Papers

We show that for any $g\geq2$, there exist uncountably many non-isomorphic smooth affine threefolds that are Stein deformation equivalent to $T^\ast\Sigma_g\times\mathbb{C}$, the product of the cotangent bundle of a genus $g$ oriented surface with the complex plane, answering a question of Ivan Smith. In fact, we prove that our family of smooth affine threefolds is parametrized by a $(6g-7)$-dimensional complex orbifold.

Let $\lambda(u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza–Guedj–Lu asked whether every maximal weak solution $\varphi_t$ of the twisted Kähler–Ricci flow satisfies $\lambda(\varphi_t,x)=\max\{\lambda(\varphi_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface $\mathbb F_e=\mathbb P_{\mathbb P^1} (\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-e))$, $e\ge2$. Let $S$ be its negative section and $F_1,\ldots,F_k$ be distinct fibres. If $a,b_i>0$, $\sum_i b_i>ea$, and the initial current is $a[S]+\sum_i b_i[F_i]$, then $\lambda(\varphi_t,x)=a-\min\{k/e,1\}t$ for $x\in S\setminus\bigcup_iF_i$ and $0<t<\min\{a,b_1,\ldots,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge2$.

For a polynomial of degree $n$ whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus $r$ to the nearest zero. If $n$ is even, the sharp radius is $\sqrt{1-r^2}$; if $n$ is odd, the sharp radius is strictly smaller for $r\in(0,1)$ and depends on $n$. Equality cases are also determined. The proof is based on the logarithmic-derivative identity and elementary geometric considerations.

We prove the transcendental Morse inequality conjecture of Boucksom-Demailly-Păun-Peternell. Among the corollaries of this result, this proves orthogonality of divisorial Zariski decompositions, shows that the pseudoeffective and movable cones are dual, and establishes the differentiability of the volume function on the big cone, on all compact Kähler manifolds.

We explore Fischer decompositions from the perspective of inframonogenic functions in Clifford analysis, a non-commutative version of harmonic functions in Euclidean spaces. We study decomposition formulas for polynomial spaces involving inframonogenic and harmonic components, providing fresh insights into their algebraic structures and further important applications.

Let $\mathcal{N}$ be the set of non-Brjuno real numbers. We determine the exact critical exponent of $\mathcal{N}$ for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges $h_\delta(r)=(\log(1/r))^{-\delta}$, the critical exponent is $2$: the $h_\delta$-Hausdorff measure is infinite locally for $0<\delta\leq2$ and vanishes globally for $\delta>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.

This paper characterizes all the singular inner eigenfunctions of the composition operators $C_\phi$ that arise from discrete measures, when $\phi$ is an automorphism of unit disk. By establishing a connection between Beurling and model invariant subspaces, we classify all the inner functions so that the corresponding Beurling subspace is invariant under the composition operators induced by non-elliptic automorphisms. This classification involves solving the eigenfunction equation for the composition operator. Further, we present some applications of the above-mentioned connection.

In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) $(c,\mu)$-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.

Assuming the validity of the recently proposed “A compact complex threefold fibred by tori over the projective line, and the six-sphere”, we construct an exotic complex structure on $\mathbb P^{3}$, distinct from the point-blowup structures of Huckleberry, Kebekus, and Peternell. Then we perform an Atiyah flop to produce a complex structure on the standard smooth manifold $S^{2}\times S^{4}$.

We prove that every Kobayashi distance isometry between $2$-dimensional Lempert manifolds having finite universal set (with atleast three elements) is holomorphic or anti-holomorphic. It is shown that every such isometry is $C_{loc}^{1,1}$ without any regularity assumption on the isometry.

We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \[ g_\sigma(z)=1+\sigma_1 z_1^2+\sigma_2 z_2^2, \qquad \sigma_1,\sigma_2\geq 0. \] A basic question is: which pairs $(\sigma_1,\sigma_2)$ can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if \[ 0\leq \sigma_1,\sigma_2<1, \qquad \sqrt{1-\sigma_1^2}+\sqrt{1-\sigma_2^2}>1. \] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ \sigma=(\sigma_1,\sigma_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_\sigma$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $\sigma$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_\sigma$. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.

