Papers

Strict negativity of holomorphic bisectional curvature need not pass from the base and fibers of a compact holomorphic fibration to its total space, even when the Kodaira-Spencer map is everywhere injective. We construct a compact relative Kähler fibration over a genus-two curve, with a smooth projective threefold as total space and an everywhere injective Kodaira–Spencer map, whose induced fiber metrics have strictly negative holomorphic bisectional curvature, whereas the total space admits no Kähler metric with this curvature property. This gives a negative answer to an open problem posed by To and Yeung. We also construct compact, effectively parametrized Monge-Ampère fibrations from universal quaternionic abelian surfaces over Shimura curves. For products of these families, the generalized Weil-Petersson metric has nonpositive holomorphic bisectional curvature and strictly negative holomorphic sectional curvature, but its mixed bisectional curvatures vanish. Thus strict bisectional negativity can fail both for the existence of a metric on the total space and for the natural metric on the parameter space, despite effective variation at every point.

We study the two-dimensional Matkowski–Sutô equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of Tóth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Daróczy and Páles, persists under holomorphy but not under monotonicity.

Let $\Lambda$ be a compact Bohr–Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

This paper is to construct deformation families of open Calabi-Yau manifolds together with some discussions on Steinness. Firstly, we construct a nontrivial compactifiable deformation of open Calabi-Yau manifolds from rational elliptic surface with global sections. Secondly, we give a condition for the existence of an elliptic fibration with global sections on the complex analytic family of rational elliptic surface with global sections, construct a holomorphic involution on the compactifiable deformation family and claim the non-Steinness for all the compactifiable deformation families constructed. Finally, we give some discussions on the non-Steinness of the quasi-projective variety from $\text{CP}^2 \text{$\#$9} \overline{\text{CP}}^2$ corresponding to a conjecture by Marco Brunella and construct deformation families whose fibers were conjectured to be Stein by Andreas Höring and Thomas Peternel.

We study the convexity of the Bergman space Berezin range of composition operators induced by unit disk and unit ball automorphisms. We obtain several new results characterizing convexity of the Berezin range for linear fractional unit disk and unit ball automorphisms composed with unitary maps.

We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$ rational and $q$ a positive integer, giving an affirmative answer to Question~6.2 in Gundersen's collection. The geometric input is Guillot's theorem on single-valued trajectories of meromorphic vector fields on surfaces. The additional argument compares the fibration provided by that theorem with the original exponential coordinate. A ramification calculation forces every finite nonzero branch value $a$ to satisfy $Da=a$, which excludes such a value and reduces the comparison to a two-point cover. Divisor divisibility and relative algebraic closedness then recover a Riccati equation over the original coefficient field. No growth hypothesis is imposed. We also identify the algebraic degree with the least integer deck period and describe the growth and value fibres of the resulting solutions.

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set $X=\{w=0\}\cup\{\Im z=\Im w=0\}\subset\C^2$ admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function $\rho(z,w)=(\Im w)^2+\sinh^2(\Im z)|w|^2$ and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka–Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in $\C^2$ with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon–Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.

In this paper, we study composition operators that act between different de Branges-Rovnyak spaces. Our main results are suggested by a paper of Mashreghi and Shabankhah concerning composition operators between model spaces. We also answer several related open questions posed by Dellepiane and Seco. To prove these results, we apply reproducing kernel Hilbert space methods, Sarason's approach to composition operators as integral operators, and Aleksandrov-Clark measures.

Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_{\beta}(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^{\beta}<+\infty\Bigg\}.$$ Given $\beta_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_\beta(\mathbb{D}^2)$ if and only if $\beta\leq \beta_{c}.$ We also show that the critical index satisfies $\beta_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$

The celebrated theorems of Shilov, Arens–Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The étale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

We study compact Kähler varieties admitting int-amplified endomorphisms from the viewpoint of positivity of tangent sheaves. Our main result shows that the tangent sheaf of such a variety is weakly positively curved; in particular, it is pseudo-effective in a strong sense. This establishes a new link between complex dynamics and the positivity theory of tangent sheaves, and provides an alternative to equivariant MMP techniques in the compact Kähler setting. As an application, we prove a Kähler analogue of Yoshikawa's structure theorem: for a compact Kähler klt variety admitting an int-amplified endomorphism, after an equivariant quasi-étale cover, it admits an equivariant flat MRC fibration with irreducible fibres onto a complex torus; in the smooth case, the fibration is smooth whose periodic fibres are of Fano type. To this end, we prove the existence of minimal models in the case of numerical dimension zero canonical divisors for non-projective Kähler varieties. We further study rationally connected manifolds with pseudo-effective tangent bundle, focusing on their relation to Fano-type properties, almost homogeneity, and fundamental groups. In the proof, we also show that every surjective endomorphism of a compact klt Kähler variety lifts to a suitable maximally quasi-étale cover.

