Papers

We construct slope-stable ample rank-two vector bundles on every Hirzebruch surface $\mathbb{F}_e$ that admit no smooth strongly pseudoconvex Finsler metric with semipositive Kobayashi curvature. These examples give a negative answer to Kobayashi's problem. The same bundles give counterexamples to the smooth strongly pseudoconvex Finsler formulation of Fulton's problem and to the Hartshorne–Schneider conjecture on the convexity of complements. For $e=1$, these counterexamples have a simply connected Fano base that is not a product.

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

We introduce sequences of generalized moments that are a generalization of sequences preserving summability and sequences of moments from Balser's theory of moment summability. We construct extensions of Laplace and Borel integral operators for these new sequences. We find integral representations of moment derivatives for generalized moments. We show the characterization of summable solutions of moment-PDEs with generalized moments. We describe singular directions for summable solutions of moment-PDEs with generalized moments and we express jumps across these singular directions in terms of hyperfunctions and in terms of kernel functions associated with sequences preserving summability, which are connected with given sequences of generalized moments.

In this paper, we generalize Guan's sharp effective strong openness theorem to the case of several plurisubharmonic weights. The proof uses a generalized Skoda-type $L^2$ division theorem for multiplier ideal sheaves. We also discuss the relationships among the $L^2$-division theorem, the $L^2$-extension theorem and the effective openness theorem.

We construct a smooth projective family of rational surfaces, an effective line bundle on its total space, and a prescribed semipositively curved singular Hermitian metric on the central fibre that has no semipositive extension to any neighbourhood of that fibre. It has analytic singularities and is of minimal singularity type. The failure persists even when the restriction is only required to have the same singularity type as the prescribed metric. The family is obtained by blowing up four disjoint sections of a product, with three points becoming collinear on the central fibre. An integrable adjoint section on that fibre cannot extend because the corresponding adjoint systems vanish on every nearby fibre. This gives a negative answer to Păun's metric-extension question, listed as Question 38 by Dinew, Guedj, and Zeriahi.

Fix a polynomial $P(x)=\sum_{j=0}^{m}a_jx^j$ of degree $m$, and let $I_kP$ be its $k$-fold antiderivative with all integration constants zero. Borwein, Chen, and Dilcher proved that, after $x=ky$, the nonzero zeros of $I_kP$ converge to the zeros of $L[P](y)=\sum_j a_j j! y^j$. That fixes a direction but neither a position nor successive differences. We compute the next two terms at a simple nonzero zero $\lambda$ of $L[P]$; these supply both. Integer-order and Riemann-Liouville integration reduce, after scaling, to one rational family $c_j(h)=\prod_{\ell\le j}(1+\ell h)^{-1}$ sampled at $h=1/k$ and $h=1/\alpha$; the scaled zero is analytic in $h$ at $h=0$. We prove $r_k=k\lambda+C_P(\lambda)+D_P(\lambda)/k+O(k^{-2})$ with explicit $C_P$ and $D_P$, whence $r_k-r_{k-1}=\lambda-D_P(\lambda)/(k(k-1))+O(k^{-3})$ and $r'(\alpha)=\lambda-D_P(\lambda)/\alpha^{2}+O(\alpha^{-3})$. So $\lambda$ is at once the scaled root location, the limiting discrete spacing, and the limiting continuous-order velocity, and $C_P(\lambda)$ is the offset of an asymptotic line. For binomials the limiting spacing is a multiple of a gamma-ratio quantity (the Factorial Fraction), and the nonzero reduced roots form an exact regular polygon at every positive order. For real quadratics the collision order of two branches is explicit and, under nondegeneracy hypotheses, positive exactly when $P$ has distinct real zeros and $L[P]$ has none; in arbitrary degree the collision discriminant carries a universal factor supported at negative orders, leaving a factor of degree at most $m(m-1)/2$. All trajectories form one plane curve of degree $m$ with leading form $L[P]$, whose collisions are discriminant values of the order projection and whose order-zero end is governed by a harmonic transform dual to the arithmetic one at infinity. The case $\lambda=0$ is rigid: the origin is pinned at every positive order.

