Papers

Let $n$ and $N$ be integers with $2\leq n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map that admits a $C^{N-n+1}$-smooth extension to the boundary. We prove that if the CR Gauss map of its boundary restriction is generically degenerate, then $F$ is holomorphically equivalent, under composition with automorphisms of the source and target balls, to $V_m\oplus 0$ for some positive integer $m$, where $ V_m(z)=\big(\sqrt{\frac{m!}{\alpha!}} z^\alpha\big)_{|\alpha|=m} $ is the degree-$m$ Veronese map. This resolves a problem posed by Xiaojun Huang.

A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is called $\alpha$-spirallike if $f(\mathbb{D})$ is an $\alpha$-spirallike domain, that is, for every $z\in f(\mathbb{D})$, the $\alpha$-spiral $ze^{-e^{i\alpha}t}$, $t\ge0$, is contained in $f(\mathbb{D})$, or equivalently, if $\operatorname{Re}\!\left(e^{i\alpha} {zf'(z)}/{f(z)}\right)>0$ for all $z\in \mathbb{D}$. The 0-spirallike functions are the usual starlike functions. The function $f$ is parabolic starlike, if $\operatorname{Re}\!\left(\frac{zf'(z)}{f(z)}\right)>\left|\frac{zf'(z)}{f(z)}-1\right|$, and lemniscate starlike if $|(zf'(z)/f(z))^2-1|<1$ respectively. In this paper, we determine the radii of parabolic starlikeness and lemniscate starlikeness of $\alpha$-spirallike functions. Our approach combines geometric characterizations of the underlying regions with the resultant method for eliminating auxiliary variables, leading to explicit equations for the sharp radii. Some known radius results are recovered as special cases.

We construct a complex structure on $T^2\times S^4$, giving an affirmative answer to the genus-one case of Calabi's problem. The resulting compact complex threefold is non-Kähler and admits no biholomorphic product decomposition.

We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in $\mathcal E_{\mathrm{loc}}$ whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in $\mathbb C^2$ whose singularities satisfy local inequalities involving $\alpha\log\|f\|$ and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.

We prove that a locally maximal plurisubharmonic function on a domain in $\bC^n$, $n\geq2$, is maximal if it has locally analytic singularities with a two-sided locally bounded remainder. Using McAdam's theorem, we show that local maximality forces the analytic spread of each defining ideal to be less than $n$. We use normalized blow-ups and Siu's analyticity theorem to compare singularities through generic divisorial Lelong numbers. We also obtain a criterion for maximality in terms of maximality outside the pole set and the analytic spread of the local defining ideals.

For every integer $r\geq9$, we construct a smooth Griffiths-positive Hermitian metric on $\mathcal O_{\mathbb P^r}(1)^{\oplus r}$ whose top Chern form is pointwise negative on a nonempty open set. This gives counterexamples on compact projective manifolds to Griffiths' conjecture on the positivity of Chern–Weil forms.

In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator $\mathcal{H}_2:A^2_\alpha\to A^2_\beta$ is obtained. More precisely, $\mathcal H_2:A^2_\alpha \to A^2_\beta$ is bounded if and only if $-1<\alpha<0$ and $\beta\geq\alpha+2$ and it is compact if and only if $-1<\alpha<0$ and $\beta>\alpha+2$. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\beta}}$ are also investigated. In particular, when $\beta=\alpha+2$, \[ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\alpha+2}} =\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\alpha+2}} =\frac{\pi\sqrt{(\alpha+2)(\alpha+3)}} {\sin(\pi(\alpha+2)/2)},\qquad -1<\alpha<0. \] When $\beta>\alpha+2$, $\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\beta}}=0$. Furthermore, for every $1\leq p<\infty$, the operator $\mathcal H_2$ belongs to the Schatten class $\mathcal S_p$ if and only if it is compact.

