Papers

Let $\varphi$ and $\psi$ be univalent functions in the unit disk $\mathbb D$ satisfying $\varphi(0)=\psi(0)=1$ and $\varphi\not\prec\psi$, and let $\mathcal P_{\varphi}$ denote the class of analytic functions $p$ in $\mathbb D$ such that $p\prec\varphi$. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of $\varphi^{-1}$ satisfying $\varphi^{-1}(1)=0$, we determine the sharp $\mathcal P_{\psi}$-radius of $\mathcal P_{\varphi}$ as \[ \mathcal R(\varphi,\psi) = \min_{|\zeta|=1} \left| \varphi^{-1}\bigl(\psi(\zeta)\bigr) \right|. \] For Ma–Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes $\mathcal{ST}(\varphi)$ and $\mathcal{CV}(\varphi)$ with respect to $\mathcal{ST}(\psi)$ and $\mathcal{CV}(\psi)$, respectively. We apply the general result to $ \varphi_{\mathrm L}(z)=\sqrt{1+z}, \varphi_{\mathrm{\leftmoon}}(z)=z+\sqrt{1+z^2}, \varphi_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $ for several choices of the Ma–Minda function $\psi$, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.

We investigate the differentiation and integration operator defined on resp. between weighted spaces of analytic functions in the special situation when the weight is given in terms of a weight sequence. More precisely, in this case the weight is defined via the associated weight function which is a crucial object in the ultradifferentiable setting and the information can be then expressed purely in terms of the underlying weight sequence. Due to the additional structure this setting is exploited in order to construct (counter-)examples, to compare conditions on weights in the different weighted frameworks, and to study and to understand in a more precise qualitative way required and admissible growth restrictions on the weights.

We resolve Erdős Problem #522, posed in 1961, by proving that for random Littlewood polynomials the proportion of zeros in the closed unit disk converges almost surely to $1/2$. More generally, for partial sums of random power series with i.i.d. coefficients that are bounded, nondegenerate, and centrally symmetric, real Gaussian, or circular complex Gaussian, we prove that the proportion of zeros in the disk of radius $1+x/n$ converges almost surely to $\frac12(1+\coth x-1/x)$, uniformly in $x\in\mathbb{R}$. Thus almost all zeros lie at distance $O(1/n)$ from the unit circle. For random Littlewood polynomials, even with arbitrary dependence between degrees, the convergence at the unit circle holds at the almost-sure rate $O(n^{-1/4}\sqrt{\log n})$. We establish a law-of-the-iterated-logarithm criterion for dependent sequences and, for real Gaussian coefficients, obtain the sharp law of the iterated logarithm over all integer degrees, variance and cross-degree covariance asymptotics, quantitative central limit theorems, joint Gaussian limits, and increment bounds. For symmetric coefficient laws with bounded density and all moments finite, the number of zeros in the closed unit disk has variance $(c_G+\kappa_4/12)n+O_\xi(n^{399/400})$, and for random signs the constant is $c_G-1/6$. Thus the radial law is universal, while the fluctuations depend on the coefficient law through its fourth cumulant $\kappa_4$. All our main results are formally verified in Lean 4, and our proof of Erdős Problem #522 has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project.

We prove that in the disk, sphere, whole-plane topology, spanning tree weighted planar maps converge to $\sqrt{2}$-Liouville quantum gravity disk, sphere or cone under circle packing and Riemann uniformization embedding as the number of faces of the map goes to infinity. As a byproduct, we also prove that the natural path on faces of the embedded tree-weighted planar maps converge to SLE$_8$. The proof is based on our earlier work on circle packing and Riemann uniformization embedding for random planar maps in ergodic scale-free environments, comparisons of circle packings in different domains in a companion paper, and the convergence of tree-weighted planar maps to $\sqrt{2}$-LQG by the first author.

Suppose $M$ is a planar map with a macroscopic boundary decorated by a spanning tree $T$, such that $(M,T)$ has been sampled from the critical Boltzman measure. We show that $M$ converges to a Liouville quantum gravity disk with parameter $\gamma=\sqrt{2}$ under the Tutte embedding as the length of the boundary of $M$ goes to infinity. Furthermore, conditioned on $(M,T)$, the simple random walk on $M$ converges to a planar Brownian motion modulo reparametrization of time.