We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of $\overline{\mathbb{R}^n}$ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.

In a previous paper, the authors introduced, by analogy with Stein manifolds, a class of complex manifolds called $\mathscr{M}$-manifolds. This class contains all complex projective manifolds. Moreover, every compact $\mathscr{M}$-manifold is Moishezon. In this note, we construct examples showing that the class of compact $\mathscr{M}$-manifolds sits properly between the classes of projective and Moishezon manifolds.

It is well-known that Douady and Hubbard proved that the Mandelbrot set is connected by constructing a uniformizing holomorphic map for its complement from the Böttcher coordinates. This strategy has a clear analogue in the setting of quasiregular Mandelbrot sets. In this paper, we show that the natural generalization of the Douady-Hubbard uniformizing map is quasiconformal in a neighbourhood of infinity, as well as, in fact, outside a certain neighbourhood of the quasiregular Mandelbrot set. On the other hand, we also show that this uniformizing map can reverse orientation at some parameters, which forces it to have folds and, in particular, prevents it from being injective. This feature cannot happen in the holomorphic setting. The connectivity of the quasiregular Mandelbrot sets therefore cannot be established along the Douady-Hubbard route, and remains open.

Let $f$ be a Hénon map of $\mathbb C^2$. We provide quantitative versions of several results involving dynamical objects associated to $f$. Our results can be interpreted as a quantitative version of Pesin theory with geometric control. As an application we show that the perdiodic points of period $n$ of $f$ equidistribute towards its equilibrium measure exponentially fast as $n$ tends to infinity.

We compute the integral Bott–Chern cohomology of a class of generalized Calabi–Eckmann manifolds of complex dimension \(3\). Although these manifolds are mutually non-biholomorphic, we show that their integral Bott–Chern cohomology rings are nevertheless isomorphic. A cohomological distinction is then obtained via a characteristic map in cohomology, connecting singular, Dolbeault, and integral Bott–Chern cohomologies.

LVMB manifolds are a large class of compact complex non-Kähler and non-algebraic manifolds, constructed from an algebro-combinatorial datum. We provide an upper bound on their algebraic dimension and prove it to be sharp. Comparing our result with the lower bound found by Meersseman, we obtain the exact algebraic dimension of an LVMB manifold in many relevant cases. Pursuing our analysis further, we also obtain a description of the algebraic reduction of an LVMB manifold. From this, it emerges that Meersseman's lower bound coincides with the dimension of the rational component. This bimeromorphic invariant of an LVMB manifold is then computed in terms of the nonrationality degree of the associated toric quasifold. This is an invariant recently introduced by Battaglia and Prato, for which we provide an explicit formula.

We show that the degree $d$ Jouanolou foliation $\mathcal J_d$ on $\mathbb P^2$ is structurally stable for every $d \geq 2$. We prove the existence of a neighbourhood $\mathcal U_d$ of $\mathcal J_d$, such that every foliation $\mathcal F \in \mathcal U_d$ satisfies: the Fatou set is a connected disc bundle over a compact Riemann surface of genus $d(d+1)/2$, no leaf is dense in $\mathbb P^2$, and all but countably many regular leaves are biholomorphic to the unit disc. Finally, we prove transverse perfectness of the Julia set on the regular locus for $d \geq 2$.

This work constructs a new function class over the algebra of dual quaternions: dual slice functions, which include one-sided dual quaternionic polynomials. The peculiar properties of dual slice functions therefore provide new tools to understand factorization of dual quaternionic polynomials, which was extensively studied in recent years for its applications to mechanism science. Dual slice functions are strictly related to the well-established theory of slice and slice regular functions over quaternions and dual quaternions.

