Schedule for: 16w5080 - Complex Analysis and Complex Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Minimal hulls and minimally convex domains were introduced in a series of papers by Harvey and Lawson. They are natural substitutes for polynomial hulls and strictly pseudoconvex domains in the context of minimal surface theory. We present a characterization of the minimal hull of a compact set $K$ in $\mathbb R^n$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense. We also study some properties of minimally convex domains. This is a report on a joint work with Alarc\'on, Forstneri\v c and L\'opez.

For holomorphic line bundles, it has turned out to be useful to not just consider smooth metrics, but also singular metrics which are not necessarily smooth, and which can degenerate. In relation to vanishing theorems and other properties of the line bundle, one considers plurisubharmonicity properties of the possibly singular metric which correspond to notions of positivity for the line bundle. In particular, having a positive singular metric means that the first Chern form associated to the metric is a closed positive \((1,1)\)-current. More recently, singular metrics on holomorphic vector bundles have been considered, Griffiths positivity of a singular metric on a vector bundle is defined in terms of plurisubharmonicity. For a vector bundle with a Griffiths positive singular metric, there is a naturally defined first Chern class which is a closed positive \((1,1)\)-current, but there are examples where the full curvature matrix is not of order 0. I will discuss joint work with Hossein Raufi, Jean Ruppenthal and Martin Sera, where we show that one can give a natural meaning to the k:th Chern form \(c_k(h)\) of a singular Griffiths positive metric h as a closed \((k,k)\)-current of order 0, as long as h is non-degenerate outside a subvariety of codimension at least \(k\). The proof builds on pluripotential theory, and in particular, one consider in the spirit of Bedford-Taylor products like \((dd^c \varphi)^q \wedge T\), where \(\varphi\) is plurisubharmonic and \(T\) is a closed positive \( (q,q) \)-current.

Studying the (long-term) behaviour of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of "degenerate parabolic complex Monge-Amp\`ere equations". The purpose of this work, is to develop a viscosity theory for degenerate complex Monge-Amp\`ere flows on compact K\"ahler manifolds. The main ingredient is the "parabolic comparison principle" which allows us to prove uniqueness of the solution to a general complex Monge-Amp\`ere flow starting from a singular metric with bounded potentials in a given K\"ahler class. Then using our previous results on the "degenerate elliptic side" of the complex Monge-Amp\`ere theory, we are able to construct barriers that allow us to prove the existence of a solution to the Cauchy problem by means of the classical method of Perron. Our general theory allows in particular to define and study the behaviour of the (normalized) K\"ahler-Ricci flow on projective varieties with canonical singularities, generalizing results of Song and Tian. In the case when the variety is Calabi-Yau or of general type, we prove that the K\"ahler-Ricci flow converges weakly in the sense of currents (strongly at the level of potentials) to the unique K\"ahler-Einstein metric on the the variety. The case of intermediate Kodaira dimension is more tricky and will be briefly sketched if time permits. This is a joint work with P. Eyssidieux and V. Guedj which will appear in Advances in Math.

In order to study the forward or backward iteration of a holomorphic self-map \(f\) of a complex manifold \(X\), it is natural to search for a semi-conjugacy of \(f\) with some automorphism of a complex manifold. Examples of this approach are given by the Schroeder, Valiron and Abel equation in the unit disc \(D\). Given a holomorphic self-map \(f\) of the ball \(B^q\), we show that it is canonically semi-conjugate to an automorphism (called a canonical model) of a possibly lower dimensional ball \(B^k\), and this semi-conjugacy satisfies a universal property. This approach unifies in a common framework recent works of Bracci, Gentili, Poggi-Corradini, Ostapyuk. This is done performing a time-dependent conjugacy of the autonomous dynamical system defined by \(f\), obtaining in this way a non-autonomous dynamical system admitting a relatively compact forward (resp. backward) orbit, and then proving the existence of a natural complex structure on a suitable quotient of the direct limit (resp. subset of the inverse limit). As a corollary we prove the existence of a holomorphic solution with values in the upper half-plane of the Valiron equation for a hyperbolic holomorphic self-map of \(B^q\).

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

We consider the question of $L^2$-estimates for the $\overline{\partial}$-problem on annuli, a simple but interesting class of non-pseudoconvex domains. We relate this question with $W^1$-Sobolev estimates on the "hole" of the annulus. We then consider special classes of non-smooth holes for which the questions can be answered. This is joint work with Mei-Chi Shaw and Christine Laurent-Thiébaut.