We prove that a holomorphic family of connected complex Lie groups is Oka if and only if its base is Oka. We establish the Oka property for moduli spaces of stable and parabolic vector bundles and characterize Oka minimal \(\delta\)-regular abelian fibrations over compact curves by the nonnegativity of their orbifold Euler characteristic. Strong dominability, analytic extension, and holomorphic descent also yield applications to mapping spaces, character varieties, and compactified Jacobians.

We prove that the mated-CRT map converges to Liouville quantum gravity under circle packing and the Riemann uniformization embedding. The proof is based on our recent work on circle packing and the Riemann uniformization embedding for random planar maps in ergodic-scale free environments, comparison of circle packing in different domains and several existing and new estimates for the mated-CRT map. As an intermediate step, we prove an extension of He and Schramm's result (1996) on convergence of circle packing maps to conformal maps to global uniform topology in several settings. The latter results are of independent interest and will be applied in our subsequent work on the convergence of spanning tree weighted planar maps to $\sqrt{2}$-LQG under circle packing and Riemann uniformization embedding.

We characterize Oka manifolds by the local approximation property (LAP), a planar approximation operation that preserves holomorphic dependence on prescribed parameter polydiscs. We prove that LAP yields convex approximation in every source dimension and propagates across sets of zero $(2n-1)$-dimensional Hausdorff measure in an $n$-dimensional target. Consequently, strong dominability characterizes Oka manifolds, convex approximation in the target dimension suffices, the Oka property is local in the Euclidean topology, and it descends under surjective holomorphic maps. Local families of entire curves give geometric applications: smooth projective rationally connected manifolds are Oka, and deleting an algebraic subset of codimension at least two from any such manifold preserves the Oka property. Structure theorems further show that compact Kähler manifolds with nonnegative holomorphic sectional curvature or nef tangent bundle are Oka. Finally, the complement of a smooth degree-$d$ hypersurface in $\mathbb{P}^n$ is Oka exactly when $d\leq n+1$.

We construct complex structures on the six-sphere $S^6$ with arbitrarily large local Kuranishi dimension. As a corollary, we find the same phenomenon holds for the smooth manifold underlying complex projective 3-space $\mathbb{P}^3$.

In this article, we study three properties of a bounded $\mathbb{C}$-convex domain $\Omega \subset \mathbb{C}^n$, $n \geq 2$, with smooth boundary, namely Gromov hyperbolicity of its Kobayashi distance, finiteness of the D'Angelo type of $\partial \Omega$, and the existence of a subelliptic estimate for the $\bar{\partial}$-Neumann problem on $(0,1)$-forms. We prove that these three properties are equivalent. The subelliptic estimates are obtained without Catlin's construction of plurisubharmonic weights. Our weights are explicit bounded transforms of the logarithm of the Bergman kernel. A single quantity controls all three properties, namely the radius of the largest disc through a point of $\Omega$ in a given complex direction. Gromov hyperbolicity forces a power bound for these radii. Finite type gives the power $1/M$, where $M$ is the maximal type, and this power cannot be improved. A subelliptic estimate gives a power bound through the canonical solution operator applied to normalized Bergman kernels. The analytic estimates need no boundary regularity. On a bounded pseudoconvex domain, a bounded weight whose induced Hessian on $(0,q)$-forms is bounded below by a negative power of the boundary distance gives a Sobolev estimate for the extension by zero of $(0,q)$-forms, and also eigenvalue bounds for the $\bar{\partial}$-Neumann operator. On smooth $\mathbb{C}$-convex domains of finite type, we show that the directional expansion exponent, the normal expansion exponent, and the supremum of the subelliptic gains are all equal to $1/M$. Next, we prove that on a Lipschitz $\mathbb{C}$-convex domain whose Kobayashi distance is Gromov hyperbolic, the Euclidean boundary and the Gromov boundary are bi-Hölder equivalent. Finally, we give a Hardy-type criterion for Gromov hyperbolicity of collar metrics.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

We provide a description of the real variable Hardy spaces $H^p$, $p\in (0,1]$ associated to a regular polyhedral open convex cone, devised as real variable analogues of the holomorphic Hardy spaces on the corresponding tube domain.

