Papers

We first prove a version of the Demailly-Paun theorem for compact weakly Kaehler spaces. Secondly, for a big and nef Bott-Chern class on a compact normal Kaehler space, we prove that the restricted non-Kaehler locus, defined using Kaehler currents with weak analytic singularities, is equal to the usual non-Kaehler locus.

The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter $a$. For $0<a\le 1/2$, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit $a\to 0$, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When $1/2<a<1$, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.

In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.

New biholomorphic invariants of a germ of a real-analytic set at a singular point (either an RC-singular point or a singular point of a smooth structure) are found: the Bloom-Graham singular type and the singular l-type. Their construction is based on the Bloom-Graham type and l-type constructions, respectively. A series of examples is considered.

We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the “general” setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly–El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.

We study transversely projectively flat regular foliations on compact Kähler manifolds.

Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge–Ampère measures associated with bounded plurisubharmonic sequences $u_j\to\varphi$ locally in $L^1$, without assuming a common bound. On a bounded domain $\Omega\subset\mathbb C^n$ ($n\geq2$), and for a bounded maximal plurisubharmonic $\varphi$, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic $\varphi$ when $\Omega$ is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure $\nu$ can also be realized by such a sequence with $(dd^cu_j)^n\to\nu$ vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic $\varphi$ and nonnegative Radon measure $\mu$, we construct $u_j\leq\varphi$ converging to $\varphi$ in $L^p(\Omega)$ for every $1\leq p<\infty$, with $(dd^cu_j)^n\to(dd^c\varphi)^n+\mu$ vaguely. The approximants can be chosen smooth when $\varphi$ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.

We study slice regular functions on the real division algebras $\mathbb{H}$ and $\mathbb{O}$ from the point of view of equivalence under automorphism and conjugation actions. Motivated by recent orbit-theoretic descriptions in terms of the invariants $(Tr,N,cdiv)$, and by the quaternionic theory of $*$-conjugation by semiregular functions, we investigate whether a slice regular function can be replaced by a canonical representative in its equivalence class. The normalization considered in this paper is algebraic: we look for representatives whose isolated nonreal zeros are aligned in a single complex slice. We call such a representative a model when it belongs to the strong-equivalence class of the original function. We prove that purely vectorial functions always admit models, while in general strong equivalence is too rigid and such representatives need not exist. We therefore introduce semi-models, which preserve the symmetrization and the real and spherical part of the zero set, but are allowed to leave the original strong-equivalence class. Our main result proves that every slice regular function $f$ with $N(f)\not\equiv 0$ admits a semi-model. The construction clarifies the different roles of the invariants $(Tr,N,cdiv)$ and the distinction between strong equivalence, weak equivalence, and semiregular conjugacy.

Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\to\pi/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $\pi/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.

We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is $n-1$, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.

Let $\Omega_{r,s}$ and $\Omega_{r',s'}$ be Type-I bounded symmetric domains with $s>r$ and $s'>r'$. Let $F:U\to M_{r',s'}(\mathbb C)$ be holomorphic, where $\overline{\Omega}_{r,s}\subset U$, and suppose that $F(S_{r,s})\subset S_{r',s'}$. If \[ k(s-r)\leq s'-r'<(k+1)(s-r) \quad\text{and}\quad r'>kr, \] then, up to target coordinates, the map contains a fixed identity block of size $r'-kr$. The proof first turns the boundary equation into a matrix identity for the first derivative. A direct dimension count gives $g(s-r)\leq s'-r'$, where $g$ is the rank of a positive semidefinite matrix arising from this identity. The extremal property of the Shilov boundary and the scalar Hopf boundary lemma then show that certain rows of the map are constant. This gives the fixed identity block. We also give examples showing that its size is optimal and explain why the proof needs $F$ on a neighborhood of $\overline{\Omega}_{r,s}$.

Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every $d\ge 2$, the multipliers of any $2d-2$ distinct periodic orbits of degree $d$ rational maps are algebraically independent over $\mathbb C$, provided that at most $d$ of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the $d+1$ fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.

Slice regular functions are a generalization of holomorphic functions where alternative real $*$-algebras are considered instead of the field of complex numbers. For such functions we show that zero sets and function values on suitable subsets may be prescribed. As a consequence, we show that for any axially symmetric domain there exist slice regular functions which (due to the nature of its zero set) can not be extended to a larger such domain.