In this talk, I will present several universality principles concerned with zero distribution of random polynomials or more generally random holomorphic sections of high powers $L^{\otimes n}$ of positive line bundle $L\to X$ defined over a projective manifold endowed with a continuous metric. In one direction, universality phenomenon indicates that under natural assumptions, asymptotic distribution of (appropriately normalized) zeros of random polynomials is independent of the choice of probability law defined on random polynomials. Another form of universality is asymptotic normality of smooth linear statistics of zero currents. Finally, if time permits, I will also describe some recent results on universality of scaling limits of correlations between simultaneous zeros of random polynomials.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is generally believed that (up to biholomorphism) very few domains have a large automorphism group and a nice boundary. For instance the Wong-Rosay Ball theorem says that a strongly pseudoconvex domain with non-compact automorphism group must be bi-holomorphic to the ball. Later, Bedford and Pinchuk proved that a convex domain of finite type and non-compact automorphism group must be bi-holomorphic to a domain defined by a polynomial. I will discuss a recent result which removes the finite type condition from the Bedford-Pinchuk result but at the cost of assuming that the automorphism group is slightly larger than non-compact. In particular, a smoothly bounded convex domain is biholomorphic to a domain defined by a polynomial if and only if an orbit of the automorphism group accumulates on at least two different complex faces of the set. The proof of this result combines rescaling arguments and ideas from the theory of metric spaces of non-positively curvature.

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\mathbb C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus).

Let $M$ be an open Riemann surface. It was proved by Alarc{\'o}n and Forstneri{\v c} that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. We prove the following substantially stronger result in this direction: for any $n\ge 3$, the inclusion of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that the above inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences. (Joint work with Finnur L{\'a}russon.)

The celebrated Gromov's non-squeezing theorem of 1985 says that the unit ball $B^n$ in $C^n$ can be symplectically embedded in the "cylinder" $rB^1 \times C^{n-1}$ of radius r only if $r\ge 1$. Hamiltonian differential equations provide examples of symplectic transformations in infinite dimension. Known results on the non-squeezing property in Hilbert spaces cover compact perturbations of linear symplectic transformations and several specific non-linear PDEs, including the periodic Korteweg - de Vries equation and the periodic cubic Schr\"odinger equation. We prove a new version of the non-squeezing theorem for Hilbert spaces. We apply the result to the discrete nonlinear Schr\"odinger equation. This work is joint with Alexander Sukhov.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We extend the basic sheaf-theoretic techniques of complex function theory to work within some algebras of holomorphic functions (joint with Alex Brudnyi)

Motivated by the observation that every continuous complex-valued function on the unit circle can be approximated by rational combinations of a single function, we will discuss some conditions under which a manifold \(M\) admits \(N\) functions whose rational combinations are dense in the space of complex-valued \(C^k\)-functions on M. As a result, we will produce an optimal bound on \(N\) in terms of the dimension of M. This is joint work with R. Shafikov.

Let $p: Z \rightarrow Z$ be a submersion from a complex manifold $Z$ to a $1$-convex manifold $X$ with an exceptional set $X.$ Let $E \rightarrow Z$ Let $a : X \rightarrow Z$ be a holomorphic section. Then there exist a conic neighbourhood $U$ of $a(X \setminus S)$ such that $U$ admits a K\"ahler metric and a metric on $E_U$ with positive Nakano curvature and with at most polynomial poles over $p^{-1}(S).$

Let $H^\infty$ be the Banach algebra of bounded holomorphic functions on the open unit disk $D\subset \mathbb C$. We extend Sundberg’s theorem on uniform approximation of functions in BMOA by $H^\infty$ functions to other classes of holomorphic functions on $D$. In our proofs we use a new characterization of meromorphic functions on $D$ that extend to continuous maps of the maximal ideal space of $H^\infty$ to the Riemann sphere.

Rescaling methods are very efficient in complex analysis or geometry because they can be combined with the theory of normal families. We will survey some typical examples of such methods and in particular those introduced by Sergey Pinchuk.

