Tootfinder

On meromorphic pencils, cusp singularities and holomorphic foliations in the complex plane
Bruno Scardua
https://arxiv.org/abs/2607.18526 https://arxiv.org/pdf/2607.18526 https://arxiv.org/html/2607.18526
arXiv:2607.18526v1 Announce Type: new
Abstract: We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $
\phi = \frac{f^p}{g^q}, $
where $f,g$ are holomorphic functions (possibly polynomials) in general position and $(p,q)=1$. We first establish a homological characterization of relative exactness: if a polynomial $1$-form $\Omega$ has vanishing periods along every closed path contained in the fibers $\phi_c$, then $\Omega$ decomposes as $\Omega = a\omega_0 dh,$ where $\omega_0 = p gdf - q f dg,$ for suitable polynomials $a$ and $h$. In the homogeneous case, degree constraints force $a$ to be constant. We then apply this integration principle to foliations leaving invariant plane curve singularities of cusp type \[ f^p g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $\Omega$ is a polynomial cusp basic form, \[ \Omega = d(f^p g^q) \lambda (p gdf - q fdg), \qquad \lambda \in \mathbb{C}. \] In particular, such foliations admit Liouvillian first integrals of hypergeometric type.
Our results provide a bridge between relative cohomology, the geometry of rational pencils, and the analytic structure of cusp foliations, yielding explicit normal forms and first integrals under homological hypotheses.
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Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature
Shuwen Chen, Junpeng Li
https://arxiv.org/abs/2608.05598 https://arxiv.org/pdf/2608.05598 https://arxiv.org/html/2608.05598
arXiv:2608.05598v1 Announce Type: new
Abstract: A long-standing conjecture in Hermitian geometry says that a compact Hermitian manifold with constant Chern holomorphic sectional curvature $c$ is K\"ahler for $c\neq 0$ and Chern flat for $c=0$. Although the conjecture has been established in complex dimension two, it remains open in general in higher dimensions. We verify the conjecture for compact balanced threefolds when $c\leq 0$. For compact locally conformally K\"ahler manifolds, Chen, Chen, and Nie established the case $c\leq 0$, while Huang and Wan recently settled the remaining case. Inspired by the approach of Huang and Wan, we investigate a generalization of the conjecture for canonical metric connections and establish it for connected compact locally conformally K\"ahler manifolds.
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Hardy spaces of discrete holomorphic functions on the upper half-lattice
Eugenio Dellepiane, Alessandro Monguzzi, Matteo Monti
https://arxiv.org/abs/2607.17726 https://arxiv.org/pdf/2607.17726 https://arxiv.org/html/2607.17726
arXiv:2607.17726v1 Announce Type: new
Abstract: We develop a theory of Hardy spaces $H^p$ of discrete holomorphic functions on the upper half-lattice, within the classical framework of discrete holomorphicity on the square lattice. We prove Cauchy and Poisson reproducing formulas, establish a boundary norm identity, and obtain Paley--Wiener type characterizations for these spaces. In the Hilbert space case, we describe the associated reproducing kernel and Szeg\H{o} projection, and we compare the discrete theory with the classical Hardy space on the upper half-plane through a family of discrete holomorphic approximants of classical $H^2$-functions. We also prove duality results for $H^p$, $1<p><\infty$, establish uniqueness and sampling results on horizontal lines, and introduce Bergman-type spaces, comparing two natural weighted scales.