Let $S$ be a minimal compact complex surface of class VII with $b_2(S)=3$. We prove that if $S$ carries two distinct singular holomorphic foliations, then $S$ contains a global spherical shell. Moreover, it is an Inoue-Hirzebruch surface. The proof is based on Teleman's theorem on the existence of a cycle of rational curves and results of Dloussky and is inspired by work of Brunella.

We introduce a new class of commutative associative algebras whose structural constants are linear operators acting in a linear topological space. An analogue of the theory of monogenic functions is developed for such algebras. It is shown that the components of monogenic functions satisfy second-order operator-differential equations. This approach provides a constructive method for obtaining solutions of various partial differential and integro-differential equations. Several illustrative examples are presented.

We introduce a new class of algebras with variable operator-valued structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural operators are established. It is shown that the components of monogenic functions satisfy second-order operator-differential equations with variable coefficients, and converse results are proved. The proposed approach provides a constructive method for obtaining solutions of various differential, integro-differential, and functional-differential equations. Several illustrative examples are presented.

We introduce a new class of algebras with variable structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural functions are established. It is shown that the components of monogenic functions satisfy certain second-order partial differential equations with variable coefficients, and converse results are proved. This approach provides a constructive method for obtaining explicit solutions of such equations. Several illustrative examples are presented.

These lectures are aimed at junior researchers seeking to enter the field of applications of complex analysis to harmonic analysis, Fourier analysis, spectral and scattering theory for differential operators and related fields. Starting with classical completeness problems posed by Bernstein, Beurling and Malliavin, Wiener, Kolmogorov and Krein, the course progresses to the solutions of the Gap and Type Problems, inverse spectral problems for Schrödinger operators and canonical Hamiltonian systems, and pointwise convergence of the non-linear Fourier transform. Each lecture is supplemented with exercises and references.

We consider a class of discrete multi-time determinantal point processes whose correlation kernels admit a double-contour integral form, containing Schur processes as simple examples. We show that the form of the correlation kernel is preserved under conditioning of the point process to a restricted domain, however, with an integrand which becomes more complicated and which is characterized by a Riemann-Hilbert problem. We apply our general result to domino tilings of reduced Aztec diamonds and to lozenge tilings of hexagons with holes. Our results show a striking analogy with the Its-Izergin-Korepin-Slavnov method for integrable kernels.

Recently, Li and Ponnusamy~\cite{LiPonnusamy2025} established the coefficient conjecture proposed by Wang et al.~\cite{Wang2024} for several prominent geometric subclasses of $\mathcal{S}^0_H(K)$, the class of sense-preserving $K$-quasiconformal univalent harmonic mappings in the unit disk. In this paper, we show that the conjecture continues to hold for a class of $K$-quasiconformal harmonic mappings defined via quasi-subordination. Further, we determine the range of $p>0$ for which such mappings belong to the Hardy space ${\bf h}^p$ and the weighted Bergman space $\mathbf{a}^{\mathbf{p}}_{\boldsymbol{\beta}}$, for $\beta>-1$. Our Hardy space result makes significant progress toward a problem posed by Pavlović, while the Bergman space result sharpens the range obtained by Das and Rasila~\cite{DasRasila}, doubling the previously known bounds. In addition, we obtain refined growth and integral mean estimates for the subclass, improving earlier results and providing further evidence toward an open problem raised by Das et al.~\cite{DasRasila2025}. Parallel results are also discussed for odd $K$-quasiconformal harmonic mappings.