Let $W$ be a domain in a complex manifold $M$. In 2008 B. J\"oricke found a way to extend holomorphic functions from $W$ to another manifold and show that it is the envelope of holomorphy of $W$ and in 2013 F. L\'arusson and the speaker used a similar approach to subextend plurisubharmonic functions from $W$ to a complex manifold. To define these manifolds the authors considered the space $A(W,M)$ of analytic disks in $M$ whose boundaries lie in $M$. The new manifolds were defined as the quotients of this space by equivalence relations, where equivalent analytic disks can be connected by a continuous path or a homotopy in $A(W,M)$. In 1983 L. Rudolph introduced quasipositive elements of braid groups that are fundamental groups of the complements to some set W of planes in Cn. He proved that these elements are boundaries of analytic disks in $A(W,\mathbb C^n)$ and form a semi- group. The talk will be divided in two parts. In the first part we will discuss general constructions of extensions of Riemann domains and subextensions of plurisubharmonic functions. In the second part we will address the notion of quasipositive elements in general situation and explain why they form a semigroup. An important question is whether this semigroup is embeddable into the fundamental group. That is equivalent of asking whether two analytic disks are homotopic as analytic disks when their boundaries are equivalent in the fundamental group. A similar problem was studied by M. Gromov and, recently, by F. Forstneri\v c and his colleagues for homotopies of submanifolds in elliptic manifolds. In our case the ambient manifold is hyperbolic and the answer is not known. In the special case is when $W$ is an analytic variety in $M$ we will show that this problem can be reduced to the problem involving only real disks.

Tuan Huynh, Ph.D. student in Orsay, obtained (IMRN 2015) examples of Kobayashi-hyperbolic hypersurfaces $\mathbb{ X}^n \subset \mathbb{P}^{n+1}(\mathbb{ C})$ of low degree $2n+2$ for $n = 2, 3, 4, 5$, and of degree $\frac{ (n+3)^2}{4}$ for $n \geqslant 6$. Song-Yan Xie, Ph.D. student in Orsay, established in 2015 the ampleness of cotangent bundles (jets of order $1$) to generic complete intersections $\mathbb{X}^n \subset \mathbb{P}^{n+c} (\mathbb{ C})$ of codimension $c \geqslant n$ with degrees $d_1, \dots, d_c \geqslant (n+c)^{(n+c)^2}$. This result answered fully a conjecture made by Debarre in 2005. The first part of the talk will present a variation of S.~Xie's proof, based on multidimensional resultants, which conducts to an improvement on the degree bound: $d_1, \dots, d_c \geqslant (n+c)^{n+c}$. In order to reach an effective generic ampleness result about higher order jet bundles, in link with Kobayashi's hyperbolicity conjecture, taking inspiration from Masuda-Noguchi (1996), the second part of the talk will focus on families of hypersurfaces $\mathbb{ X}^n \subset \mathbb{ P}^{n+1} ( \mathbb{ C})$ having homogeneous polynomial defining equations of the form: \[ 0 \,=\, \sum_{\alpha_0+\alpha_1+\cdots+\alpha_{n+1} \,=\, {\sf fmn}}\, A_{\alpha_0,\alpha_1,\dots,\alpha_{n+1}}(X)\,\, \big((X_0)^d\big)^{\alpha_0} \big((X_1)^d\big)^{\alpha_1} \cdots \big((X_{n+1})^d\big)^{\alpha_{n+1}}, \] with {\sl Fermat-Masuda-Noguchi index} ${\sf fmn} \geqslant n^2 + n$, and with polynomials $A_\bullet \big( X_0 \colon X_1 \colon \cdots \colon X_{n+1}\big)$ homogeneous of relatively low degree ${\sf deg}\, A_\bullet =: {\sf a} \geqslant n$, compared with the dominant degree $d \geqslant n^{n^2}$. Mainly, some appropriately truncated order-$n$ jet bundles will happen to admit (a wealth of) global holomorphic sections, by means of a new process of forming (huge) Macaulay-type matrices, thanks to an application of Hartogs' theorem, a bit similarly as was performed by Siu-Yeung (Invent. 1996) and by Siu (Invent. 2015). The end of the talk will conclude by presenting a link between the geometry of complex vector bundles and the first complete effective computations of CR curvatures of CR manifolds up to dimension $\leqslant 5$ performed by Samuel Pocchiola (ex-Ph.D. student in Orsay) and Masoud Sabzevari (Shahrekord).

This is a joint work with Filippo Bracci. We try to extend the Carath\'eodory prime ends theory in higher dimension, defining the "horosphere boundary" of complete hyperbolic (in the sense of Kobayashi) manifolds. We prove that a strongly pseudoconvex domain together with its horosphere boundary, endowed with the horosphere topology, is homeomorphic to its Euclidean closure, whereas the horosphere boundary of a polydisc is not even Hausdorff. As an application we study the boundary behaviour of univalent mappings.