Fix integers $k,d\geq2$. Let $\mathrm{End}_d^k$ denote the parameter space of endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\mathrm{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every dominant rational self-map of $\mathbb{P}^k$ commuting with an iterate of $f$ is itself an iterate of $f$. The key intermediate result classifies dominant rational semiconjugacies to iterates of $f$ from normal projective varieties of dimension $k$. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of $f$. An analogous centralizer theorem holds generically for regular polynomial endomorphisms of $\mathbb{C}^k$. These results extend several theorems of Pakovich to higher dimensions. We give two applications. We first classify periodic dominant correspondences between a general endomorphism and an arbitrary endomorphism of degree at least two. We then show that, for a general $f\in\mathrm{End}_d^k$, an irreducible hypersurface of $\mathbb{P}^k$ is $f$-special in the sense of Ghioca–Tucker and DeMarco–Mavraki if and only if it is $f$-preperiodic.

We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.

We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,\pi]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\,d\sigma(\eta), \] where $\sigma$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $\lambda_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta)=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\le\omega_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.

We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\geq2$ and let $\Sigma$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_\ell\to+\infty$ such that, for every $\ell$, there exists a complex manifold of complex dimension $2d_\ell+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $\Sigma$ into $\mathbb S^{2n}$ of degree $d_\ell$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z} = \{z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\} \subset \mathbb{C}^4$. These metrics have tangent cone $\mathbb{C} \times (\mathbb{C}^2 / \mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}^2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.

Let $(a_n)_{n\geq0}$ be a sequence of complex numbers, and define \[ b_n=\sum_{k=0}^n \binom{n}{k} a_k, \qquad c_n=\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}a_k. \] Let $\beta>1$, put $\alpha=\sqrt{\beta^2-1}$, and suppose that $b_n,c_n=O(\beta^n)$. Mashreghi and Ransford proved that \[ \limsup_{n\to\infty}\frac{|a_n|}{\alpha^n} \leq \kappa \left(\limsup_{n\to\infty}\frac{|b_n|}{\beta^n}\right)^{\!1/2}\! \left(\limsup_{n\to\infty}\frac{|c_n|}{\beta^n}\right)^{\!1/2} \] with a universal constant satisfying $2/\sqrt{3} \leq \kappa\leq2$. We prove that the optimal constant is indeed $\kappa=2/\sqrt{3}$. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group $\mathcal{H}(S)$, has finite index in the asymptotically conformal modular group $\text{Mod}_0(X)$, where $X$ is a hyperbolic structure on $S$. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in $\mathcal{H}(S)$ of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if $S$ has $n\geq 3$ ends, then $\text{Mod}_0(X)$ has type $F_{n-1}$ but not $FP_n$. We also establish analogous results for $L^p$ modular groups.

We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space $A^2_1$. The correspondence sends an inner function $I$ to the invariant subspace $[I']$, and in the opposite direction, assigns to a non-zero function $H \in A^2_1$ the Liouville map $I_H$ associated to the canonical solution of the Gauss curvature equation $\Delta u = |H|^2 e^{2u}$. We prove that $I'_H$ generates the same cyclic subspace as $H$. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence $I \to [I']$ is a bijection.

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao–Deng–Matsumura. In contrast, for compact simply connected complex manifolds $X$, we prove that the answer is affirmative if $H^1(X,\mathcal O_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

In this paper, we continue the development of the foundations of a quaternionic function theory that is associated to a fractional proportional Cauchy-Riemann type operator acting on quaternionic functions. We introduce a new class of fractional hyperholomorphic functions adapted to affine transformations of the domain, and study the corresponding quaternionic right module structure. In particular, we establish versions of the Stokes, Borel-Pompeiu and Cauchy formulas for this function theory. A covariance property of the fractional operator under affine transformations is proved, showing that the composition with an affine map, weighted by a suitable quaternionic factor, preserves the class of fractional hyperholomorphic functions. The result is shown to include the case considered previously by the authors, and extends their covariance property to the setting of affine transformations.