For five complex points $z_1,\ldots,z_5$ satisfying $\sum_i|z_i|^2=5$, Mordell's discriminant inequality states that $\prod_{i<j}|z_i-z_j|^2\leq5^5$, with equality precisely for a unit regular pentagon, up to rotation and permutation. We organize its proof around a direct comparison of discriminant values. A global maximizer gives a centered stationary configuration, whose coefficient equations reduce the generic case to two quartic branches. On each branch, the absolute value of a cubic polynomial gives the original discriminant objective after accounting for the normalization scale. Exact rational interval bounds exclude both branches from the maximum, and the exceptional coefficient cases leave the regular pentagon. The argument determines the maximizing configuration without reconstructing all stationary shapes. We have completed a full Lean formalization of the five-point inequality and both directions of its equality condition, including all reductions and finite checks used in this proof.

Alongside three established presentations of elementary complex analysis, we propose an approach centered on one organizing question: when is contour convolution well defined on homotopy classes? From this starting point, we develop the foundational results of the subject. A supporting conceptual distinction identifies those elements of real and complex analysis that share the same operational and symbolic rules. We refer to this collection as the "Toolbox." Examples and structural connections illustrate the explanatory value and reach of the approach across different aspects of the theory. We also examine the relationship of our approach to the established presentations.

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree $d$ holomorphic endomorphisms of $\mathbb P^2=\mathbb P^2(\mathbb C)$ satisfying suitable expansion and domination conditions. Write $\lambda_1>\lambda_2>0$ for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that ${\mathrm dim}_H \mu_F>\log d/\lambda_1+\log d/\lambda_2$ when $\lambda_2>\log d$, and that ${\mathrm dim}_H \mu_F=2\log d/\lambda_2$ when $\lambda_2\ge2\log d$. Applying these results to an explicit quadratic family, we disprove the Binder–DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li–Pan–Tong–Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier–Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.

We prove a canonical quantization-commutes-with-reduction theorem for CR functions. Under the geometric hypotheses of Hsiao–Ma–Marinescu, if the zero moment level is nonempty and the strictly pseudoconvex reduction is globally CR embeddable of real dimension at least three, the CR Guillemin–Sternberg map is a bounded isomorphism at every real Sobolev order. The central new step is an algebraic and complex-analytic elimination of its finite-dimensional Fredholm defect: multiplicativity removes the kernel, while the conductor of the smooth range, integral dependence, and holomorphic removability produce global lifts. Conductor nonvanishing follows from the reduced Szegő singularity; we also prove it using global CR peak functions. The map is an isomorphism of Fréchet algebras, the reduction is connected, and the corresponding maps for holomorphic line bundles positive near the zero level are isomorphisms in every nonnegative tensor degree. Invariant holomorphic sections vanish in every negative degree; when the reduced base has positive complex dimension, the section maps are isomorphisms in every integer degree.

Winkelmann conjectures that every smooth flexible complex affine variety of dimension at least two is a Rosay–Rudin space. We verify this for every positive-dimensional quotient $G/H$ with $G$ a connected complex semisimple algebraic group and $H$ a closed connected reductive subgroup. Weak and strong tameness coincide, every injection between tame discrete sets extends to a holomorphic automorphism, and complements of tame, finite, or empty sets are Oka. Every sufficiently sparse enumerated sequence can be sent to a fixed sequence by a composition of $n$ complete holomorphic flow maps, with $n\le6$; injective self-maps of the fixed sequence admit such a realization with $n\le2$. These flows preserve an invariant algebraic volume form. The proof uses entire interpolation and regular functions invariant under two solvable subgroups to construct the automorphisms. If $G$ is simple of rank at least two and $G/H$ is spherical and admits a closed equivariant embedding into an irreducible module, the normalization bound improves to $n\le4$. We also prove that $\SL_2/N(T)$, where $T$ is a maximal torus, is an RR-space. This quotient is the complement of a smooth conic in $\mathbf P^2$.