In 1986, Shea and Sons obtained the following bound for a meromorphic function $f$ of finite order $\rho$ in the unit disc, under the hypothesis $0<\lambda(f)\leq+\infty$, and for every positive integer $n$: \[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n(\rho+1)}{\lambda(f)}. \] Under the condition $0<\alpha(f)\leq+\infty$, they also obtained \[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n}{\alpha(f)}. \] Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<\lambda(f)\leq+\infty$, one has \[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n(\rho+1)}{\lambda(f)}, \] and under the condition $0<\alpha(f)\leq+\infty$ one has \[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n}{\alpha(f)}. \] The coefficient $n$ in each of $n(\rho+1)/\lambda(f)$ and $n/\alpha(f)$ is best possible for every $n$, and the dependence on $\rho$ and $\alpha(f)$ in these terms is also essential. At the end, we extend both estimates to finite subsets $A$ of an $n$-dimensional complex vector space $V\subset\mathbb{C}(z)$, with $f^{(n)}$ replaced by the monic differential operator $D_Vf$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.

We prove the Second Main Theorem with truncation level one for holomorphic maps with Zariski-dense image intersecting a reduced effective Cartier divisor: Let $A$ be a semi-abelian variety with an equivariant compactification $A\subset\overline A$, and let $f:\mathbb C^p\to A$ be a holomorphic map with Zariski-dense image. Let $D$ be a reduced effective divisor on $A$ extending to a divisor $\overline D$ on $\overline A$. After possibly replacing $\overline A$ by another equivariant compactification depending only on $D$ and independent of $f$, for every $\varepsilon>0$, \[T_f(r,\overline D)\leq_{\mathrm{exc}}N_f^{[1]}(r,D)+\varepsilon T_f(r,\overline D).\]

We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\tilde f)$ is not integrable, where $\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $Y(\tilde f)$ fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a $Y$-integrability version of Zygmund's result.

We associate normal toric transverse analytic envelopes with the singular orbits of diagonal holomorphic actions, and construct their canonical flat transport. At independent singular supports, the semiglobal first-integral algebra is determined by periodic resonances; complete integrability is equivalent to full resonance rank and finite envelope monodromy. We classify labelled residual actions with a fixed envelope by Grassmannian families and reconstruct the ambient weights from the labelled quotient and logarithmic residues at full resonance rank. For commuting vector fields in convergent simultaneous semisimple normal form, we identify the algebra of holomorphic first integrals with the common kernel of the derivations induced on the toric envelope. Under compatible flat transport this yields a semiglobal criterion. Polynomial examples show that losses of first integrals may be invisible to any prescribed finite jet.

In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact Kähler space $X$, then, after passing to a finite quasi-étale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of Höring–Peternell, originally established for projective varieties, to the Kähler setting. The proof relies on two main ingredients, both of which are new even in the projective case. The first is a flatness theorem for stable sheaves: we show that a slope-stable pseudo-effective sheaf with vanishing first Chern class is Hermitian flat. This is obtained by combining Hermitian–Einstein theory with the subharmonicity properties of direct image sheaves. The second is a singular Kähler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $\overline{Z}$ a nilpotent orbit closure in the projectivization $\mathbb{P}(\mathfrak{g})$. We investigate the space of tangent directions of lines on $\overline{Z}$ passing through a general point $z$, denoted by $F(\overline{Z},\,z)$. We first prove that every irreducible component of $F(\overline{Z},\,z)$ is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of $F(\overline{Z},\,z)$ in two cases: the case where $\overline{Z}$ admits a Springer resolution; the case where $\overline{Z}$ is square-zero in a projectivized simple Lie algebra of classical type. As an application, two corollaries are presented: a characterization of Richardson orbits among square-zero orbits in terms of $F(\overline{Z},\,z)$; a description of $F(\overline{Z},\,z)$ for $\overline{Z}$ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.

Several upper bounds as well as one lower bound for the operator norm of the generalized Hilbert operator $\mathcal{H}_b$ acting on weighted Bergman spaces $A_{\alpha}^p$ are established. Moreover, under some mild assumptions, we obtain the exact norm of $\mathcal{H}_b$ on $ A_{\alpha}^p$.