The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\to\alpha\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

We provide a geometric description of Lee-Yang tensors, building upon several recent results of Bravyi, Gosset, Liu, and Wong. By applying tools from complex Perron-Frobenius theory, we generalize one of their key lemmas to bound the spectral ratio of non-normal operators with Lee-Yang radius greater than one. As a direct consequence, we provide a local-to-global principle for bounding spectral gaps of Suzuki-Fisher Hamiltonians, a family of $2$-local Hamiltonians, using only the Lee-Yang radius of each local operator to bound the gap from below. On the other hand, we develop a simple perturbative technique to show that the Gibbs states of $k$-local Hamiltonians, $k \geq 3$, cannot have Lee-Yang radius at least one for all inverse temperatures $\beta \geq 0$. We apply this same method to conclude that Suzuki-Fisher Hamiltonians are precisely those Hamiltonians whose Gibbs states are Lee-Yang at all inverse temperatures $\beta \geq 0$.

Let $L$ be a smooth Hermitian holomorphic line bundle on a compact complex manifold $X$ of dimension $n$, and let $r$ be its maximal curvature rank. For every fixed holomorphic vector bundle $E$, we prove $h^q(X,L^p\otimes E)=O_\varepsilon(p^{r+\varepsilon})$ for every $\varepsilon>0$ and every Dolbeault degree, with the sharp bounds $h^0,h^n=O(p^r)$ at the endpoints. The curvature may have variable rank and arbitrary signature, and no global foliation or Kähler assumption is imposed. The all-degree estimate follows from local rescaling and holomorphic restriction estimates that remain uniform across rank changes. An independent maximum-principle argument locates the maximal rank of a section map at a semipositive curvature point; it yields $\kappa(L|_{X_0})\le r$ on each connected component $X_0$ and gives the endpoint bounds.

In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors–Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $\mathbb{C}^n$. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors–Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.

In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$ denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak $\partial\bar\partial$-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at $E_1$. These results provide non-Kähler extensions of the theorems of Li–Pan and Iacono–Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.

In this paper, we study analytic multiplier ideal sheaves of toric plurisubharmonic functions on normal affine toric complex analytic spaces equipped with torus-invariant rational divisors, assuming that a positive integer multiple of the log canonical divisor is Cartier. We describe these sheaves in terms of Newton convex bodies and show that they are globally generated by finitely many meromorphic monomials. Our results generalize the description of An and the formulas of Howald and Blickle for monomial ideals, and recover the descriptions of Rashkovskii and Guenancia on affine space. We also extend this description to weights obtained by subtracting positive rational multiples of monomial ideal weights.

Zilber's celebrated quasiminimality conjecture states that a subset of $\mathbb C$ defined by polynomials and exponentials is either countable, or has countable complement. Koiran has asked whether the same could be true replacing $\exp$ with any unary entire function. We give a negative answer to Koiran's question, using results from the theory of holomorphic approximation.

Let $\Omega=G/K$ be an irreducible bounded symmetric domain other than the unit disk, and let $S$ be its Shilov boundary. We study the two-parameter family of Shilov–Poisson transforms between homogeneous line bundles over $S$ and $\Omega$. For a transform $F$ of $L^2$ boundary data, we determine what finite Euclidean regularity up to the full boundary $\partial\Omega$ forces in the interior. Outside five explicit exceptional parameter families, sufficiently high finite regularity forces $F=0$. In each exceptional family, the conclusion is an interior covariant differential equation. In the scalar Poisson–Szegő case, it yields a finite-order Graham type theorem: the stated boundary regularity forces a Bergman-harmonic function to be pluriharmonic.

