The workshop will focus on

• adapted complex structures / Grauert tubes
• hyperkähler structures on (co)tangent bundles
• positivity notions for vector bundles (especially tangent bundles)
• adapted complex structures and Kählerian reduction
• Bergman/Szego kernels and geometry of Grauert tubes
and aims at stimulating interaction between researchers approaching these topics from different angles and with different techniques.

## Speakers

Roger Bielawski
Judith Brinkschulte
Robert Chang
Xiaoyang Chen
Alix Deleporte
Laura Geatti
Hendrik Herrmann
Michael Hitrik
Andreas Höring
Andrea Iannuzzi
Masataka Iwai
Stefan Kebekus
Shin-ichi Matsumura
Thomas Peternell
Robert Szőke
John Toth
Xiaojun Wu
Zhixin Xie
Steven Zelditch

## Abstracts

#### Masanori Adachi: On weighted Bergman spaces of a domain with Levi-flat boundary

The maximal Grauert tube of a closed hyperbolic surface can be realized as a domain with Levi-flat boundary in a ruled surface. It is known that this domain does not admit non-constant bounded holomorphic function and its Hardy space is trivial. Nevertheless, we show that its weighted Bergman spaces are infinite dimensional for any weight order. We also discuss the comparison between the Grauert tube and its complementary domain in this realization. Although they are only "half" biholomorphic, we observe that they share common complex analytic properties.

#### Roger Bielawski: The hyperkähler geometry of $T^*G$ and generalisations

Kronheimer's hyperkähler metric on the cotangent bundle of a complex reductive Lie group is one of the basic examples of the Feix-Kaledin construction, and one of the few where the resulting hyperkähler metric is complete. I shall describe various ways of understanding this metric, and what they tell us about the hyperkähler geometry of the Moore-Tachikawa varieties arising in the 2d TQFT.

#### Judith Brinkschulte: On meromorphic connections and exceptional minimal sets of holomorphic foliations

We present a proof of Brunella’s conjecture concerning exceptional minimal sets of holomorphic foliations based on a residue formula for meromorphic connections of a holomorphic line bundle whose curvature extends across a simple normal cros sing divisor. This is joint work with M. Adachi and S. Biard.

#### Robert Chang: Szego kernel asymptotics on Grauert tubes

The Grauert tube $M_\tau$ of a real analytic Riemannian manifold $(M, g)$ is a strongly pseudoconvex domain whose defining function is derived from the Riemannian metric $g$ . In particular, the complexified Riemannian exponential map is a $C^\omega$ diffeomorphism between $M_\tau$ and the co-ball bundle consisting of co-vectors of length at most $\tau$. Grauert tubes therefore offer a bridge between the setting of geometric quantization (where the dual disk bundle is a strongly pseudoconvex domain) and the setting of standard quantization on Riemannian manifold. In this talk, I will discuss my joint work with Abraham Rabinowitz (Northwestern) on the asymptotic expansion of the Szego kernel on Grauert tubes. As a corollary, we derive sharp $L^p$ estimates on the mapping norm of the Szego kernel, as well as on complexified Laplace eigenfunctions (Husimi distributions).

#### Xiaoyang Chen: Rational ellipticity of Riemannian manifolds

It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold with nonnegative sectional curvature is rationally elliptic, i.e., it has finite dimensional rational homotopy groups. We will discuss some recent progress on this conjecture.

#### Alix Deleporte: Semiclassical Bergman kernels in analytic regularity

There is a recent surge of activity towards the understanding of the asymptotics of Bergman kernels over a Kähler manifold (in a large exponential weight/large curvature/large tensor power limit) in the case of real-analytic regularity of the data. Using tools from analytic semiclassical analysis, one can provide an asymptotic formula for the Bergman kernel up to an exponentially small error. In this talk, I will review the basic techniques of analytic semiclassical analysis, the different strategies that have been used to establish this result, and its different applications.

