Papers

We prove that every Gromov elliptic domain $\Omega$ in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood $V$ such that $\Omega\cap V$ becomes Stein after removing a set which is, locally in $\Omega\cap V$, the union of the polar sets of finitely many meromorphic functions on $\Omega\cap V$. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain $\Omega=\operatorname{int}\overline\Omega$ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete Kähler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds $\mathbb C^n\setminus\overline{\mathbb B^n}$ and $\mathbb C^n\setminus\mathbb R^n$, $n\geq2$, are not elliptic, and that no proper domain with Lipschitz boundary in $\mathbb P^n$, $n\geq2$, is elliptic.

For a meromorphic function $f$ in the complex plane, we determine when $f$ and $f^{(k)}$ are algebraically dependent. This happens precisely when $f$ belongs to the class $W$ of elliptic functions and their degenerates, or satisfies a linear differential equation $f^{(k)} + af + b = 0$ with constant coefficients. The case of functions having a pole was settled previously. We complete the proof by treating entire solutions.

We investigate the notion of holomorphic $p$-contact structure, that we introduced recently in the scalar case with H. Kasuya and L. Ugarte and in the line-bundle case with K. Broder, in the context of a holomorphic family of compact complex manifolds equipped with a holomorphic family of holomorphic line bundles. Firstly, we show that the existence of a holomorphic $p$-contact structure is deformation-open when a non-jumping assumption is made on the $\bar\partial$-cohomology of bidegree $(p,\,0)$. This ought to become a source of further examples of holomorphic $p$-contact manifolds. Secondly, we obtain a Berndtsson-type (semi-)positivity result for the holomorphic vector bundle over the base induced, as a direct image, by the line bundle on the total space that supports the p-contact structures. Thirdly, we introduce a moduli space of relative holomorphic p-contact structures and we show that it is Hausdorff and open in a certain quotient topological space.

We prove two height inequalities for rational points on abelian varieties over function fields of characteristic 0. The first is the GCD inequality, which shows that the height with respect to a closed subscheme of an abelian variety of codimension at least 2 is much smaller than the usual height function. This can be viewed as a compact and geometric analogue of a theorem of Levin. The second is a special case of the (geometric) abc conjecture proposed by Vojta, which compares the truncated counting function and the usual counting function. This is a compact analogue of a theorem of Guo, Nguyen, Sun and Wang, and also resolves a conjecture of Buium, which was only known for traceless abelian varieties or constant abelian varieties previously. The proofs of these inequalities follow a common three-step pattern: from microscopic local estimates to local estimates, and then to global estimates. Building on the canonical Xie-Yuan heights, we control discrete counting functions by continuous integrals of Betti forms. The main new tools are uniform Nevanlinna-type estimates for families of linear entire curves on abelian varieties; combined with covering arguments based on Besicovitch theory and averaging, these handle overlaps and gaps precisely.

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $\Omega=\mathbb{D}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $\Omega(1)$ admits a transitive subgroup of automorphisms with NC lifts and $\Omega(1)$ has the bounded $H^\infty$-Gleason property at the origin, then $\|f(T)\|\leq C_{\Omega,n}\|f\|_{H^\infty(\Omega(1))}$ for every commuting tuple $T\in\Omega(n)$, with $C_{\Omega,n}=(2\rho_{\Omega}K_{\Omega})^{n-1}$; here $K_{\Omega}$ is the Gleason constant and $\rho_{\Omega}$ is the sum of the suprema of the coordinate functions on $\Omega(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where $\rho_{\Omega}=d$, yielding an unconditional estimate $(2dK_{\Omega})^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^\infty$-Gleason property at the origin, which we do not establish when $\min(p,q)\geq 2$.

To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman–Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate–Shafarevich twists. We give a systematic interpretation of known results about Tate–Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda–Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to $U^2\oplus\langle-2\rangle^2$ to a smooth cubic surface with a choice of a Schläfli double-six. We also review some other approaches towards the construction of complete families of O'Grady six-manifolds with Lagrangian fibrations, and show how they may still be useful even though they do not quite work.

