CyberSec Research

In this paper, we establish a structure theorem for a compact K\"{a}hler manifold $X$ of rational dimension $\mathrm{rd}(X)\leq n-k$ under the mixed partially semi-positive curvature condition $\mathcal{S}_{a,b,k} \geq 0$, which is introduced as a unified framework for addressing two partially semi-positive curvature conditions -- namely, $k$-semi-positive Ricci curvature and semi-positive $k$-scalar curvature. As a main corollary, we show that a compact K\"{a}hler manifold $(X,g)$ with $k$-semi-positive Ricci curvature and $\mathrm{rd}(X)\leq n-k$ actually has semi-positive Ricci curvature and $\mathrm{rd}(X)\geq \nu(-K_X)$. Of independent interest, we also confirm the rational connectedness of compact K\"{a}hler manifolds with positive orthogonal Ricci curvature, among other results.

We develop an analog of the notion of a character variety in the context of algebraic correspondences. It turns out that matings of certain Fuchsian groups and polynomials are contained in this ambient character variety. This gives rise to two different analogs of the Bers slice by fixing either the polynomial or the Fuchsian group. The Bers-like slices are homeomorphic copies of Teichm\"uller spaces or combinatorial copies of polynomial connectedness loci. We show that these slices are bounded in the character variety, thus proving the analog of a theorem of Bers. To produce compactifications of the Bers-like slices, we initiate a study of degeneration of algebraic correspondences on trees of Riemann spheres, revealing a new degeneration phenomenon in conformal dynamics. There is no available analog of Sullivan's 'no invariant line field' theorem in our context. Nevertheless, for the four times punctured sphere, we show that the compactifications of Teichm\"uller spaces are naturally homeomorphic.

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $\omega_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $\omega$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |\omega_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $\alpha<1/2$, $n_F(r_j)=O(r_j^{\alpha})$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$.

We provide and discuss complex analytic methods for overcoming the formal character of formal deformation quantization. This is a necessity for returning to physically meaningful statements, and accounts for the fact that the formal parameter $\hbar$ carries the interpretation of Planck's constant. As formal star products are given by a formal power series, this naturally leads into the realm of holomorphic functions and analytic continuation, both in finite and infinite dimensions. We propose a general notion of strict deformation quantization and investigate how one can use established results from complex analysis to think about the resulting objects. Within the main body of the text, the outlined program is then put into practice for strict deformation quantizations of constant Poisson structures on locally convex vector spaces and the strict deformation quantization of canonical mechanics on the cotangent bundle of a Lie group. Numerous auxiliary results, many of which are well-known yet remarkable in their own right, are provided throughout.

We investigate properties of holomorphic extensions in the one-variable case of Whitney's Approximation Theorem on intervals. Improving a result of Gauthier-Kienzle, we construct tangentially approximating functions which extend holomorphically to domains of optimal size. For approximands on unbounded closed intervals, we also bound the growth of holomorphic extensions, in the spirit of Arakelyan, Bernstein, Keldych, and Kober.

In this paper, we expand the theory of Weierstrassian elliptic functions by introducing auxiliary zeta functions $\zeta_\lambda$, zeta differences of first kind $\Delta_\lambda$ and second kind $\Delta_{\lambda,\mu}$ where $\lambda,\mu=1,2,3$. Fundamental and novel results pertaining to these functions are proven. Furthermore, results already existing in the literature are translated in terms of auxiliary zeta functions. Their relationship to Jacobian elliptic functions and Jacobian functions are given.

In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a special case. Furthermore, we give novel two-term addition, three-term addition, duplication and triplication formulas. New identities for elliptic invariants are also proven.

We prove an equidistribution result for the zeros of polynomials with integer coefficients and simple zeros. Specifically, we show that the normalized zero measures associated with a sequence of such polynomials, having small height relative to a certain compact set in the complex plane, converge to a canonical measure on the set. In particular, this result gives an equidistribution result for the conjugates of algebraic units, in the spirit of Bilu's work. Our approach involves lifting these polynomials to polynomial mappings in two variables and proving an equidistribution result for the normalized zero measures in this setting.

