From Complex Analysis to Quantization

Title: On Levi-flat real hypersurfaces with positive normal bundle

Abstract: Two families of 3-dimensional Levi-flat hypersurfaces with positive normal bundles are known: certain flat circle bundles in ruled surfaces, and hyperbolic torus bundles in hyperbolic Inoue surfaces. These are of particular interest because they resemble the hypothetical Levi-flat hypersurface in the complex projective plane, whose existence remains an open question. In this talk I will report on mostly computational observations of these examples, including those related to weighted Bergman spaces.

Title: Dynamics of Hénon-like maps

Abstract: Hénon-like maps are invertible holomorphic maps, defined on some convex bounded domain of $\mathbb C^k$, that have (non-uniform) expanding behaviour in $p$ directions and contracting behaviour in the remaining $k-p$ directions. They form a large class of dynamical systems in any dimension. In dimension 2, they contain the Hénon maps, which are among the most studied dynamical systems. In this talk, I will give an overview of the main dynamical properties of these maps. In particular, I will focus on how tools from pluripotential theory can allow one to go beyond the algebraic setting of the Hénon maps. The talk is based on joint works with Tien-Cuong Dinh and Karim Rakhimov.

Title: The positivity of the relative canonical bundle in the Kähler setting and applications

Abstract: Let $p: X \to Y$ be a fibration between two compact Kähler manifolds, and assume that the canonical bundle of the generic fiber is pseudo-effective. An important conjecture states that the relative canonical bundle $K_{X/Y}$ is also pseudo-effective. In the first part, we will discuss progress toward this conjecture. The main tool is the variation of relative Bergman kernel. In the second part, we will present an application to the algebraicity of holomorphic foliations with positive minimal slope. The proof relies on the positivity of the relative canonical bundle and a recent breakthrough due to Wenhao Ou on the algebraic criterion in the Kähler setting. It is a joint with Mihai Paun.

Title: Exponential equidistribution of periodic points for endomorphisms of $\mathbb{P}^k$

Abstract: Let $f$ be a non-invertible holomorphic endomorphism of $\mathbb{P}^k$. We show that the periodic points equidistribute towards the equilibrium measure of $f$ exponentially fast as the period tends to infinity. This quantifies a theorem of Lyubich and of Briend-Duval. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers. The talk is based on a joint work with Henry de Thelin and Lucas Kaufmann.

Title: Zeros of square-integrable random holomorphic sections

Abstract: We present some recent progress regarding the distributions of the zeros of square-integrable random holomorphic sections from a probabilistic point of view. Our focus is on the asymptotic distributions of these zeros in the semi-classical limit, in particular, we prove equidistribution results, large deviation estimates, as well as a central limit theorem of the random zeros on the support of the given function. This talk is based on joint works with Bingxiao Liu and George Marinescu.

Title: Small eigenvalues of Toeplitz operators and Kähler geometry

Abstract: Alexander Drewitz, Bingxiao Liu and George Marinescu recently established that for an arbitrary non-negative function, which is not zero almost everywhere, the smallest eigenvalue of the associated Toeplitz operator decays at most exponentially with respect to the semiclassical parameter. They further raised the question of whether such exponential decay always occurs when the function is properly supported on the manifold. We answer this question affirmatively and go further by providing a precise description of the logarithmic asymptotics of the small eigenvalues of Toeplitz operators. This description draws on the theory of Mabuchi geodesics — originally developed in connection with constant scalar curvature Kähler metrics — and the concept of plurisubharmonic envelopes from complex pluripotential theory. The aim of the talk is to reveal how these tools from Kähler geometry naturally emerge in the asymptotic analysis of quantization phenomena.