In this paper, we introduce two types of two-sided slice regular functions, which are left slice regular in the first variable and right slice regular in the second variable. We prove that these two classes of functions coincide on axially symmetric domains. We further investigate fundamental properties of two-sided slice regular functions and define a star-product that preserves this regularity. In addition, we establish the Cauchy integral formula for these functions.

Let $M$ be a complete Kähler manifold of complex dimension $n\geq 1$ whose Kähler form is $d$-bounded in the sense of Gromov. We prove that its Bergman space, namely the Hilbert space of $L^2$ holomorphic $n$-forms, is infinite-dimensional. The proof combines Gromov's construction of auxiliary holomorphic line bundles, parameter-uniform $\db$ estimates, and $L^2$ methods with singular weights.

In this paper, we investigate Bohr-type inequalities for certain integral transforms of some bounded holomorphic operator-valued functions defined on a shifted disc. More precisely, we consider functions $f\in H^{\infty}(\Omega_\gamma,\mathcal{B}(\mathcal{H}))$, where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded linear operators on a complex Hilbert space $\mathcal{H}$, and study the Bohr phenomena associated with the series representations in \eqref{e1.2a} and \eqref{e2.1}. Moreover, we determine the sharp radii $R_\gamma>0$ for which the corresponding majorant series satisfy the required inequalities for all $|z|\leq R_\gamma$. One of our main results extends some recent result due to Kumar and Sahoo \cite{Kumar-Sahoo-Meditarr-2023} to a large extent, and the other one is new in the literature

In this short note, we study the behavior of the algebraic entropy of birational automorphism groups in families of projective hyper-Kähler manifolds.

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre–Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre–Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

In this paper, we present an operator representation for Herglotz-Nevanlinna functions in several variables using self-adjoint linear relations. Moreover, a description of the reproducing kernel Hilbert space corresponding to a Herglotz-Nevanlinna function of several variables is also presented, along with a comparison to some existing result concerning the class of Loewner functions.

Let $\Omega\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\Omega$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be Kähler–Einstein. Consequently, extremality, constant scalar curvature, and the Kähler–Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using Hörmander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and Kähler–Einstein throughout the locus where $K_\Omega(z,z)>0$.

We study how the topology and geometry of a compact Kähler manifold relate to the zeros of its one-forms. We determine all implications among several closely related conditions. In particular, we construct a smooth projective variety for which the Aomoto complex is exact for every nonzero holomorphic one-form and every semisimple local system, although every closed real one-form, and hence every holomorphic one-form, has a zero.

We show that any bounded strictly convex domain in $\Bbb C^N$, endowed with the Kobayashi distance, is visible.

The generalization of the Kähler metric in the complex Finsler setting admits four natural versions. Among these, the strongly Kähler–Finsler condition is known to coincide with one of the intermediate conditions. In this paper, we establish the equivalence between the weakly Kähler–Finsler condition and the remaining intermediate condition. Moreover, we obtain new necessary and sufficient conditions for a weakly Kähler–Finsler metric to be strongly Kähler–Finsler. Using these criteria, we construct, for the first time, smooth examples of weakly Kähler–Finsler metrics which are not Kähler–Finsler, via the Kobayashi metrics on smooth bounded strongly convex domains.

Let $L$ be a line bundle of negative degree on an elliptic curve $E$. While the ruled surface $M := \mathbb P(\mathcal O_E \oplus L)$ is one of the simplest varieties in algebraic geometry, its tangent bundle $T_M$ is surprisingly complex: we first show that the ring of symmetric tensors is not finitely generated. We then construct a holomorphic function on the canonical extension $Z_M$ which allows us to show that $Z_M$ is not Stein.

Let $M$ be a closed manifold equipped with a collection of smooth real vector fields $X_1,\ldots, X_k$ and a smooth measure $\mu$. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of $M$ can be connected by a path obtained by joining integral curves of the vector fields, the equation \[ Pu=f \] admits unique zero-average solutions $u$ in the natural energy space, for all zero average data $f\in L^2$. We say that exact regularity holds for the above equation if $u\in H^k(M)$ whenever $f\in H^k(M)$, where $H^k(M)$ is the $L^2$ based Sobolev space of any order $k\in \mathbb{N}$. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields $X_j$, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich–Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the $\bar\partial$-Neumann problem.