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product of $\mathbb{C}^2$ of degree $d\geqslant 2$ and let $\mu$ and $\mu_p$ be the Green measures associated to $f$ and $p$ respectively. Let $\lambda_1 > \lambda_2$ and $\lambda_p$ be the Lyapunov exponents of $(f,\mu)$ and $(p,\mu_p)$ respectively. We show that if $\lambda_2 = \lambda_p$, then $\mu$ is exact dimensional and: \begin{equation*}\dim_H\mu =\frac{\log d}{\lambda_1}+\frac{\log d}{\lambda_2}.\end{equation*} The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that $\mu$ has constant Jacobian $d^2$, and on a fibered version of a normal form theorem for the inverse branches of the iterates of $f$.

We prove that a bounded pseudoconvex domain with piecewise $C^2$-smooth boundary which has a finite volume quotient for the complete Kähler-Einstein metric is biholomorphic to a product of complex unit balls. More generally, the same conclusion holds for a bounded domain possessing a boundary point satisfying Kodama's conditions (C.1)-(C.5), provided the quotient has finite volume for a complete $\operatorname{Aut}(D)$-invariant Kähler metric. For piecewise Levi-flat boundary the product is a polydisk and in the smooth pseudoconvex case it is a single ball.

We study weighted composition–differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces. Explicit dyadic criteria yield boundedness, compactness and two-sided norm and essential-norm estimates, including endpoint and quasi-Banach cases. We also obtain multiplier and approximation results, Schatten criteria on weighted Dirichlet spaces, and exact singular values for monomial weights and linear symbols.

It is known that the dynamical properties of Kleinian groups with parabolic elements are much more complex than those without parabolic elements. In this paper, we mainly study the quasi-conformal deformation properties of geometrically finite Kleinian groups with parabolic elements. We show that, for a Kleinian group $G$ with parabolic elements, there exists a subset $K$ of the limit set $\Lambda(G)$ such that the first return map $f_{K}$ is (eventually) expending. Furthermore, the expending property is invariant under quasiconformal deformations. As an application, we show that the geometrically finite Kleinian groups have Ruelle's property. Finally, we use eigenvalue algorithm to calculate the precise Hausdorff dimensions of Hecke groups, which serves as an example to support our results.

In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity $> 2$ of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.

The Briot–Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, elliptic, a rational function of $e^{cz}$ with $c\neq 0$, or satisfies $f^{(n)}=af+b$. In particular, every meromorphic solution with $P$ irreducible of degree at least two belongs to the Weierstrass class. Solutions with poles were classified by Eremenko, Liao and Ng. We settle the remaining entire case with a rigidity theorem for polynomial compositions: if $h$ is nonconstant entire and $D^nR(h)=S(h)$, where $D=d/dz$ and $\deg R=\deg S\ge 2$, then $h(z)=Ae^{bz}+Be^{-bz}+C$. No growth assumption is needed. The proof turns the cancellation of leading terms into a contraction of indicator diagrams and obtains incompatible zero counts near an exposed edge. As an arithmetic application, an algebraic Laurent expansion at one algebraic point and an algebraic value at another force a solution to lie in $\overline{\mathbb{Q}}(z)$. Hence any two distinct derivatives of a function in Mahler's class (transcendental meromorphic functions with algebraic Laurent coefficients at every algebraic point) are algebraically independent over $\mathbb{C}$.

We study the structure of the centralizer of a non-elliptic holomorphic self-map of the unit disc and its relation with continuous dynamical structures. We prove that if the identity is not isolated in the centralizer of a non-elliptic self-map $\varphi \in \mathrm{Hol}(\mathbb D)$, then $\varphi$ is univalent and its centralizer contains a non-trivial continuous one-parameter semigroup of holomorphic self-maps. This result provides a general criterion for the existence of continuous dynamics arising from discrete iteration. Our approach is based on the theory of holomorphic models and on simultaneous linearization techniques for commuting self-maps. In particular, we exploit recent developments on simultaneous linearization for parabolic self-maps with positive hyperbolic step. As a consequence, we obtain new sufficient conditions for the embeddability of holomorphic self-maps into continuous semigroups.

This note grew out of two questions arising in different settings. The first concerns recognizing holomorphic functions among continuous functions on an annulus. The second concerns what self-reproducing subsets can tell us about semigroups of functions arising in analysis. Our analytic starting point is a variation on the classical relation between holomorphy and invariance of contour integrals. It identifies the holomorphic functions as the largest class containing the constant function 1 for which a contour convolution is independent of the chosen curve for every pair of functions. On this class, convolution amounts to multiplying Laurent coefficients. We then study families $B$ satisfying $B\diamond B=B$. We show that expressing a function as a convolution product of $m$ holomorphic functions corresponds precisely to its extension to a wider annulus determined by $m$. This identifies the largest self-reproducing family as the functions extending holomorphically to the punctured plane. We also prove that uniform boundedness on a fixed subannulus surrounding the unit circle forces all members of such a family to have Laurent expansions supported on one common finite set. A general semigroup theorem describes the minimal self-reproducing families.