A self-homeomorphism $h$ of the unit circle $\mathbb{T}$ is strongly symmetric if it is absolutely continuous and $\log h'\in\text{VMO}(\mathbb{T})$. Let $P_h^-$ be the anti-analytic component of the pullback operator $P_h: F\mapsto F\circ h$ on $\text{BMOA}(\mathbb{D})$, where $\mathbb{D}$ is the unit disk. P. Jones proved $P_h$ is bounded on BMO if and only if $h$ is strongly quasisymmetric. Fan, Hu, and Shen showed that the strong symmetry of $h$ yields the compactness of $P_h^-$, and raised the question of whether the converse is true. We answer this question affirmatively, establishing that $P_h^-$ is compact if and only if $\log h'\in\text{VMO}(\mathbb{T}))$, which completes the VMO theory of the universal Teichmüller space.

In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set $X\subset \mathbb{C}^n$ that is definable in an o-minimal structure on $\mathbb{R}$ must be an algebraic set. This result is known nowadays as the o-minimal Chow theorem. In this article, we present some geometric and metric versions of Chow's theorem that generalize the o-minimal Chow's theorem. For instance, we prove that if $X\subset \mathbb{C}^n$ is a pure $d$-dimensional entire complex analytic set, then the following statements are equivalent: (1) $X$ is a complex algebraic set; (2) $\mathcal{H}^{2d+1}(C(X,\infty))=0$, where $\mathcal{H}^{k}(A)$ denotes the $k$-dimensional Hausdorff measure of $A$, and $C(X,\infty)$ denotes the tangent cone at infinity of $X$; (3) $\mathcal{H}^{2d+2}(C_{\mathbb{C}}(X,\infty))=0$, where $C_{\mathbb{C}}(X,\infty)$ denotes the complex tangent cone at infinity of $X$; (4) $\mathcal{H}^{2d}(Z(X,\infty))=0$, where for $A\subset \mathbb{C}^k$, $Z(A,\infty)=\{[v]\in \mathbb{C}P^{k-1};v\in C_{\mathbb{C}}(A,\infty)\}$; (5) For any $k\in\{d+1,...,n\}$ and for any projection $\pi\colon \mathbb{C}^{n}\to \mathbb{C}^{k}$ such that $\pi^{-1}(0)\cap C_{\mathbb{C}}(X,\infty)=\{0\}$ and $Y=\pi(X)$, $\mathcal{H}^{2d}(Z(Y,\infty))<\mathcal{H}^{2d}(\mathbb{C}P^{d})$.

Let $U$ be a connected open subset of the complex unit ball $B^n$, $n\geq2$, and let $F_i: U \to \Omega_i$ be non-constant holomorphic maps into irreducible bounded symmetric domains. We prove that if $F=(F_1, \cdots, F_m)$ is a holomorphic isometric map from $U$ to the product of $\Omega_i$, then every $F_i$ extends to a proper holomorphic isometric embedding up to a positive constant from $B^n$ into $\Omega_i$.

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre–Ostermann–Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

We investigate some of the properties of the Levi map and related objects for nondegenerate CR submanifolds in $\mathbb{C}^N$ of codimension d.

For the purposes of proving Smale's mean value conjecture, one may restrict consideration to an explicitly described family of so-called Schlicht normalized polynomials. For Schlicht normalized polynomials whose leading coefficient decays sufficiently slowly, we show that the conjectured upper bound $1$ becomes asymptotically valid as the degree $d$ tends to infinity. The key and novel input is a refined Koebe $\frac{1}{4}$-type theorem of Cunningham, tailored to Schlicht functions whose images have bounded logarithmic capacity. For the dual mean value conjecture, we obtain an improvement of Eremenko's Markov-type inequality for regions bounded by polynomial lemniscates on which the polynomial is univalent. As a consequence, we improve the best known unconditional lower bound $\frac{1}{d^2}$ of Dubinin to $\frac{(d-\frac{1}{2})^{1/d}}{d^2}$ for all $d \ge 2$. The strengthened Markov-type inequality constitutes one of the main technical contributions of this paper. Its proof combines the solution to a related extremal problem for the logarithmic capacity of polynomial lemniscates, with the quasi-conformal deformation method introduced by Eremenko and Hayman to establish the connectedness of extremal polynomial lemniscates arising in the Erdos, Herzog and Piranian's problem on maximal lemniscate length.

