arXiv subjects

Zhangchi Chen

Publications and source records attributed to Zhangchi Chen.

14 recordsLinked to original sources

arXiv2026

Nakano-positive determinants outside the Hodge-Riemann cone

Dinh and Nguy\^en asked whether the determinant of a Griffiths positive matrix with $(1,1)$-form entries belongs to the Hodge--Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, we present an explicit Nakano positive $k\times k$ matrix of constant $(1,1)$-forms on $\C^n$ whose determinant has a singular Lefschetz map in bidegree $(1,n-k-1)$. This gives a negative answer throughout this range. The boundary cases $n=k\geqslant4$ and $n=k+1\geqslant5$ remain open. We next study the question under the simultaneous diagonalizability (SD) condition. Under this condition, we prove the Hodge--Riemann property in every bidegree $(p,q)$ with $p+q=n-k$ and $\min(p,q)\leqslant1$. Consequently, SD gives an affirmative answer when $1\leqslant k=n-2$ or $n-3$. For every $n\geqslant6$ and $2\leqslant k\leqslant n-4$, however, we present SD examples whose Lefschetz map in bidegree $(2,n-k-2)$ is singular, showing that SD alone does not imply the Hodge--Riemann property in all bidegrees. The positive result uses the theory of dually Lorentzian polynomials developed by Ross, S\"u\ss, and Wannerer, in particular their generalized Alexandrov--Fenchel inequality and its equality characterization. Exact Python verification programs accompany the constructions.

math.AG↗

arXiv2026

Distribution modulo one of linear recurrent sequences

We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance between two limit values. Our results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas.

math.NT↗

arXiv2025

On a question of Astorg and Boc Thaler

Astorg and Boc Thaler studied the dynamics of certain skew-products $f$ tangent to the identity on $\mathbb{C}^2$, with two real parameters $\alpha>1$ and $\beta$ derived from its coefficients. They proved that if there exists a strictly increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}:=(n_{k+1}-\alpha n_k-\beta\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $\alpha>1$ with the Pisot property, the condition that $\theta:=\frac{\beta\ln\alpha}{\alpha-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $\alpha$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$\alpha$, we prove that $\theta\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one.

math.DS↗

arXiv2024

Frequently hypercyclic meromorphic curves with slow growth

We construct entire curves in projective spaces that exhibit frequent hypercyclicity under translations along countably many prescribed directions while maintaining optimal slow growth rates. Furthermore, we establish a fundamental dichotomy by proving the impossibility of such curves simultaneously preserving frequent hypercyclicity for uncountably many directions under equivalent growth constraints. This result reveals a striking contrast with classical hypercyclicity phenomena, where entire functions can achieve hypercyclicity over some uncountable direction set without growth rate compromise. Our methodology is rooted in Nevanlinna theory and guided by the Oka principle, offering new insights into the relationship between dynamical properties and growth rates of entire curves in projective spaces.

math.CV↗

arXiv2024

Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries

The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Dinh-Nguyên proved the mixed HLT, HRR and LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive $k\times k$ matrices with $(1,1)$-form entries in $\bc^n$ satisfies these theorems in the linear case. This paper answered their question positively when $k=2$ and $n=2,3$. Moreover, assume that the matrix only has diagonalized entries, for $k=2$ and $n\geqslant 4$, the determinant satisfies HLT for bidegrees $(n-2,0)$, $(n-3,1)$, $(1,n-3)$ and $(0,n-2)$. In particular, for $k=2$ and $n=4,5$ with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive $2\times 2$ matrix with $(1,1)$-form entries, if all entries are $\mathbb{C}$-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension $\leqslant 5$, the determinant of a Griffiths positive $2\times 2$ matrix with diagonalized entries satisfies these theorems.

math.CV↗

arXiv2023

Entire holomorphic curves into $\mathbb{P}^n(\mathbb{C})$ intersecting $n+1$ general hypersurfaces

Let $\{D_i\}_{i=1}^{n+1}$ be $n+1$ hypersurfaces in $\mathbb{P}^n(\mathbb{C})$ with total degrees $\sum_{i=1}^{n+1} °D_i\geqslant n+2$, in general position and satisfying a generic geometric condition: every $n$ hypersurfaces intersect only at smooth points and the intersection is transversal. Then, for every algebraically nondegenerate entire holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, we show a Second Main Theorem: $$ \sum_{i=1}^{n+1} δ_f(D_i) < n+1 $$ in terms of defect inequality in Nevanlinna theory. This is the first result in the literature on Second Main Theorem for $n+1$ general hypersurfaces in $\mathbb{P}^n(\mathbb{C})$ with optimal total degrees.

math.CV↗

arXiv2023

Universal Entire Curves in Projective Spaces with Slow Growth

We construct explicit universal entire curves in projective spaces whose Nevanlinna characteristic functions grow slower than any preassigned transcendental growth rate. Moreover, we can make such curves to be hypercyclic for translation operations along any given countable directions.