We study the relationships between several varieties parametrizing marked curves with differentials in the literature. More precisely, we prove that the space $\mathcal{B}_n$ of multiscale differentials of genus 0 with $n+1$ marked points of orders $(0,\ldots,0,-2)$ is a wonderful variety. This shows that the Chow ring of $\mathcal{B}_n$ is generated by the classes of a collection of smooth boundary divisors with normal crossings subject to simple and explicit linear and quadratic relations. Furthermore, we realize $\mathcal{B}_n$ as a subvariety of the space $\mathcal{A}_n$ of multiscale lines and prove that $\mathcal{B}_n$ can be realized as the normalized Chow quotient of $\mathcal{A}_n$ by a natural $\mathbb{C}^*$-action.

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that generates a hyperbolic surface whose lengths of boundary components are prescribed positive numbers. Furthermore, a generalized combinatorial Yamabe flow is introduced in the same geometry setting, with the long time existence and convergence established. This result yields an algorithm for searching bordered surfaces, which may accelerate convergence speed.

In this note, we study rigid complex manifolds that are realized as quotients of a product of curves by a free action of a finite group. They serve as higher-dimensional analogues of Beauville surfaces. Using uniformization, we outline the theory to characterize these manifolds through specific combinatorial data associated with the group under the assumption that the action is diagonal and the manifold is of general type. This leads to the notion of a $n$-fold Beauville structure. We define an action on the set of all $n$-fold Beauville structures of a given finite group that allows us to distinguish the biholomorphism classes of the underlying rigid manifolds. As an application, we give a classification of these manifolds with group $\mathbb Z_5^2$ in the three dimensional case and prove that this is the smallest possible group that allows a rigid, free and diagonal action on a product of three curves. In addition, we provide the classification of rigid 3-folds $X$ given by a group acting faithfully on each factor for any value of the holomorphic Euler number $\chi(\mathcal O_X) \geq -5$.

Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence. On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C. These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: * for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k; * for the classical groups of (right/left-right/contact) equivalence of Singularity Theory; * for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic. The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k. Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff their zero sets are ambient isomorphic". This study involves the contact equivalence of maps with singular targets, which seems to be not well-established. We write down the relevant part of this theory.

Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a different method) a result of Bangert on the existence of complex lines. We show that Ahlfors currents control the asymptotic behavior of families of pseudoholomorphic curves, refining a result of Demailly. Lastly, we show that the space of Ahlfors currents is convex.

We prove subelliptic estimates for ethe complex Green operator $ K_q $ at a specific level $ q $ of the $ \bar\partial_b $-complex, defined on a not necessarily pseudoconvex CR manifold satisfying the commutator finite type condition. Additionally, we obtain maximal $ L^p $ estimates for $ K_q $ by considering closed-range estimates. Our results apply to a family of manifolds that includes a class of weak $ Y(q) $ manifolds satisfying the condition $ D(q) $. We employ a microlocal decomposition and Calder\'on-Zygmund theory to obtain subelliptic and maximal-$ L^p $ estimates, respectively.

Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8 and 10. The result is proved by applying Gromov's h-principle in order to adapt a method of Haefliger, originally used to study foliations, to the holomorphic setting. For dimension 12 and greater, the conjecture remains open.

A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and some of their consequences. The mentioned results are the Caratheodory-Toeplitz criterion of continuing a polynomial to a Caratheodory class function, and the Riesz-Fejer theorem about trigonometric polynomials. This is an English translation of a preprint originally published in Russian: https://preprints.ru/article/1799

In this paper we give sharp bounds of the difference of the moduli of the second and the first logarithmic coefficient for Bazilevi\v{c} class of univalent functions.