We analyze exponential mixing for analytic observables and show that mixing at a double exponential rate occurs for genuinely nonlinear systems. Neither its presence nor its sharp exponent is stable under arbitrarily small area-preserving perturbations in the real-analytic topology. We construct two families of area-preserving Anosov Blaschke product diffeomorphisms that converge in the real-analytic topology to the same hyperbolic toral automorphism. For nonzero parameters, both deformations are topologically conjugate to the linear model but not $C^1$-conjugate. For all nonzero parameter values, the maps in the first family are mixing at a double exponential rate with the same sharp exponent, strictly smaller than that of the linear map. The sharp exponent is therefore not lower semicontinuous at the linear map. In the second family, mixing at a double exponential rate is lost altogether, although the maps remain exponentially mixing. We also establish double exponential bounds for every ergodic toral endomorphism and uniform bounds of the same type for totally ergodic higher-rank toral actions. Our method combines the Fourier escape argument with Koopman operator estimates from a Hilbert space of analytic functions to its dual space of hyperfunctions.

We use Loewner-Kufarev evolution to decompose the Brownian loop soup into a Poisson point process on Brownian bubbles which we call the Brownian bubble tea driven by $\rho$, where $\rho$ is the driving measure for the Loewner chain. The only assumption on $\rho$ is that the evolving boundaries are locally connected. This generalizes a similar disintegration result of Lawler and Werner to essentially optimal regularity, and includes foliations with finite Loewner-Kufarev energy as well as radial and whole-plane $\mathrm{SLE}_\kappa$ for all $\kappa \geqslant 0$. As applications, we give Brownian loop interpretations of the conformal distortion formula for Loewner-Kufarev energy and the relationships between the Loewner energy, Loewner-Kufarev energy, and the Schwarzian action.

This article investigates random Dirichlet series within the framework of Hardy and mixed norm spaces of Dirichlet series. We completely characterize when the random symbol space of the Hardy space $\mathscr{H}^p$ coincides with $\mathscr{H}^2$. In the setting of mixed norm spaces $\mathscr{H}^{p,q}_{\alpha}$, the corresponding characterizations for the random symbol space of $\mathscr{H}^{p,q}_{\alpha}$ coinciding with $\mathscr{H}^{2,q}_{\alpha}$ are given when $q\geq4$, and some sufficient and necessary conditions are established when $q<4$. We also obtain some integrability results when Dirichlet series belong to the corresponding random symbol spaces.

The pentablock is a domain in $\mathbb C^3$ arising in the study of the $\mu$-synthesis problem and is defined by \[ \mathcal P = \left\{ (a_{21},\operatorname{tr}(A),\det(A)): A=[a_{ij}]_{i,j=1}^2\in M_2(\mathbb C),\ \|A\|<1 \right\}. \] We introduce the Hardy space $H^2(\mathcal P)$ associated with the canonical measure $\mu$ on the distinguished boundary $b\mathcal P$. We identify $H^2(\mathcal P)$ with a closed subspace of $L^2(b\mathcal P,\mu)$ and show that it is a reproducing kernel Hilbert space. We establish a new description of $\mathcal P$ and its closure and obtain a uniqueness set. In particular, we prove that $\mathcal P$ is the interior of the polynomial convex hull of $\Psi(\overline{\mathbb D}^3)$, where \[ \Psi(\xi,\lambda_1,\lambda_2) = \left( \frac{\xi(\lambda_1-\lambda_2)}{2}, \lambda_1+\lambda_2, \lambda_1\lambda_2 \right) \] is the parametrization map for $\mathcal P$. We also show that $H^2(\mathcal P)$ embeds isometrically as a closed subspace of $H^2(\mathbb D^3)$. We then obtain a Brown-Halmos type characterization of Toeplitz operators on $H^2(\mathcal P)$ and a commutant lifting theorem for $\mathcal P$-isometries and their minimal $\mathcal P$-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize $\mathcal P$-unitaries as images of commuting triples of unitaries under $\Psi$ and present a class of $\mathcal P$-contractions from triples having $\overline{\mathbb D}^3$ as a spectral set.

