developments on Oka manifolds

Abstract: I shall describe recent developments in the theory of Oka manifolds, focusing on new examples of Oka domains in Euclidean and projective spaces. In particular, I will present recent joint work with Erlend F. Wold in which we found surprisingly small Oka domains in complex Euclidean spaces of dimension n>1 at the very limit of what is possible. Under mild geometric assumptions on a closed unbounded convex set E in Cn we show that the complement of E is an Oka domain; this holds in particular if E does not contain any affine real line. Hence, there are Oka domains which are only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic.

Title: Characterization of the unit ball via properties of the Kobayashi metric.

Abstract: We prove that the Kobayashi metric of a bounded strictly pseudoconvex domain in Cd, d>1, is K\ahler if and only if the universal cover of the domain is biholomorphic to the unit ball in $\mathbb C^d$. This result is a consequence of a more general result concerning the extension of a locally symmetric Hermitian metric, defined outside a compact set of a Stein manifold with complex dimension greater than one. This is a joint work with Andrew Zimmer.

Title: Polynomially convex embeddings of compact real manifolds

Abstract: A compact subset of Cn is polynomially convex if it is defined by a family of polynomial inequalities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known. In this talk, we will discuss the methods used to obtain the best known bound on this quantity, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.

Title: Discrete and continuous versions of the continuity principle

Abstract: In this talk we shall present a certain generalization of the classical Kontinuit ̈atssatz of Behnke for holomorphic/meromorphic functions in terms of their lift to the envelope of holomorphy. We consider two non-equivalent formulations: “discrete” and “continuous” ones. Giving a proof of the “discrete” version we, somehow unexpectedly, construct a counterexample to the “continuous” one when convergence/continuity of analytic sets is considered in Hausdorff topology or, even in the stronger topology of currents. But we prove the “continuous” version of the Kontinuit ̈atssatz if continuity is understood with respect to the Gromov topology. Our formulations seem to be not yet existing in the literature. A number of relevant examples and open questions will be given as well.

Title: Holomorphic Linearization

Abstract: The holomorphic linearization problem asked whether a reductive group G acting holomorphically on Cn is conjugate in the holomorphic automorphism group Aut(Cn) to a subgroup of the general linear group GLn (C). In that case we would call the action of G linearizable. We give some overview over that subject starting with positive results, most importantly our work with Heinnzner, explaining the counterexamples found with Derksen, and going over to recent positive results jointly with Larusson-Schwarz, Schwarz and Kuroda-Pelka. The proofs of all positive results depend on establishing certain Oka Principles.

Title: A new family of compact non-Kähler manifolds

Abstract: In my talk I will explain the construction of new compact complex non-Kähler manifolds that generalize Inoue surfaces of type S_N^+. This is joint work with Karl Oeljeklaus.

Title: On rational convexity of totally real sets

Abstract: Totally real sets are defined as zero loci of nonnegative strictly plurisubharmonic functions. Locally such sets are contained in totally real manifolds. Under a mild technical assumption, we prove a necessary and sufficient condition for a compact totally real set in Cn to be rationally convex. This generalizes a classical result of Duval–Sibony.

Title: Hyperbolicity of some unbounded domains in Cn

Abstract: The main topic of the talk is Kobayashi and Bergman hyperbolicity of some natural classes of unbounded domains in ${\mathbb C}^n$. In particular, the question of dependence of Kobayashi hyperbolicity of Model domains (i.e. pseudoconvex domains of the form $\mathfrak{A} = \{(z, w) \in \mathbb C^{n-1}_z \times \mathbb C_w : v >F(z, u)\}, w = u +iv$) on the dimension $n$ of the ambient space will be discussed. If the time permit, we will also indicate the construction of a smoothly bounded strongly pseuodoconvex domain in ${\mathbb C}^2$ which is Kobayashi and Bergman hyperbolic and complete, but has no nonconstant bounded holomorphic functions.

Title: Géométry of the Levi flat singularitiés
Abstract: TBA

Title: Dolbeault and Bott-Chern harmonic forms on almost-Hermitian manifolds

Abstract: On a complex manifold $X$ the exterior derivative $d$ decomposes as the sum of two differential operators, namely $d=\partial+\overline\partial$ satisfying $\partial^2=0$, $\overline\partial^2=0$ and $\partial\overline\partial+\overline\partial\partial=0$.
Once an Hermitian metric on $X$ is fixed one can define several natural elliptic differential operators, e.g., the Dolbeault Laplacian $\Delta_{\overline\partial}$ and the Bott-Chern Laplacian $\Delta_{BC}$; if $X$ is compact the kernel of such operators has a cohomological interpretation, i.e., it is isomorphic to the Dolbeault and Bott-Chern cohomology of $X$, respectively. If we do not assume the integrability of the almost-complex structure, i.e., $(X,\,J)$ is an almost-complex manifold, $\Delta_{\overline\partial}$ and $\Delta_{BC}$ are still well-defined and elliptic but they have no more a cohomological meaning. In particular, the dimension of their kernels depends on the metric.
We will discuss some new results concerning such operators and the associated spaces of harmonic forms on compact almost-Hermitian manifolds.

These are joint works with Andrea Cattaneo, Riccardo Piovani and Adriano Tomassini.

Title: Entropy of maps on non-Archimedean fields

Abstract: For complex dynamics of holomorphic and meromorphic maps over compact Kahler manifolds, there are connections between topological entropy and cohomological invariants (Gromov-Yomdin's equality and Dinh-Sibony's inequality). For dynamics of rational maps over non-Archimedean fields, and the corresponding maps on the associated Berkovich spaces, Dinh-Sibony's inequality still holds but surprisingly many interesting and non-trivial maps have zero topological entropy. This talk will present results obtained exploring these topics, joint with Charles Favre and Junyi Xie.