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective linear isometry $T$ of $H^p_\omega(D)$ has the rigid form \( Tf=T(1)(f\circ\varphi), \) where $\varphi\in\operatorname{Aut}(D)$. This extends the classical Forelli-type classification beyond highly symmetric or polynomially convex domains to arbitrary smoothly bounded pseudoconvex domains. The principal difficulty is not the construction of a holomorphic symbol, but proving that this symbol takes values in $D$ and is in fact biholomorphic. We overcome this difficulty by combining equimeasurability methods of Rudin and Schneider with boundary uniqueness, holomorphic approximation, plurisubharmonic exhaustion functions, and removable-singularity arguments across analytic sets. We also solve the complementary geometric problem of determining when an automorphism of $D$ gives rise to an isometry. The answer depends decisively on the boundary measure. We construct two natural measures for which every automorphism induces an isometry: one obtained from an invariant defining function when $\operatorname{Aut}(D)$ is compact, and the other given by Fefferman's invariant surface measure. In sharp contrast, we exhibit domains with noncompact automorphism group—including domains biholomorphic to the unit ball—for which the analogous conclusion fails for ordinary Euclidean surface measure. Thus the isometric structure of Hardy spaces detects not only the biholomorphic geometry of the domain, but also the finer interaction between that geometry and the chosen boundary measure.

We determine the radii of concavity for the classes $\mathcal{S}_e^*$ and $\mathcal{C}_e$ of univalent functions associated with the exponential mapping $e^z$. Under the geometric framework of Avkhadiev–Wirths for conformal mappings with unbounded convex complements of opening angle $\pi A$ ($A \in (1,2]$), the radii are characterized by explicit transcendental equations. Sharpness is established globally by constructing explicit univalent extremal functions related to the disk automorphism $\omega_0(z) = -z$, and the strict monotonicity of these radii with respect to the parameter $A$ is verified numerically.

We study two notions of positivity for smooth currents on singular spaces. We show that they are not equivalent by establishing a necessary condition on the fourth Whitney cones. We also construct an explicit compact normal projective variety $X$ and a real smooth $(1,1)$-form $T$ on $X$ such that $T$ has local smooth ddc-potentials and is a Kähler current, but $T$ is not a positive form (hence not Hermitian).

Let $\Sigma$ denote the class of functions $f(z) = z + b_0 + b_1 z^{-1} + \cdots$ that are analytic and univalent in the exterior unit disk $\Delta^* = \{z \in \mathbb{C} : |z| > 1\}$. In 1962, Ch. Pommerenke proved that if $F$ and $G$ are convex functions in $\Sigma$, then every convex linear combination $H = \lambda F + (1-\lambda)G$ ($0 < \lambda < 1$) remains univalent and belongs to $\Sigma$. In Hayman's problem collection (Research Problems in Function Theory, Problem 6.10), Pommerenke raised the question of whether $H$ is necessarily also a convex function. We resolve this question in the negative by constructing an explicit counterexample in $\Sigma$ with parameters in $\mathbb{Q}(i)$. The argument is self-contained and has been formally certified in the Lean 4 proof assistant.

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

We continue our study of the conformal loop ensemble (CLE) with parameter $\kappa=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

We consider the conformal loop ensemble (CLE) with the parameter $\kappa=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_\kappa$ loops as $\kappa \downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

A basic property of holomorphic functions $f:D\to \mathbb{C}$ defined on domains $D$ of $\mathbb{C}$ is that $f$ is uniquely determined by its real part up to an additive constant. The same is true for slice regular functions defined on circular slice domains of the division algebra of quaternions or octonions. The aim of this paper is to extend the latter result to more general classes of algebras, including, among many others, the Clifford algebras $\mathbb{R}_{p,q}$ and the split octonions $\mathbb{S}\mathbb{O}$. New phenomena appear, as well as unexpected connections with graph theory and the theory of slice-Nash functions.

We consider probability measures on compact subsets of the complex plane arising as limiting distributions of Galois conjugates of certain algebraic integers. These measures are characterized by infinitely many integral inequalities, one for each nonzero integer polynomial. We study a broad family of measures given by particular convex combinations of harmonic and equilibrium measures. We then show that, for measures in this family, it suffices to verify two conditions to guarantee all the required inequalities. Under additional arithmetic constraints on the underlying compact sets, we further show that the infinite collection of inequalities is equivalent to checking the inequalities associated with two explicitly specified integer polynomials.