Title: Distribution of Zeros of Random Holomorphic Functions on Strictly Pseudoconvex Domains

Abstract: We consider a relatively compact, smoothly bounded, strictly pseudoconvex domain of dimension greater than or equal to two. Given a contact form $\alpha$ on the boundary, let $T$ denote the CR Toeplitz operator associated with the Reeb vector field and a Reeb invariant volume form. We consider holomorphic functions on the domain arising from holomorphic extensions of random CR functions on the boundary, related to eigenvalues of $T$. It turns out that their zeros concentrate near the boundary according to $\alpha$ as the eigenvalues become large. This is a joint work with Chin-Yu Hsiao, George Marinescu and Wei-Chuan Shen.

Title: George Marinescu's mathematical work

Abstract: George Marinescu has made many significant and influential contributions in the fields of complex analysis, complex geometry, and geometric quantization. I In this talk, I will present some of his work in these areas from my perspective. I will also mention our recent collaboration, in which we introduced semi-classical analysis into several complex variables.

Title: Asymptotics of unitary matrix elements on the spaces of homogeneous polynomials

Abstract: In this talk, we will show how to estimate the matrix elements of the unitary group acting on the space of homogeneous polynomials as the degree tends to infinity. We will then explain how to obtain this result as a consequence of the off-diagonal asymptotic expansion of the Bergman kernel established by Ma and Marinescu.

Title: Projections of connected Stein open subsets

Abstract: We show that for every open surjective holomorphic map with infinite fibers $f: X \to Y$ between connected complex spaces, there exists a connected Stein open subset of $X$ whose image through $f$ is $Y$. Based on joined work with Mihnea Coltoiu.

Title: Quantum hall states on hyperbolic surfaces

Abstract: I will talk about many-body wave functions in quantum Hall effect, notably Laughlin states and their generalizations. We will construct Laughlin on compact higher genus surfaces using free fields and also talk about some recent results for integer QHE state on the hyperbolic cylinder, based on work to appear with P. Wiegmann.

Title: Semi-classical asymptotics of partial Bergman kernels on R-symmetric complex manifolds with boundary

Abstract: Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M’$. Let $(L,h) \rightarrow M’$ be a positive holomorphic line bundle over $M’$. Suppose $M’$ admits a holomorphic $R$-action which preserves the boundary of $M$ and lifts to $L$. We establish the asymptotic expansion of a partial Bergman kernel associated to a package of Fourier modes of high frequency with respect to the $R$-action in the high powers of $L$. As an application, we establish an $R$-equivariant analogue of Fefferman's and Bell-Ligocka's result about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains in $C^n$. Another application concerns the embedding of pseudoconcave manifolds. This talk is based on a joint work with George Marinescu and Chin-Yu Hsiao.

Title: Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

Abstract: Let $(X,\omega)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=\omega$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections. This is a joint work with Turgay Bayraktar, Dan Coman and George Marinescu.

Title: Harmonic analysis meets ergodic theory and Kähler geometry: rigidity of $\Gamma$-equivariant bounded holomorphic maps