Let $D\subseteq \mathbb{C}^n$ be a bounded $\mathbb{C}$-convex domain with $C^1$ boundary whose outward unit normal admits a modulus of continuity $\omega$ satisfying $\lim_{t\to 0^+}\dfrac{\omega(t)}{\sqrt t}$. We prove that $D$ admits an increasing exhaustion by bounded $C^{\infty}$ strongly linearly convex domains. This, in particular, answers a question posed by Azinberg \cite{azin} in affirmative for a class of domains $D$.

We study the hard-core model on recursively defined sequences $(G_n)_{n\geq0}$ of graphs with a fixed number $k\geq 1$ of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number $m\geq 2$ of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing $k$ labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs $G_n$ are uniformly bounded and the distances between the labeled vertices in $G_n$ diverge, the complex zeros of the univariate independence polynomials $Z_{G_n}(\lambda)$ avoid a neighborhood of the non-negative real axis. By the Lee–Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph $G_0$. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps $F_\lambda$ on the $(2^k-1)$-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.

Let $\mathcal E$ be a separable Hilbert space, $B\in H^\infty_1(L(\mathcal E))$, and $\alpha>-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A^2_{\alpha,\mathcal E}$ are the sub-Bergman spaces $\mathcal A_\alpha(B)$ and $\mathcal A_\alpha(B^*)$. In this paper, we prove that \[ \mathcal A_\alpha(B)\supset A^2_{\alpha-1,\mathcal H(B(0))} \quad\text{and}\quad \mathcal A_\alpha(B^*)\supset A^2_{\alpha-1,\mathcal H(B(0)^*)}. \] This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function $B$ such that $\|B(0)\|<1$ and \[ \mathcal A_\alpha(B)\approx \mathcal A_\alpha(B^*)\approx A^2_{\alpha-1,\mathcal E}, \] but $B$ is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.

We prove a rigidity theorem: a globally symmetric complex Finsler space $(M, J, F)$ is necessarily a Kähler-Berwald space, namely $F$ must be a Kähler-Berwald metric. We also obtain a Schwarz lemma for holomorphic mappings $f$ from an arbitrary bounded symmetric domain $\mathfrak{D}$ into itself whenever $\mathfrak{D}$ is endowed with an $\mbox{Aut}(\mathfrak{D})$-invariant Kähler-Berwald metric $F$ such that its holomorphic sectional curvature is bounded below and above by negative constants $-K_1<0$ and $-K_2<0$, respectively. The novelty of this Schwarz lemma is that the Lu constant of $(\mathfrak{D},F)$ is optimal for each given $F$ whenever rank$(\mathfrak{D})\geq 2$.

We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining \(1\)-form.

We construct a two-parameter family of complex structures on the standard six-sphere, starting from a rational elliptic surface with singular fibers $III^*,I_1,I_1,I_1$. The resulting torus fibration over $\mathbb{P}^1$ has one multiple fiber of multiplicity four, one hexagonal fiber, and two rank-one degenerations. These complex structures are not biholomorphic to the original Alpöge–Claude examples.

We show that on every smooth hypersurface $X\subset\mathbb P^4$ the Demailly–Semple bundle $E_{3,5q}T^*_X$ of invariant jet differentials of order $3$ and weighted degree $5q$ is not ample, for every integer $q\ge1$. This contradicts the Diverio–Trapani conjecture in the form “$E_{k,m}T^*_X$ is ample for all $m\gg0$” for hypersurfaces in $\mathbb P^4$ at $k=3$. We then extend the obstruction to complete intersections of any dimension, to jet differentials of order $4$, and to all sufficiently divisible weights, and we discuss how ampleness of $E_{k,m}T^*_X$ depends on $m$.

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

We prove a converse to the original Skoda $L^2$ division theorem, asserting that Skoda $L^2$ division estimates for a single nonzero holomorphic function force the weight to be plurisubharmonic. In addition, we establish corresponding results for higher-rank quotient bundles. Finally, we derive an optimal $L^2$ extension theorem directly from the division estimates.