math.CV↗

arXiv2021

Minkowski dimension of the boundaries of the lakes of Wada

The lakes of Wada are three disjoint simply connected domains in $S^2$ with the counterintuitive property that they all have the same boundary. The common boundary is a indecomposable continuum. In this article we calculated the Minkowski dimension of such boundaries. The lakes constructed in the standard Cantor way has $\ln(6)/\ln(3)\approx 1.6309$-dimensional boundary, while in general, for any number in $[1,2]$ we can construct lakes with such dimensional boundaries.

math.GN↗

arXiv2020

On nonsingularity of circulant matrices

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a prime, or a product of two distinct primes. For any other integer $2k+1$ we construct circulant matrices having determinant $0$. The smallest singular matrix appears when $2k+1=45$. The possibility for such matrices to be singular is rather low, smaller than $10^{-4}$ in this case.

math.AC↗

arXiv2020

Directed harmonic currents near non-hyperbolic linearized singularities

Let $(\mathbb{D}^2,\mathcal{F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb{D}^2$ defined by the linear vector field \[ z \,\frac{\partial}{\partial z}+ λ\,w \,\frac{\partial}{\partial w}, \] where $λ\in\mathbb{C}^*$. Such a foliation has a non-degenerate linearized singularity at $0$. Let $T$ be a harmonic current directed by $\mathcal{F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ and whose the trivial extension $\tilde{T}$ across $0$ is $dd^c$-closed. The Lelong number of $T$ at $0$ describes the mass distribution on the foliated space. In 2014 Nguyen proved that when $λ\notin\mathbb{R}$, i.e. $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. For the non-hyperbolic case $λ\in\mathbb{R}^*$ the article proves the following results. The Lelong number at $0$: 1) is strictly positive if $λ>0$; 2) vanishes if $λ\in\mathbb{Q}_{<0}$; 3) vanishes if $λ<0$ and $T$ is invariant under the action of some cofinite subgroup of the monodromy group.

math.DS↗

arXiv2020

Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants

Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential vector fields. Inspired by Olver's recurrence formulas, we study the algebra of $A(3)$ differential invariants of surfaces. We obtain necessary conditions for homogeneity of algebraic nature. Solving these conditions, we organise homogeneous models in inequivalent branches.

math.DG↗

arXiv2020

On Differential Invariants of Parabolic Surfaces

The algebra of differential invariants under $SA_3(\mathbb{R})$ of generic parabolic surfaces $S^2 \subset \mathbb{R}^3$ with nonvanishing Pocchiola $4^{\text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomials. The proof is based on Fels-Olver's recurrence formulas, pulled back to the parabolic jet bundles.

math.DG↗

arXiv2020

Normal Forms for Rigid $\mathfrak{C}_{2,1}$ Hypersurfaces $M^5 \subset \mathbb{C}^3$

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathcal{C}^ω$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, ζ, w = u + iv)$: \[ u = F\big(z,ζ,\bar{z},\barζ\big). \] The Gaussier-Merker model $u=\frac{z\bar{z}+ \frac{1}{2}z^2\barζ+\frac{1}{2} \bar{z}^2 ζ}{1-ζ\barζ}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone $\{x_1^2+x_2^2-x_3^2=0\}$. Another representation is the tube $u=\frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: \[ (z,ζ,w) \longmapsto \big( f(z,ζ), g(z,ζ), ρ\,w+h(z,ζ) \big) =: (z',ζ',w'). \] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiola's formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\bar{z}+\frac{1}{2}\,z^2\barζ +\frac{1}{2}\,\bar{z}^2ζ}{ 1-ζ\barζ} + \sum_{a,b,c,d \atop a+c\geqslant 3}\, G_{a,b,c,d}\, z^aζ^b\bar{z}^c\barζ^d, \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {\rm Im}\, G_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0, 2, 3, 0}$, ${\rm Re} G_{3,0,1,1}$. With this, a brige Poincaré $\longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.

math.CV↗

arXiv2017

A counterexample to Hartogs' type extension of holomorphic line bundles

Consider a domain $\varOmega$ in $\mathbb{C}^n$ with $n\geqslant 2$ and a compact subset $K\subset\varOmega$ such that $\varOmega\backslash K$ is connected. We address the problem whether a holomorphic line bundle defined on $\varOmega\backslash K$ extends to $\varOmega$. In 2013, Fornæss, Sibony and Wold gave a positive answer in dimension $n\geqslant 3$, when $\varOmega$ is pseudoconvex and $K$ is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for $K$ of general shape, we construct counterexamples in any dimension $n\geqslant 2$. The key is a certain gluing lemma by means of which we extend any two holomorphic line bundles which are isomorphic on the intersection of their base spaces.

math.CV↗