We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic. This is a joint work with J.E. Fornaess.

Following Henri Poincare, numerous results in Dynamics establish the curious phenomenon saying that two smooth objects (e.g., vector fields), which can be transformed into each other by means of a formal power series transformation, can be also transformed into each other by a smooth map. This is a kind of analogue of Borel Theorem on smooth realizations of formal power series. In CR-geometry, similar phenomena hold for real-analytic CR-manifolds, and the usual outcome is that two formally equivalent CR-manifolds are also equivalent holomorphically. However, in our recent work with Shafikov we proved that there exist real-analytic CR-manifolds, which are equivalent formally, but still not holomorphically. On the other hand, in our more recent work with Lamel and Stolovitch we prove that the following is true: if two 3-dimensional real-analytic CR-manifolds are equivalent formally, then they are $C^\infty$ CR-equivalent. In this talk, I will outline the latter result.

I will give a brief introduction to the density property for Stein manifolds, which is a notion to express that a manifold has ``many'' holomorphic automorphisms. \smallskip Although large classes of Stein manifolds with the density property are known, e.g.\ most of the homogeneous spaces of Stein Lie groups, they include only very few surfaces, namely $\mathbb{C}^2$, $\mathbb{C} \times \mathbb{C}^\ast$ and the smooth Danielewski surfaces. The lack of examples of such surfaces is due to the absence of the so-called ``compatible pairs'' of complete vector fields, which are usually the main tool for proving the density property. \smallskip Smooth Gizatullin surfaces provide a good class of candidates for surfaces with the density property, and they include the examples mentioned above. The next natural step is the investigation of Gizatullin surfaces of type $[[0,0,-r_2,-r_3]], \; r_2, r_3 \geq 2$, which can be described by the equations \[ \left\{ \begin{array}{lcl} y u &=& x P(x) \\ x v &=& u Q(u) \\ y v &=& P(x) Q(u) \end{array} \right. \] in $\mathbb{C}^4$ with coordinates $(x,y,u,v)$, where $P$ and $Q$ are polynomials of degree $r_2-1$ resp.\ $r_3-1$. We establish the density property for smooth Gizatullin surfaces of this type and describe a dense subgroup of the identity components of their holomorphic automorphism groups. Joint work with Frank Kutzschebauch and Pierre-Marie Poloni.

The study of analytic discs attached to a totally real submanifold M of $\mathbb C^n$ leads to the consideration of a regular Riemann-Hilbert problem of a special form. Following this approach, Forstneric, and later on Globevnik, characterized the existence and dimension of a family of deformations of a given analytic disc attached to M in terms of certain indices. However, in case M admits some complex tangencies, the indices mentioned above are no longer well-defined and the Forstneric-Globevnik method falls apart. In this talk, I will focus on a class of such singular Riemann-Hilbert problems. We will see that they can be solved by a factorization technique that reduces them to regular Riemann-Hilbert problems with geometric constraints. In particular, we will deduce the existence of stationary type discs attached to finite type hypersurfaces.

I will discuss a joint work in progress with Mats Andersson, in which we study and develop a calculus for direct images of semi-meromorphic currents. In my talk I will focus on regularity properties of these and in particular show that the sheaf of such currents is stalkwise injective.

The standard CR structure on the three dimensional sphere can be deformed in such a way that the deformed structures have no (CR) umbilical points. A 1-parameter family of such deformations was essentially discovered by E.~Cartan (and later studied by Cap, Isaev, Jacobowitz). The CR manifolds in this family, however, cannot be embedded in $\mathbb C^2$. It is an open question whether the unit sphere can be perturbed in $\mathbb C^2$ such that no umbilical points remain on the perturbed CR manifolds. In this talk, we shall discuss an approach to this problem, and describe some recent results. One of the results that will be described guarantees stable (in a sense to be made precise in the talk) umbilical points on generic perturbations of ”almost circular type”. This complements a previous result by the speaker and Son Duong on existence of umbilical points on circular three-dimensional CR manifolds.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

We discuss optimal regularity of geodesics in the space of K\"ahler metrics of a compact K\"ahler manifold, as well as the space of volume forms on a compact Riemannian manifold. They are solutions of nonlinear degenerate elliptic equations: homogeneous complex Monge-Amp\`ere equation and Nahm's equation (introduced by Donaldson), respectively. The highest regularity one can expect is $C^{1,1}$.