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Replaced article(s) found for hep-th. https://arxiv.org/list/hep-th/new
[3/3]:
- Distinct finite-temperature phase diagrams of non-invertible Kennedy--Tasaki duals
Weiguang Cao, Haruki Watanabe
https://arxiv.org/abs/2607.24231 https://mastoxiv.page/@arXiv_condmatstrel_bot/116996702950349969
- Renormalizations in holomorphic field theories on K\"ahler manifolds
Minghao Wang, Junrong Yan
https://arxiv.org/abs/2608.00546 https://mastoxiv.page/@arXiv_mathph_bot/117036123885412792
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A Rudin-Carleson theorem with uniform approximation for manifold-valued maps
Benedikt Steinar Magn\'usson, Tyson Ritter
https://arxiv.org/abs/2607.16843 https://arxiv.org/pdf/2607.16843 https://arxiv.org/html/2607.16843
arXiv:2607.16843v1 Announce Type: new
Abstract: Given a closed set $E \subset \partial {\mathbb D}$ of measure zero and a continuous function $\varphi : E \to {\mathbb C}$, the classical Rudin-Carleson interpolation theorem states that there exists a continuous function $F : \overline {\mathbb D} \to {\mathbb C}$ that is holomorphic on ${\mathbb D}$ and satisfies $F\rvert_E = \varphi$. In this paper we obtain a generalisation of the Rudin-Carleson theorem for maps $\varphi : E \to X$ into arbitrary connected complex manifolds $X$ that combines interpolation of $\varphi$ on $E$ with uniform approximation on compact subsets of $\overline {\mathbb D} \setminus E$ of another given continuous map $f:\overline {\mathbb D} \to X$ that is holomorphic on ${\mathbb D}$. Under the further assumption that $X$ is an Oka manifold we obtain a corollary that combines Rudin-Carleson interpolation of a continuous map $\varphi : \overline {\mathbb D} \to X$ on $E$ with Runge approximation of $\varphi$ on a compact set $K\subset {\mathbb D}$ without any holes on which $\varphi$ is holomorphic.
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Replaced article(s) found for math.DG. https://arxiv.org/list/math.DG/new
[3/3]:
- A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-t...
Tianzhi Hu, Runze Zhang
https://arxiv.org/abs/2609.08757 https://mastoxiv.page/@arXiv_mathCV_bot/117240229130532493
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Metric entropy of the space of holomorphic functions
Siarhei Finski
https://arxiv.org/abs/2607.18890 https://arxiv.org/pdf/2607.18890 https://arxiv.org/html/2607.18890
arXiv:2607.18890v1 Announce Type: new
Abstract: By Montel's theorem, the continuous functions on a compact subset of a complex manifold that admit uniformly bounded holomorphic extensions to the manifold form a compact set in the uniform topology. We render this compactness quantitative by determining the asymptotics of the associated metric entropy, thereby giving a new solution to a problem of Kolmogorov for domains in $\mathbb{C}^n$ and extending the solution to domains in arbitrary Stein manifolds.
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Crosslisted article(s) found for nlin.SI. https://arxiv.org/list/nlin.SI/new
[1/1]:
- Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic tra...
Urei Miura
https://arxiv.org/abs/2608.04850 https://mastoxiv.page/@arXiv_quantph_bot/117047618895534612
- Machine-Learning Search for Lax Connections
Osamu Fukushima, Tomohiro Shigemura, Ryosuke Suda, Norihiro Tanahashi, Kentaroh Yoshida
https://arxiv.org/abs/2608.05146 https://mastoxiv.page/@arXiv_hepth_bot/117047572101225603
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Polarization complete left invariant connections
Bal\'azs Forman, R\'obert Sz\H{o}ke
https://arxiv.org/abs/2609.15673 https://arxiv.org/pdf/2609.15673 https://arxiv.org/html/2609.15673
arXiv:2609.15673v1 Announce Type: new
Abstract: Let $(M,\nabla)$ be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood $N$ of the zero section in $TM$ is a complex structure on $N$ such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization $P$ on $N$ is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to $P$. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization.
The connection is called entire (resp. polarization complete or simply $\mathcal P$-complete) if the ac-structure (resp. the ac-polarization) exists on $TM$. Although on a small enough $N$ an ac-structure always exists, entire connections are rear and if a maximal domain of definition $N_{max}$ of an ac-structure exists (different from $TM$), $N_{max}$ is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole $TM$. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map.
As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are $\mathcal P$-complete.
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Crosslisted article(s) found for nlin.SI. https://arxiv.org/list/nlin.SI/new
[1/1]:
- Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic tra...
Urei Miura

Strictly paired ${\rm CR}$ linear automorphisms of $\R^4$
Ioannis D. Platis
https://arxiv.org/abs/2607.19016 https://arxiv.org/pdf/2607.19016 https://arxiv.org/html/2607.19016
arXiv:2607.19016v1 Announce Type: new
Abstract: This paper investigates the algebraic and differential geometric properties of smooth mappings between domains in $\mathbb{C}^2$, classifying them according to the constant ranks of their holomorphic and antiholomorphic derivative components. Particular emphasis is placed on strictly paired CR (SPCR) linear automorphisms of $\mathbb{R}^4$, where both block matrices $A$ and $B$ have rank 1. We characterise the set of SPCR linear automorphisms as a 12-dimensional submanifold of $\mathrm{GL}(2,\mathbb{C})^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.