Abstract: Let $\Omega$ be a bounded symmetric domain of rank $\ge 2$ and $\Gamma \subset {\rm Aut}(\Omega)$ be a torsion-free irreducible lattice, and write $X_\Gamma := \Omega/\Gamma$. Let $D \Subset \mathbb C^N$ be any bounded domain, $\Gamma'\subset {\rm Aut}(D)$ be a discrete subgroup such that $Y_{\Gamma'} := D/\Gamma'$ is of finite volume with respect to the Kobayashi-Royden volume form. Let $F: \Omega \to D \Subset \mathbb C^N$ be a holomorphic map which is $\Gamma$-equivariant with respect to a group homomorphism $\Phi: \Gamma \to \Gamma'$. In a joint work with Kwok-Kin Wong, we proved that $F: \Omega \to D$ must be a biholomorphic map provided that $\Phi: \Gamma \to \Gamma'$ is a group isomorphism. We call this the Isomorphism Theorem.
Denote by $\beta: \Omega\hookrightarrow \widehat{\Omega}$ the Borel embedding of $\Omega$ into its compact dual $\widehat{\Omega}$, which is a Hermitian symmetric space of the compact type, in particular a uniruled projective manifold. The proof of the Isomorphism theorem relies on harmonic analysis, ergodic theory and Kähler geometry. An irreducible bounded symmetric domain $\Omega$ of rank $r \ge 2$ can be realized as a Siegel domain holomorphically fibered over an irreducible bounded symmetric domain $\Omega_0$ of rank $r-1\ge 1$ corresponding to a partial Cayley transforms whose fibers are holomorphically and isometrically embedded copies of a complex unit ball $\mathbb B^{p+1}$ where $p$ is the dimension of the variety of minimal rational tangents (VMRT) $\mathscr C_0(\widehat{\Omega})$ of $\widehat{\Omega}$. To prove that $F: \Omega\overset{\cong}\longrightarrow D$ is a biholomorphism it suffices to be able to invert the holomorphic map. To do this we construct a holomorphic map $R: D \to \Omega$ such that $R\circ F$ is the identity map which implies that $F: \Omega \to F(\Omega) \subset D$ is a holomorphic embedding whose inverse we denote by $\varphi$ and that $R = \varphi\circ\varpi$ for a holomorphic retraction $\varpi: D \to F(\Omega)$. To construct $R$ we introduce an averaging process on bounded holomorphic functions on $\Omega$ belonging to ${\bf H} := F^*H^\infty(D)$. Harmonic analysis on the complex unit ball $\mathbb B^n$ enters when one studies non-tangential limits of restrictions to Cayley fibers $\cong \mathbb B^{p+1}$ of bounded holomorphic functions belonging to ${\bf H}$, and Moore's ergodicity theorem on semisimple groups allows us to (a) obtain elements of ${\bf H}$ of the form $\rho_{\Phi,\Psi}^*s$ where $s \in H^{\infty}(\Phi)$ is a bounded holomorphic function on some maximal face (boundary component) $\Phi \subset \partial\Omega$ biholomorphic to $\Omega_0$, and $\rho_{\Phi,\Psi}$ is the Cayley projection determined by a one-parameter hyperbolic flow from a maximal face $\Psi \subset \partial\Omega$ to $\Phi$ satisfying $\overline{\Psi} \cap \overline{\Phi} = \emptyset$, and (b) introduce an averaging process on ${\bf H}$ in order to reconstruct componentwise the identity map ${\it id}_\Omega$ from ${\bf H}$. Kähler geometry enters first in the realization that fibers of the Cayley projection are biholomorphic to the complex unit ball $\mathbb B^{p+1}$ in view of the asymptotic behavior of holomorphic (bi)sectional curvatures, and more significantly in the proof of the triviality of fibers of the holomorphic retraction $\varphi: D \to F(\Omega)$ which relies on analysis on complete Kähler manifolds of finite volume. Such a metric is made available on an extension of $X_\Gamma$ via a schlicht enlargement of $D$ to its hull of holomorphy $\check{D}$, which allows us to exploit the existence theorem of the canonical complete Kähler-Einstein metric on any bounded domain of holomorphy in a complex Euclidean space.

Title: Restricted spaces of holomorphic sections vanishing along subvarieties

Abstract: Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$. This is a joint-work with Dan Coman and George Marinescu.

Title: Bochner Laplacians and Bergman kernels for families

Abstract: We generalize the results of Marinescu-Savale to families of Bochner Laplacians. This particularly leads to the fiberwise expansion for families Bergman kernels of horizontally semi-positive index bundles. Our results include the case of highest weight families. The proofs are based on Ma-Zhang's description for the curvature of the index bundle as a fiberwise Toeplitz operator as well as on methods from sub-Riemannian geometry. Based on joint work with X. Ma and G. Marinescu.