#### Laura Geatti: The adapted hyper-Kähler structure on  the tangent bundle of a Hermitian symmetric space

The cotangent bundle of a compact Hermitian symmetric space $X=G/K$ (a tubular neighbourhood of the zero section, in the non-compact case) carries a unique $G$-invariant hyper-Kähler structure compatible with the Kähler structure of $X$ and the canonical complex symplectic form of $T^*X$ (cf. Biquard and Gauduchon). The tangent bundle $TX$, which is isomorphic to $T^*X$, carries a canonical complex structure $J$, the so called adapted complex structure", and admits a unique $G$-invariant hyper-Kähler structure compatible with the Kähler structure of $X$ and with $J$. The two hyper-Kähler structures are related by a $G$-equivariant fiber preserving diffeomorphism of $TX$ (cf. Dancer and Szöke). The fact that the domain of existence of $J$ in $TX$ is biholomorphic to a $G$-invariant domain in the complex homogeneous space $G^{\bf C}/K^{\bf C}$ enables us to use Lie theoretical tools and moment map techniques to explicitly compute the various quantities of the adapted hyper-Kähler structure". This is joint work with Andrea Iannuzzi.

#### Hendrik Herrmann: Embedding Results for CR Manifolds of High Codimension

Any compact strongly pseudoconvex codimension one CR manifold of dimension greater than or equal to five is CR embeddable into the complex Euclidean space by a classical result due to Boutet de Monvel. Based on the methods used in his work we present an embedding result for the high codimension case under the hypothesis of a CR Lie group action. This is a joint work with Kevin Fritsch and Chin-Yu Hsiao.

#### Michael Hitrik: Positivity, complex FIOs, and Toeplitz operators

We study Toeplitz operators acting on Bargmann spaces, with Toeplitz symbols that are exponentials of complex quadratic forms, from the point of view of Fourier integral operators in the complex domain. We show that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols, in agreement with the Berger-Coburn conjecture, relating Toeplitz and Weyl quantizations. Sufficient conditions are also established for the composition of two such operators to be a Toeplitz operator. This is joint work with Lewis Coburn and Johannes Sjöstrand.

#### Andreas Höring: The cotangent bundle of a $K3$ surface of degree two

$K3$ surfaces of degree two are among the most well-understood complex manifolds, they can be constructed as a double cover of $P^2$ along a smooth sextic. Nevertheless the positivity properties of their cotangent bundle are rather mysterious, for example we don't know how to compute the pseudo effective threshold (which I will introduce in the talk). In this talk I will present the surprisingly intricate geometry of the projectivised cotangent bundle and explain the central role of a certain covering family of singular genus one curves.

#### Andrea Iannuzzi: Maximal adapted complexifications of homogeneous Riemannian manifolds

The maximal domain for the existence of the adapted complex structure in the tangent bundle of a symmetric space is a Riemannian invariant which was firstly detected by Akhiezer and Gindikin. In the context of homogeneous Riemannian manifold, under certain extensibility assumptions on the geodesic flow such a maximal adapted complexification" can be explicitly determined. The classes of generalized Heisemberg Lie groups and of naturally reductive homogeneous manifolds meet such assumptions and provide interesting examples. Their explicit realization show that maximal adapted complexifications may not enjoi nice complex-analytic properties, e.g. need not be Stein. From a collaboration with Stefan Halverscheid at the beginning of the century.

#### Masataka Iwai: Abundance theorem for minimal compact Kahler manifolds with vanishing second Chern class

In this talk, I will explain the structure of generically nef vector bundles with nef first Chern class and vanishing second Chern class. As an application, I will talk about the abundance theorem and structure theorem for compact Kahler manifolds with nef canonical line bundles and vanishing second Chern class. This is a joint work with Shin-ichi Matsumura.

#### Stefan Kebekus: An Albanese construction for Campana's C-pairs

Almost twenty years ago, Campana introduced C-pairs to complex geometry. Interpolating between compact and non-compact geometry, C-pairs capture the notion of "fractional positivity" in the "fractional logarithmic tangent bundle". Today, they are an indispensible tool in the study of hyperbolicity, higher-dimensional birational geometry and several branches of arithmetic geometry. This talk reports on joint work with Erwan Rousseau. Aiming to construct a "C-Albanese variety", we clarify the notion of a "morphism of C-pairs" and discuss the beginnings of a Nevanlinna theory for "orbifold entire curves".

