For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty. $$ This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor.
We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide...
Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{\xi_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ \xi_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $\zeta \in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz ang...
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-m...
We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence $\{ x_j \}_{j\in\mathbb{N}}$ with Bessel bound $1$ in a Hilbert space, we show that there exists functions $\{ f_j \}_{j\in\mathbb{N}}$ belonging to the unit ball of $L^\infty([0,1])$ such that for all $j,k \...
Lukas Liehr, Mitchell A. Taylor, Peiyang Yu· 1 citation
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville n...
Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat...
A. Giaquinto, J. Irving, A. Lauve et al.· 2 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