In this paper, we investigate refined Bohr-type inequalities involving Schwarz mappings for holomorphic mappings defined on the unit ball \( B_X \) of a complex Banach space. We first establish refined Bohr inequalities for scalar-valued holomorphic mappings \( F : B_X \to \mathbb{U} \). We then extend some results to vector-valued holomorphic mappings involving Schwarz mappings of the form \( F : B_X \to \overline{\mathbb{U}}^{\,n} \), \( n \ge 2 \), where improved Bohr radii are obtained under suitable norm conditions. Furthermore, we study the Bohr radius involving Schwarz mappings for holomorphic mappings whose ranges are contained in the closure of the unit ball of a JB$^*$-triple. Our results generalize the results of Jia et al. \cite{JLP}, Liu et al. \cite{LLP2021} and Liu et al. \cite{LSX2018}.

We first revisit the Nevanlinna condition for representing measures of Herglotz-Nevanlinna functions in two variables and give a general operator-theoretic criterion for extremality. We then derive an explicit formula for the representing measure of a real rational Herglotz-Nevanlinna function in two variables, and combine it with the extremality criterion to obtain a family of extreme elements in the full cone of Nevanlinna measures. The family includes examples with reducible denominators, and these yield counterexamples to the irreducibility assertion in a conjecture of Knese. We then characterize reducibility within this family both algebraically and geometrically.

We prove the Koumandos–Ruscheweyh conjecture for every $0<\rho\leq1$. If $\nu(\rho)$ is the unique root in $(0,1]$ of $\int_0^{(1+\rho)\pi}t^{\mu-1}\sin(t-\rho\pi)\,\mathrm{d}t=0$, then $(1-z)^\rho s_n^\mu(z)\prec((1+z)/(1-z))^\rho$ for every $n\geq0$, $0<\mu\leq\nu(\rho)$ and $z\in\mathbb{D}$, where $s_n^\mu(z)=\sum_{k=0}^n(\mu)_kz^k/k!$. The parameter $\nu(\rho)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.

We construct, through Hodge-theoretical methods, what we call the $E_r$-Jacobian torus, the $E_r$-Albanese torus and the $E_r$-Albanese map of any compact complex manifold that is either {\it page-(r-1)-$\partial\bar\partial$} or {\it $E_r$-sGG}. These two classes of manifolds, the former of which is contained in the latter, have been introduced recently by both authors jointly with J. Stelzig, respectively by the first-named author. A Hodge theory is also developed for the latter class of manifolds. We then apply our results to give structure theorems and an identity of algebraic dimensions in terms of the $E_r$-Albanese map and torus. Other applications result in cohomological and metrical theorems for the $6$-dimensional sphere when it is equipped either with a hypothetical complex structure or with the complex structures very recently claimed to exist in the literature. For example, we show that in the latter case no strongly Gauduchon metric exists on $S^6$.

We introduce a Monte Carlo approach for estimating isolated polynomial zeros through their contour integral representations. For a simple zero enclosed by an isolating contour, we show that the zero can be expressed exactly as the expectation of a complex-valued random variable obtained by uniformly sampling the contour. This yields an unbiased estimator without perturbing the polynomial coefficients. We establish its variance and derive finite sample probabilistic error bounds based on concentration inequalities, with the bounds explicitly reflecting the geometry of the contour and its separation from the zeros. The framework also provides a stochastic formulation of contour based root counting and root isolation. Numerical results illustrate the estimator's behavior under varying polynomial degrees, and we further provide comparisons to existing methods.

In this paper, we prove Borcea's 2-variance conjecture. The Lean 4 formalization of the main results are also provided.