Title: Bergman kernels and geometric quantization on complex manifolds with boundary

Abstract: In this talk I will first review the principle that “quantization commutes with reduction” ([Q, R]=0) for symplectic manifolds. Then I will discuss the [Q, R]=0 principle for complex manifolds with boundary. An important difference between the complex manifolds with boundary setting and the symplectic setting is that the quantum spaces in the case of compact symplectic manifolds are finite dimensional, whereas the quantum spaces consisting of holomorphic functions smooth up to the boundary for the compact complex manifolds with boundary are infinite dimensional. We will present that under natural pseudoconvexity assumptions that the Guillemin-Sternberg map is Fredholm. The main ingredient is asymptotics of G-invariant Bergman kernels.

Title: Bergman functions associated with measures on totally real submanifolds

Abstract: I present my recent joint work with George Marinescu on Bergman kernel functions associated with measures on totally real submanifolds. This is a far-reaching generalisation of the standard Bergman kernel function in complex geometry, and also generalises the fundamental notion of Christoffel functions in approximation theory. If time permits, I will try to outline some open questions.

Title: Moment map and convex functions

Abstract: The concept moment map plays a central role in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this talk, we propose a theory of moment maps coupled with an $\textup{Ad}_K$-invariant convex function $f$ on $\mathfrak{k}^\ast$, the dual of Lie algebra of $K$, and study the structure of the stabilizer of the critical point of $f$ composing with the moment map. As an outcome, we are able to obtain a general Calabi-Matsushima decomposition based only on the convexity of $f$ so that all existing Calabi-Matsushima type of decomposition theorems fall into this new framework. This work is motivated by the work of Donaldson $\textup{[Donaldson2017]}$ together with the goal of finding a natural interpretation of Tian-Zhu's Calabi-decomposition for Käher-Ricci solitons in $\textup{[TianZhu2002]}$, which are examples of infinite dimensional version of our setting. (This is a joint work with King-Leung Lee and Jacob Sturm)

Title: On the Guedj-Rashkovskii’s zero mass conjecture

Abstract: Plurisubharmonic functions play important roles in the research field of several complex variables, complex geometry and algebraic geometry. There are many invariants to study the singularity of plurisubharmonic functions, for instance, Lelong numbers, multiplier ideal sheaves and so on. However, these invariants are insensitive to the singularities of zero Lelong number. A famous conjecture, named Guedj-Rashkovskii’s zero mass conjecture, states that if the Lelong number of an isolated singularity is zero, then the top Monge-Ampere mass at this singularity is zero. There are many groups made important progress on this conjecture. In this talk, we will present our recent progress towards this conjecture by introducing the concept log truncated threshold (lt for short) and establishing a sharp estimate on the Monge-Ampere mass for isolated singularity with finite lt number, which provides a new approach to the zero mass conjecture, unifying and strengthening well-known results well-known results about this conjecture. This work is joint with Yinji Li, Quunhuan Liu, Professors Fusheng Deng and Xiangyu Zhou.

Title: Bundles with singular metrics of positive curvature via $L^2$ estimates

Abstract: In this talk, we start with some basic properties of multiplier ideal sheaves associated to pseudoeffective line bundles,e.g., a solution of Demailly's strong openness conjecture (Guan-Zhou), then present our characterization of Nakano positivity via solving $\bar{\partial}$ equations with $L^2$ estimates (Deng-Ning-Wang-Zhou), which is a converse proposition of Hörmander-Demailly's $L^2$ existence theorems. As an application of the criterion, we give an affirmative answer to Lempert’s problem (Liu-Yang-Zhou), which asks whether the limit metric of an increasing sequence of hermitian metrics with Nakano semi positive curvature on holomorphic vector bundles is still Nakano semi-positive. As another application, one may define singular metric of positive curvature in the sense of Nakano on holomorphic vector bundles. Finally, we present our recent results on the strong openness of the multiplier submodule sheaves (vector bundle version of multiplier ideal sheaves) by Liu-Xiao-Yang-Zhou and Le Poiter type isomorphism theorem between cohomology of the vector bundles twisted with the multiplier submodule sheaves and cohomology of the associated line bundles twisted with the multiplier ideal sheaves (Liu-Liu-Yang-Zhou).