## SCV seminar 2016

### Severine Biard: An introduction to the complex Green operator

**Friday, 16. December, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: With the closure of the range of the tangential CR operator in hands, we have a well-behaved L^2 theory and an associated Hodge decomposition. This allows me to introduce the complex Green operator, an operator similar to the d-bar Neumann operator for the tangential CR operator. Similar properties hold as well as the interest to study its compactness.

### Benedikt Magnússon: Random polynomials and global extremal functions — Probabilistic viewpoint

**Friday, 9. December, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: We will look at the probabilistic preliminaries needed to show how random polynomials converge to the global extremal function.

### Benedikt Magnússon: Random polynomials and global extremal functions — Pluripotential viewpoint

**Friday, 2. December, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: I will introduce the global extremal function and describe its properties, most importantly its connection with polynomials given by the Siciak-Zaharijuta theorem.

### Severine Biard: Lecture on CR submanifolds of hypersurface type – the boundary of a complex manifold, II

**Friday, 18. November, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: I will continue tomorrow on CR submanifolds of \(C^n\) of hypersurface type of CR dimension m-1.

After quickly recalling the main ingredients introduced last week, I will introduce the tangential CR complex on those CR submanifolds. Then I will give the idea of the proof of the one-sided complexification allowing to consider some of those CR submanifolds as the boundary of a complex manifold.

### Severine Biard: Lecture on CR submanifolds of hypersurface type – the boundary of a complex manifold

**Friday, 11. November, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: I will recall some background of Cauchy-Riemann (CR) geometry, mainly embedded CR submanifolds and the objects that are associated with them. Focusing on CR submanifolds of hypersurface type – a generalization of real hypersurfaces in \(C^n\) -, I will focus on an extrinsic point of view that relates the \(\overline partial_b\) complex to the \(partial_b\) complex on the ambiant \(C^n\).

In 2012, Baracco proved that such compact, smooth, orientable and pseudoconvex CR submanifolds can be seen as the boundary of a complex manifold. I will give the idea of the proof and one of the most important applications: the range of \(\overline partial_b\) is closed.

### Severine Biard: Nonexistence of smooth Levi-flat hypersurface with positive normal bundle in compact Kähler manifolds of dimension \(\geq 3\), II

**Friday, 28. October, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: Among results of nonexistence of Levi-flat hypersurface in CPn, \(\geq 3\), conjectured by D. Cerveau in 1993, there are some generalizations to compact Kähler manifolds, particularly the conjecture given by Marco Brunella in 2008: there is no smooth Levi-flat hypersurface such that the normal bundle to the Levi foliation is positive along the leaves in compact Kähler manifolds of dimension \(\geq 3\). In a joint work with Andrei Iordan, we obtained a positive answer to this conjecture by using \(L^2\)-weighted estimates for d-bar. I will introduce first the problem of nonexistence, then I will explain the idea of the proof of our result and I will finish by talking about interesting questions around this subject.

### Severine Biard: Nonexistence of smooth Levi-flat hypersurface with positive normal bundle in compact Kähler manifolds of dimension \(\geq 3\)

**Friday, 21. October, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: Among results of nonexistence of Levi-flat hypersurface in CPn, \(\geq 3\), conjectured by D. Cerveau in 1993, there are some generalizations to compact Kähler manifolds, particularly the conjecture given by Marco Brunella in 2008: there is no smooth Levi-flat hypersurface such that the normal bundle to the Levi foliation is positive along the leaves in compact Kähler manifolds of dimension \(\geq 3\). In a joint work with Andrei Iordan, we obtained a positive answer to this conjecture by using \(L^2\)-weighted estimates for d-bar. I will introduce first the problem of nonexistence, then I will explain the idea of the proof of our result and I will finish by talking about interesting questions around this subject.

### Auðunn Skúta Snæbjarnarson

**Friday, 21. October, 10:00-11:30, Tgv227 in Tæknigarður. **

### Severine Biard: On the Chen’s proof of the Demailly’s weak openess conjecture

**Friday, 7. October, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: I will explain the proof of the weak Openess conjecture based on the Chen’s paper: https://arxiv.org/pdf/1506.01146v2.pdf

The main ingredient is a weighted \(L^2\)-approximation theorem whose the proof is slightly different to the one in his paper.

### Auðunn Skúta Snæbjarnarson:

**Friday, 30. September, 10:00-11:30, Tgv227 in Tæknigarður. **

### Auðunn Skúta Snæbjarnarson:

**Monday, 26. September, 10:00-11:30, Tgv227 in Tæknigarður. **

### Severine Biard: On Diederich-Fornaess exponent and its application to L^2 estimates

**Friday, 16. September, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: A Diederich-Fornaess exponent for a domain \(\Omega\) is an exponent \(\eta\) such that \(-(-\rho)^{\eta}\) is a bounded strictly plurisubharmonic function for a defining function \(\rho\) of \(\Omega\). Such an exponent was introduced in 1977 by Diederich and Fornaess on smooth pseudoconvex domains in Stein manifolds. Its existence is related to the geometry of the ambiant space and of the domain and has many applications. I will focus on one such application: I will explain how this exponent allows to choose a less plurisubharmonic weight function via Berndtsson-Charpentier method, yielding \(L^2\) estimates for the d-bar equation on pseudoconvex domains.

### Mitja Nedic, Stockholm University: On the class of Nevanlinna function in two variables

**Friday, 9. September, 10:00-11:30, Tgv227 in Tæknigarður.**

Abstract: Nevanlinna functions are holomorphic functions mapping the poly-upper half-plane to the closed upper half-plane. In the case of one variable we have classical results due to Herglotz, Nevanlinna, Pick, Cauer and others regarding integral representations of such functions. In the case of several variables the most work has been done by Vladimirov in the 1970s, but his results are incomplete in the sense that they do not provide an “if and only if” characterization.

In this talk we will discuss the statement and proof of an integral representation that provides such an “if and only if” characterization. We will also look closer at the class of representing measure of Nevanlinna functions as well as some examples. In the end we will present some open problems and how they relate to applications of Nevanlinan functions in electromagnetic engineering.

### Ragnar Sigurðsson: The openness theorem

**Friday, 2. September, 10:00-11:30, Tgv227 in Tæknigarður.**

Abstract: In the lecture we will go through a proof of Guan and Zhou Math. Ann. 182 (2015), of the so called openness conjecture, which was originally stated by Demailly in 2000.