Given a reproducing kernel Hilbert space $\mathcal H$ and a bounded operator $T$ on $\mathcal H$, we say that $T$ obeys the strong reproducing kernel thesis (SRKT) if $\|T\|=\sup_k\|Tk\|$, where $k$ ranges over the normalized reproducing kernels in $\mathcal H$. Two problems are posed. One of these is to determine which Hankel operators on the Hardy space $H^2$ obey the SRKT. The other concerns the geometry of the unit ball of $BMO$ with respect to the Garsia norm.

The primary objective of this paper is to investigate sharp constant problems in harmonic Hardy spaces. We first establish a sharp asymptotic constant for a Littlewood-Paley type inequality applicable to weak harmonic $(K, K_0)$-quasiregular mappings, where $K \geq 1$ and $K_0 \geq 0$. Subsequently, we derive sharp Hardy-Littlewood type constants for harmonic mappings.

The main purpose of this paper is to address an open problem posed by Huang, Rasila, and Zhu in [Anal. Math. 48(2022), 1069-1080]. We first establish a sharp Lipschitz continuity result for locally univalent harmonic Bloch mappings with respect to the pseudo hyperbolic metric. This result is then applied to demonstrate that harmonic $(K,K_0)$-quasiregular Bloch type mappings are Lipschitz continuous under the same metric, where $K\geq1$ and $K_{0}\geq0$. Moreover, the estimates obtained are asymptotically sharp as $K \to 1^+$ and $K_0 \to 0^+$. In particular, when $K_0 = 0$, our theorem provides a solution to the aforementioned open problem in the asymptotically sharp sense.

For complex Hessian equations on compact Kähler manifolds, we give an alternative proof of the gradient estimate without using blow-up argument, and clarify its dependence on the right-hand side. In addition, using the similar approach, the case of Kähler manifolds with boundary is also investigated.

We characterize the continuous mappings of the unit disk that preserve every Carleson interpolating sequence. Preservation in one direction is equivalent to being a disk homeomorphism whose inverse is uniformly continuous in the pseudohyperbolic metric and whose associated weighted pushforward is bounded on positive Carleson measures. Preservation in both directions is equivalent to being a disk homeomorphism that, together with its inverse, is uniformly continuous in that metric and has a strongly quasisymmetric boundary extension. Within this class, comparability of the boundary defects is equivalent to a bi-Lipschitz boundary map. Building on the canonical factorization established in our earlier work, we give explicit criteria for membership in the continuous boundary kernel. For a homeomorphism of the closed disk with identity boundary values and a uniform boundary expansion of first order, kernel membership is equivalent to positivity of the determinant of the real boundary differential, or equivalently of the inward normal derivative of the boundary defect. No differentiability in the interior is required. The $C^1$ criterion on the closed disk follows as a corollary, while a counterexample shows that pointwise boundary differentiability does not suffice. We also characterize factorizations with a quasiconformal kernel and record the exact maximal dilatation of its radial factor.

Let $f_{\lambda}(z)=\lambda e^z$, where $0<\lambda<1/e$. In 1984, Devaney and Krych showed that the Julia set consists of pairwise disjoint hairs. Viana further in 1988 proved that these hairs are $C^{\infty}$. In this paper we show that hairs are not analytic except for trivial ones. This solves a long-standing open question.

Little is known about the dynamics of Toeplitz operators on the Bergman space, and even in the case of harmonic symbols, few results are available. In this paper, we study the dynamical properties of Toeplitz operators with symbols of the form $a \bar{z} + p (z)$ on the Bergman space, where $a\neq 0$ and $p$ is analytic on the closed unit disk. We obtain some necessary conditions and some sufficient conditions for such operators to be hypercyclic, weakly mixing, mixing, chaotic, or frequently hypercyclic. As an application, we obtain a complete characterization of hypercyclicity for Toeplitz operators with harmonic linear polynomial symbols. We also show that Toeplitz operators with the same symbol may exhibit different hypercyclic behavior on the Hardy and Bergman spaces.