In this paper, we study natural determinantal point processes on Stein manifolds defined by canonical sections and plurisubharmonic weight functions. We present a natural condition on the weights under which these processes satisfy a large deviation principle with a good rate function given by the Fenchel–Legendre transform of the pluripotential equilibrium energy. Along the way, we prove some pluripotential theoretic results on Stein manifolds, such as the $C^{1,1}$ regularity of appropriate envelopes. These results are noncompact analogues of results of Berman and Berman–Boucksom on compact Kähler manifolds.

In this paper, we introduce a new class of unbounded Reinhardt domains in $\mathbb C^n$ ($n\geq 2$),which we call horn domains. We study the dimensions of the Bergman spaces,the holomorphic sectional curvature of the Bergman metric, and the holomorphic equivalence problem for such horn Reinhardt domains.

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \geq 2$. If $G$ is a finite group, then its purely non-free genus $\sigma^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and ${\mathfrak S}_{4}$. For general $G$, we provide the value of $\sigma^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\rangle,$ where $n \geq 3$ and $2 \leq s \leq n-1$ is such that $s^{2} \equiv 1 \mod(n)$.

Let $\mathcal{S}_c$ be the subgroup of $Aut(\mathbb{C}^2)$ generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. $\mathbb{D}^*$, $\mathbb{C}^*$ and $\mathbb{C}$, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.

We study algebraic relations among functions in star-invariant subspaces. Let $\theta$ be an inner function and suppose that $f,g,h\in K_{\theta}^p$ satisfy $P(f,g,h)=0$, in which $P$ is a homogeneous polynomial and the associated projective plane curve is smooth. Then $\dim\operatorname{span}\{f,g,h\}\leq1$ if either $\operatorname{deg} P\geq 4$ and the boundary spectrum of $\theta$ has logarithmic capacity zero, or $\operatorname{deg} P = 3$ and the boundary spectrum contains at most one point. This settles new cases of a problem of Dyakonov, subsumes the previously known results, and recovers the classical fact that polynomial parametrizations of a smooth projective plane curve of degree at least three are projectively constant.

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.

The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to $\C^N$, with prescribed finite nerves, or are unions of Fatou–Bieberbach domains whose nonempty finite intersections are again Fatou–Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some $\C^N$, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in $\C^5$. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.

We prove that the classical Bloch constant is strictly smaller than the Ahlfors–Grunsky value, disproving the famous Bloch constant conjecture. The Lean 4 formalization of the main results are also provided.

We study several function-theoretic aspects of the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$. We introduce and investigate the notions of $\Gamma_{E(3;3;1,1,1)}$-inner functions and $\Gamma_{E(3;2;1,2)}$-inner functions, and establish their relationships with $\Gamma_{E(2;2;1,1)}$-inner functions and $\Gamma_{E(2;1;2)}$-inner functions. We provide explicit constructions and representations of rational $\Gamma_{E(3;3;1,1,1)}$-inner functions and rational $\Gamma_{E(3;2;1,2)}$-inner functions. As an application of these results, we obtain partial solutions to the associated interpolation problems and construct explicit rational inner interpolants for interpolation data lying in suitable subsets of $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, respectively.

We study the Fatou components of holomorphic skew products $F \colon \mathbb{C}^2 \to \mathbb{C}^2, F(z,w) = (f(z) + r(z,w), g(w))$ near an attracting invariant fiber, introducing a refined classification of components on the fiber: bulging, partial bulging, and nowhere bulging. For polynomial skew products, we give a complete characterization of when the basin at infinity is non-bulging, and show that in this case non-bulging implies nowhere bulging, ruling out partial bulging entirely. In the transcendental setting, the behavior is richer: we construct partially bulging wandering domains and Baker domains, show that any orbitally unbounded Fatou component can be made nowhere bulging by choosing an appropriate function $r$, and prove that for any nonconstant $r$ there exists an entire function $f$ with an escaping wandering domain that fails to be bulging. Bulging thus depends on the interplay between fiber dynamics and perturbation.

We establish sharp estimates for the generalized Zalcman functional corresponding to the index $n=2$ for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space and on the unit polydisk in $\mathbb{C}^n$. We also derive sharp Fekete-Szegö inequalities for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space. Our results extend classical coefficient estimates to higher-dimensional holomorphic mappings and provide a partial confirmation of the generalized Zalcman conjecture in higher dimensions.

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.