In this article, we investigate refined and generalized versions of the Bohr and Bohr–Rogosinski inequalities for a normalized subclass $\mathcal{P}_{\mathcal{H}}^{0}(\alpha)$ ($0 \le \alpha < 1$) of univalent close-to-convex harmonic mappings $f = h + \overline{g}$ defined on the open unit disk $\mathbb{D} \subset \mathbb{C}$. By incorporating non-negative monotone increasing functions associated with the planar area integral $S_r/\pi$ of the image domain $f(\mathbb{D}_r)$, we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance $d(f(0), \partial f(\mathbb{D}))$. Furthermore, we establish sharp Bohr–Rogosinski-type inequalities involving powers of the modulus of the mapping $|f(z)|^p$ ($p \ge 1$). All associated radii are proven to be sharp, and extremal functions realizing the equality cases are explicitly identified. As applications, our results generalize and unify several well-known classical and recent theorems in geometric function theory.

Let $H$ be the Hilbert transform, let $h$ be a quasisymmetric homeomorphism of the unit circle $S^1$, and set $V_hu=u\circ h$ and $J_h=V_hHV_h^{-1}$, defined on the Sobolev space $H^{1/2}(S^1)$. We prove that $h\mapsto V_h$ is nowhere continuous in operator norm, although it is continuous in the strong operator topology. By contrast, the induced map $h\mapsto J_h$ is a real-analytic diffeomorphism onto its image in the operator-norm topology. Based on this, we further compute the differential at the identity and show that it is precisely the Calderón commutator. In graph coordinates, the tangent map admits a weighted Hankel matrix representation, whose Hilbert-Schmidt norm recovers the Weil-Petersson tangent quadratic form.

Exceptional CR-quadrics are studied. An example of an exceptional (4,4)-quadric is constructed; it realizes the minimum exceptional type with respect to both n and k. Its graded Lie algebra is described. The available information on exceptional types is summarized, and the lattice of CR-types is decomposed into the disjoint union of three sets: A, the types for which exceptional quadrics are impossible; B, the types for which examples of exceptional quadrics are known; and C, the types whose status is currently unknown (neither an example nor a nonexistence proof is known). Several questions are posed.

The Dinew–Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the Kähler metrics in that class, providing a variational approach to Kähler geometry. Dinew and Popovici proved that every critical point is Kähler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-Kähler critical metrics on products of two Kähler surfaces $(S_1,\eta_1)$ and $(S_2,\eta_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-Kähler critical products exist in the Aeppli class $[\eta_1+\eta_2]_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed Kähler background when this background metric has nonnegative holomorphic bisectional curvature.

Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of $(1,1)$-forms belongs to the Hodge–Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, we present counterexamples constructed by AI: Nakano positive $k\times k$ matrices of constant $(1,1)$-forms on $\mathbb{C}^n$ whose determinants have singular Lefschetz maps in bidegree $(1,n-k-1)$. Under the simultaneous diagonalizability (SD) condition, we prove the Hodge–Riemann property in every bidegree $(p,q)$ with $p+q=n-k$ and $\min(p,q)\leqslant1$. The proof uses the generalized Alexandrov–Fenchel inequality of Ross, Süß, and Wannerer. For every $n\geqslant6$ and $2\leqslant k\leqslant n-4$, we also present SD examples whose determinants have singular Lefschetz maps in bidegree $(2,n-k-2)$. Their underlying idea comes from Ross–Toma's construction. The remaining cases $n=k\geqslant4$ and $n=k+1\geqslant5$ are open for general Griffiths positive matrices. They reduce to the top-Chern case of Griffiths' positivity question, equivalently to Finski's double mixed discriminant problem. We also ask whether a $2\times2$ counterexample on $\mathbb{C}^4$ can be both Nakano positive and dual Nakano positive.

In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus $(\xi_j)_{j=1}^{n}$ and $(\zeta_j)_{j=1}^{n}$ the inequality: $$\bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m\bigg|^2+ \bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m^{-1}\bigg|^2\leqslant 4n^2\sin^2\frac{\pi}{n}$$ holds for $n\geqslant 4$. As a consequence, for $n=4,$ we disprove Hall's conjecture on Heinz type inequality for harmonic self-mappings of the unit disk. The connection with the theory of elliptic billiards via the Mather beta function will also be given.

Jarnicki and Pflug constructed a bounded balanced domain of holomorphy in $\mathbb C^n$, $n\ge 3$, with continuous Minkowski function which is not Kobayashi complete, and explicitly asked whether the same phenomenon can occur in dimension two. We give such a domain in $\mathbb C^2$.