{For a matrix function $A(t)$, we study the normalisation of its Wiener–Hopf factorisation $A(t)=A_-(t)D(t)A_+(t)$ which guarantees the uniqueness of the factors $A_\pm(t)$. Previously, this problem was investigated for second-order matrix functions, the stable factorisation, and the factorisation of matrix functions with distinct partial indices. The main tools for solving the problem of normalisation are the theorem of I.C. Gohberg and M.G. Krein on a general form of the factors $A_\pm$, a Birkhoff factorisation of matrix functions, and a block PLU factorisation of the numerical matrix $A_-(\infty)$. To solve the normalisation problem, a concept of P-normalised factorisation is introduced. The Birkhoff factorisation of matrix functions plays a key role in defining this concept. The permutation matrix $P$ is determined by PLU factorisation of the numerical matrix $A_-(\infty)$. We proved that any invertible matrix function $A(t)$ admits P-normalisation. This normalisation guarantees the uniqueness of the Wiener–Hopf factorisation. The P-normalised factorisation of $A(t)$ allows us to find the Birkhoff factorisation of $A(t)$. Without normalisation, it is impossible to obtain explicit estimates of the absolute errors in the approximate computation of the factorisation factors.}

A normalized analytic function $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ defined on the unit disk $\mathbb{D}$ is called Ma–Minda starlike or Ma–Minda convex if $zf'(z)/f(z)$ or $1+zf''(z)/f'(z)$, respectively, is subordinate to a Ma–Minda function $\varphi$. In this paper, we investigate the radii of Ma–Minda starlikeness and convexity for functions satisfying $|f''(z)|<M$ in $\mathbb{D}$. We first develop a unified approach for determining these radii and then apply it to obtain explicit radius estimates for several important choices of the Ma–Minda function~$\varphi$.

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion whose total space $X$ is a Kähler manifold. We prove that $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This was conjectured by Y.-T. Siu in 2002.

We disprove the Clunie–Sheil-Small conjecture $\bigl||a_n|-|b_n|\bigr|\le n$ for normalized univalent harmonic functions $f=h+\overline g$ in the unit disk $\mathbb{D}$, where $a_n$ and $b_n$ are the Taylor coefficients of $h$ and $g$, respectively. For every $K>1$, we construct a harmonic $K$-quasiconformal map which has $|a_n|-|b_n|\asymp n^{1+\varepsilon_K}$ as $n\to\infty$, with $\varepsilon_K>0$. Nevertheless, this map satisfies both individual coefficient bounds in the conjecture. Next, we show that the same map belongs to the harmonic Hardy space $h^p$ if and only if $0<p<1/(2+\varepsilon_K)$. Thus, harmonic quasiconformal maps do not retain the full conformal Hardy range $p<1/2$. Interestingly, the latter range is recovered when the oscillation of the analytic dilatation on disks of fixed hyperbolic radius tends to zero near the boundary. This holds, in particular, when the dilatation has finite Dirichlet energy. For general $K$-quasiconformal maps, we impose the additional assumption that their Beltrami coefficients belong to the Sobolev space $W^{1,s}(\mathbb{D})$ for some $s\ge1$. Every such map has bounded $p$-integral means for $0<p<1/(2K)$ when $1\le s<2$, and for $0<p<1/2$ when $s\ge2$. Both these ranges are sharp. For $s\ge2$, each map also has the same critical Hardy exponent as a conformal map onto the same image.

We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert–Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.

Consider a system of equations $A(x)\cdot F(x) = B(x)$ on a subset $X$ of $\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\mathcal C^m$ vector-valued function $f$ on $\mathbb R^n$, and ask whether we can find a $\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \subset \mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb R$.

We prove that for any $p>4$ we can find a Schatten class Hankel operator $\Ha_\phi$ defined on the Hardy space of the infinite torus $H^2(\mathbb{T}^\infty)$ that belongs to the Schatten class $S^p$ but does not admit a bounded symbol, extending the previously known range $p>(1-\log\pi /\log 4)^{-1}=5.7388...$.

We establish an a priori $C^2$-estimate for complex Hessian equations on compact Hermitian manifolds. As an application, we obtain a regularity result for solutions to complex Hessian equation with densities of the right hand side depending on the solution. This generalizes Székelyhidi–Tosatti's result to the setting of complex Hessian equations on Hermitian manifolds.

We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.

We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of $k$-Hessians and, in the homogeneous case, precisely the $k$-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the $\mathcal{J}$-equation.