It is not a priori clear which of the Gauduchon connections is better suited to LCK and Vaisman manifolds. We thus investigate the geometry of these connections through Einstein problems and more general analytic and cohomological conditions on their Ricci tensors. Our results single out the Bismut connection as the privileged one for the non-Kähler lcK and Vaisman geometry. We therefore study the {second-Bismut–Einstein} equation and provide a characterization of these structures on Hopf manifolds.

We study properties of the transcendental numerical dimension on compact Kähler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact Kähler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao–Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(\Omega_X^1)^{\otimes m},~m \geq 1\). We prove that \[ \nu(L,X)\leq\nu(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.

In this paper, we characterize compactness in terms of the Berezin transform and obtain an essential norm estimate for operators in the Toeplitz algebra on Bergman spaces of the Siegel upper half-space.

Let E be a polar compact set in $\mathbb C$. Let $f_\infty$ be a germ at $\infty$ that can be analytically continued along an arbitrary path $\gamma$ lying in $\widehat{\mathbb C}\setminus E$ and starting at the point $\infty$. In 1985–1986 Herbert Stahl presented his proof of a fundamental theorem on the convergence of diagonal Padé approximants constructed from such germ $f_\infty$. Since then, this theorem has borne his name. A crucial role in his proof is played by the existence of a compact set $S_{f_\infty}$ of minimal logarithmic capacity among all compact sets $K$ such that the germ $f_\infty$ extends as a single-valued meromorphic function to $\widehat{\mathbb C}\setminus K$. Unfortunately, the proof of this fact presented by H. Stahl in 1985 contains a crucial mistake. It is surprising that this mistake was made in the original Stahl's paper in 1985 and repeated in his last preprint in 2012, and, as far as we know, no one has pointed it out before! In this paper we explain this serious Stahl's mistake and present a correct proof of the existence of a compact set $S_{f_\infty}$. We emphasize that our proof will use ideas completely different from Stahl's ideas. Also we discuss the more general Stahl's conjecture about the existence of a compact set $S_{f_\infty}$ for an arbitrary germ $f_\infty$ without any assumption on the paths along that the germ $f_\infty$ can be continued.

Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $\mathcal{O}^{\mathfrak{r}\,G}_X$ on $X$ by \begin{align*} \mathcal{O}^{\mathfrak{r}\,G}_X(V):=\{u\in \mathcal{C}(V,G)|\ &u \hbox{ has regularity class $\mathfrak{r}$ on $V$}, \\ &u|_{V\cap\mathrm{int}(X)} \hbox{ is holomorphic}\}. \end{align*} Let $X_0\Subset Z_0\Subset \mathcal{X}$ be a strictly pseudoconvex extension in $\mathcal{X}$, and let $X:= \bar X_0$, $Z:= \bar Z_0$ be the corresponding compact manifolds with boundary in $\mathcal{X}$. Let $K\subset X_0$ be a compact set and $\mathcal{P}$ a holomorphic principal $G$-bundle of class $\mathfrak{r}$ (i.e. an $\mathcal{O}^{\mathfrak{r}\,G}_X$-torsor) on $X\setminus K$. Assuming that (H1) the given strictly pseudoconvex extension $X_0\Subset Z_0$ is non-critical, or that (H2) the underlying topological bundle of $\mathcal{P}$ extends to $Z\setminus K$, we prove that $\mathcal{P}$ admits an extension to $Z\setminus K$. This gives a new proof and a new generalisation of the extension problem stated in the article S. Donaldson, Boundary value problems for Yang-Mills fields, Journal of Geometry and Physics 8, (1992) and studied with different methods in a previous article of the author.

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d \gamma)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.