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Holonomy Asymptotics along Quartic Differential Rays
Weihan Ma
https://arxiv.org/abs/2608.04729 https://arxiv.org/pdf/2608.04729 https://arxiv.org/html/2608.04729
arXiv:2608.04729v1 Announce Type: new
Abstract: Let \(X\) be a closed Riemann surface and let \(q\in H^0(X,K^4)\) be a nonzero holomorphic quartic differential on \(X\). For \(t >0\), the ray \(tq\) determines a family of Hitchin representations in the \(\operatorname{PSp}(4,\mathbb R)\)-Hitchin component. We study, as \(t\to \infty\), the asymptotic behavior of their holonomy along closed curves. We obtain explicit asymptotic formulas for all singular values and for the absolute values of all eigenvalues of the holonomy. Their logarithmic growth rates are given by integrating the local fourth roots of \(q\) along the saddle connections forming the geodesic representative of the curve with respect to the singular flat metric \(\lvert q\rvert^{1/2}\). No restriction is imposed on the orders of the zeros of \(q\).
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The Dimension Conjecture for 2-Nondegenerate Hypersurfaces
M. A. Stepanova
https://arxiv.org/abs/2607.19186 https://arxiv.org/pdf/2607.19186 https://arxiv.org/html/2607.19186
arXiv:2607.19186v1 Announce Type: new
Abstract: The dimension conjecture in CR geometry is proved for 2-nondegenerate real-analytic hypersurfaces with sign-definite Levi form. Namely, it is proved that, in the indicated class of hypersurfaces in $\mathbb{C}^{N}$, the maximum dimension of finite-dimensional Lie algebras of infinitesimal holomorphic automorphisms is strictly smaller than the corresponding dimension for nondegenerate hyperquadrics in $\mathbb{C}^{N}$.
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Crosslisted article(s) found for math.CV. https://arxiv.org/list/math.CV/new
[1/1]:
- $m$-Positive Stability of Holomorphic Vector Bundles and Moduli Spaces
Dan Popovici
https://arxiv.org/abs/2607.17203 https://mastoxiv.page/@arXiv_mathDG_bot/116956936619519985
- Universal Correlators on Exponentially Ramified Spectral Curves
Mohamad Alameddine, Alexander Hock
https://arxiv.org/abs/2607.17711 https://mastoxiv.page/@arXiv_mathph_bot/116956963922457688
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Certain real surfaces in $\mathbb{C}^2$ with degenerated CR singularities
Sushil Gorai, Suman Karak, Golam Mostafa Mondal
https://arxiv.org/abs/2607.17016 https://arxiv.org/pdf/2607.17016 https://arxiv.org/html/2607.17016
arXiv:2607.17016v1 Announce Type: new
Abstract: In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as
$\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k o(|z|^{k})\}$
or
$M_t
:=
\left\{
(z,w)\in\mathbb{C}^2 :
w=(z t\overline{z})^k o(|z|^k)
\right\}$,
for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$.
We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\frac{\pi}{k}\right)$.
On the other hand, for $0<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<\min\left\{\sin\left(\frac{\pi}{k}\right),\frac{1}{k-2}\right\}$.
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Moduli of the Sourceless Framed Beltrami-Vekua Normal Form
Daniel Alay\'on-Solarz
https://arxiv.org/abs/2608.05459 https://arxiv.org/pdf/2608.05459 https://arxiv.org/html/2608.05459
arXiv:2608.05459v1 Announce Type: new
Abstract: We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and M\"obius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo M\"obius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=\Delta\arg B$, whose hyperbolic density at $C^2$ regularity is $\kappa=\Delta_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive H\"older density and every phase current arising as the Laplacian of a H\"older phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,d\eta,d{\star}\eta)$, where $\eta=\operatorname{Im}(dB/B)$. On the tame sector, $d\eta$ is the atomic charge measure, while $d{\star}\eta$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.