#### Shin-ichi Matsumura: Structure theorem for projective klt pairs with nef anti-canonical divisor

In this talk, I would like to discuss structures of projective klt pairs with nef anti-log canonical divisor. After reviewing some related topics, including the successive works by Cao, Höring, Paun, and Zhang,  I will explain a structure theorem for maximal rationally connected fibrations of such projective klt pairs, which decomposes them into Fano-like varieties and Calabi-Yau varieties.  In particular, this talk will focus on the meaning and applications of this structure theorem. This talk is based on joint work with Juanyong WANG.

#### Thomas Peternell: Stein and affine complements in projective manifolds

Given a projective manifold and a smooth hypersurface $Y$ in $X$, we study conditions under which the complement $X \setminus Y$ is Stein or affine. This is particularly interesting when X is the projectivization of the so-called canonical extension of the tangent bundle of a projective manifold $M$ and $Y$ being the projectivization of the tangent bundle of $M$. There is an interesting conjectural connection to the nefness of the tangent bundle of $M$. This is joint work with Andreas Höring.

#### Robert Szőke: Generalizations of a theorem of Wiegerinck

A theorem of Wiegerinck says that the Bergman space over any domain in the complex plane is either trivial or infinite dimensional. In the talk we discuss possible generalizations of this theorem when D is a domain in a compact Riemann surface.

#### John Toth: Bounds for Steklov eigenfunctions on compact, real-analytic manifolds with boundary

Let $(\Omega,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial \Omega.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions to the interior of $\Omega$ are called Steklov eigenfunctions. In the talk, I will describe recent results (joint with J. Galkowski) on sharp exponential upper and lower bounds for Steklov eigenfunctions in the interior of $\Omega.$

#### Xiaojun Wu: Pseudo-effective vector bundle and numerical flatness

In this talk, I will discuss the generalisation of the notion of pseudo-effective line bundle to the higher rank case. In particular, I will talk about the proof of the following result: a strongly pseudo-effective vector bundle over a compact Kähler manifold with vanishing first Chern class is numerically trivial. The proof is based on a natural construction of closed positive current in the first Chern class called Segre current. The projective case of this result has been proven by Campana-Cao-Matsumura and Hosono-Iwai-Matsumura. As a geometric application, the tangent bundle or cotangent bundle of a Calabi-Yau manifold or a symplectic irreducible holomorphic manifold is not strongly pseudo-effective. If times permits, I will talk about some more recent work.

#### Zhixin Xie: The Nonvanishing problem for varieties with nef anticanonical bundle

Let $X$ be a complex projective threefold with mild singularities. The result of Miyaoka from the 80s showed that if the canonical bundle $K_X$ is nef, then some multiple of $K_X$ is effective. In this talk, we will discuss an analogous result for the anicanonical bundle $-K_X$, which is the semipositive curvature counterpart of Miyaoka's result towards the Nonvanishing problem. We will prove that if $-K_X$ is nef, then the numerical class of $-K_X$ is effective. This is joint work with Vladimir Lazić, Shin-ichi Matsumura, Thomas Peternell and Nikolaos Tsakanikas.

#### Steven Zelditch: Husimi distributions on Grauert tubes and their sup norms

In complex geometry, in the setting of positive Hermitian holomorphic line bundles, Husimi distributions are just norm-squares $|s(z)|_h^2$ of holomorphic sections. Many papers are concerned with their sup-norms and weak limits as one takes powers of the line bundle. The analogue in the Grauert tube setting is to analytically continue Laplace eigenfunctions of a Riemannian manifold $(M,g)$ to the Grauert tube and to L2 normalize it. It then defines a positive measure. The square root of the eigenvalue is analogous to the power of the line bundle. The main results give universal bounds on the sup norms, and more significantly, necessary conditions under which they are achieved. They are derived from a 2-term pointwise Weyl law for sums of Husimi distributions, of a type initiated by Y. Safarov. The second term is dynamical and we show that it is maximal only along an elliptic closed geodesic. This result can be integrated over the fibers of the natural projection to M, and shows that there must exist a positive measure of closed geodesics going through a point in order that this pushforward be of maximal size. The pushforward is not the square of the eigenfunction but the result is related to the long-standing conjecture that eigenfunctions only obtain the universal sup norm bound at points through which pass a positive measure of closed geodesics.