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Dirichlet symbols and the nonlinear wave equation
Ramlal Debnath, Haakan Hedenmalm
https://arxiv.org/abs/2608.04666 https://arxiv.org/pdf/2608.04666 https://arxiv.org/html/2608.04666
arXiv:2608.04666v1 Announce Type: new
Abstract: We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the disk or the exterior disk. Those coordinates allow us to characterize $\log\psi'$ for $\psi$ in the class $\Sigma$ of normalized univalent functions without explicitly touching the univalence property. Moreover, those manifolds extend the universal Teichm\"uller space of Lipman Bers beyond the quasicircle boundary setting, allowing for more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effectiveaverage amplitude of $(1-|z|^2)^2|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $72/5=14.4$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $\varphi\in\mathscr{S}$, and the analogous statement is valid for $\psi\in\Sigma$ as well.
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An Infinitesimal Circular Morera Theorem
Qiteng Guo, Ao Xiao
https://arxiv.org/abs/2608.04540 https://arxiv.org/pdf/2608.04540 https://arxiv.org/html/2608.04540
arXiv:2608.04540v1 Announce Type: new
Abstract: We prove an infinitesimal circular version of Morera's theorem. Let $D\subset\mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^ $, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circular integrals implies holomorphicity. The proof uses a local distributional $\partial$-primitive, a circular identity for weak $\partial$-derivatives, and a pointwise asymptotic mean-value criterion for harmonicity.
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Explicit Estimates for the Bergman Kernel Form on Polarized Riemann Surfaces for General Tensor Powers
Johannes Testorf
https://arxiv.org/abs/2608.03493 https://arxiv.org/pdf/2608.03493 https://arxiv.org/html/2608.03493
arXiv:2608.03493v1 Announce Type: new
Abstract: Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and put $\omega=\ddbar\phi$. We obtain effective pointwise estimates for the Bergman form of $H^0(X,K_X\otimes L^m)$. If $\Ric\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[
K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega, \] and the constant is sharp on $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. A local version, depending on an upper curvature bound and the injectivity radius, recovers the first two terms of the Bergman expansion when the curvature is constant. Under the two-sided bound $-\omega\leq\Ric\omega\leq\omega$ and the same closed-geodesic hypothesis, we also prove \[ K_{m\phi}\leq \frac{m\omega}{2\pi}
\left(1 \frac{54.8\log(2m)}{m-\frac{1}2}\right). \] The lower estimates use the deformation-to-the-tangent-space form of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound combines a weighted submean inequality with quantitative isothermal coordinates which was obtained in recent work by Eilat.
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Replaced article(s) found for math.CV. https://arxiv.org/list/math.CV/new
[1/1]:
- Singularity of non-pluripolar cohomology classes
Duc-Bao Nguyen, Shuang Su, Duc-Viet Vu
https://arxiv.org/abs/2508.14669 https://mastoxiv.page/@arXiv_mathCV_bot/115065656430154433
- Analytic structure of $q$-pseudoconcave subsets of continuous graphs
Filippo Valnegri
https://arxiv.org/abs/2603.04967 https://mastoxiv.page/@arXiv_mathCV_bot/116181309294452296
- Non-symmetric vector dyson equations
Jiaoyang Huang, Zhonggen Su, Ruizhe Xu
https://arxiv.org/abs/2607.16333 https://mastoxiv.page/@arXiv_mathCV_bot/116956844212485469
- Cofinite Zeros of High Derivatives
Eric Hou
https://arxiv.org/abs/2607.20816 https://mastoxiv.page/@arXiv_mathCV_bot/116973837268195335
- Coefficient Problems for a Ma-Minda Convex Class Associated with the Normalized Arcsine Mapping
Shantanu Panja, Abhijit Banerjee, Jhilik Banerjee, Sujoy Majumder
https://arxiv.org/abs/2607.27293 https://mastoxiv.page/@arXiv_mathCV_bot/117013548331556229
- Skoda division theorem on compact K\"ahler manifolds for line bundles with singular hermitian met...
Jaehoon Jeong
https://arxiv.org/abs/2607.29669 https://mastoxiv.page/@arXiv_mathCV_bot/117030527056701741
- Classes of Holomorphic Multicomplex-Valued Functions Generated by Elliptic-Admissible Involutions
Nicolas Doyon, Pierre-Olivier Paris\'e, William Verreault
https://arxiv.org/abs/2211.13875
- Fast vanishing cycles on perturbations of complex weighted-homogeneous complete intersection germs
Dmitry Kerner, Rodrigo Mendes
https://arxiv.org/abs/2311.13423 https://mastoxiv.page/@arXiv_mathAG_bot/111458171750230260
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