Recent zbMATH articles in MSC 30F30https://www.zbmath.org/atom/cc/30F302022-05-16T20:40:13.078697ZWerkzeugSpecial cases of the orbifold version of Zvonkine's \(r\)-ELSV formulahttps://www.zbmath.org/1483.140922022-05-16T20:40:13.078697Z"Borot, GaĆ«tan"https://www.zbmath.org/authors/?q=ai:borot.gaetan"Kramer, Reinier"https://www.zbmath.org/authors/?q=ai:kramer.reinier"Lewanski, Danilo"https://www.zbmath.org/authors/?q=ai:lewanski.danilo"Popolitov, Alexandr"https://www.zbmath.org/authors/?q=ai:popolitov.aleksandr"Shadrin, Sergey"https://www.zbmath.org/authors/?q=ai:shadrin.sergeySummary: We prove the orbifold version of Zvonkine's \(r\)-ELSV formula in two special cases: the case of \(r=2\) (completed 3-cycles) for any genus \(g\geq 0\) and the case of any \(r\geq 1\) for genus \(g=0\).Quadratic differentials and signed measureshttps://www.zbmath.org/1483.300812022-05-16T20:40:13.078697Z"Baryshnikov, Yuliy"https://www.zbmath.org/authors/?q=ai:baryshnikov.yuliy-m"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovichSummary: In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary, we introduce several more general classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess certain properties of Strebel differentials and often appear in applications. We discuss the relation between gradient differentials and special signed measures supported on their set of critical trajectories. We provide a characterization of gradient differentials for which there exists a positive measure in the latter class.On existence of quasi-Strebel structures for meromorphic \(k\)-differentialshttps://www.zbmath.org/1483.300822022-05-16T20:40:13.078697Z"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Tahar, Guillaume"https://www.zbmath.org/authors/?q=ai:tahar.guillaumeA meromorphic differential \(\Psi\) of order \(k\geq 2\) on a compact orientable Riemann surface \(Y\) without boundary is a meromorphic section of the \(k\)-th tensor power \((T^{*}_{\mathbb{C}}Y)^{\otimes k}\) of the holomorphic cotangent bundle of \(Y\). Zeros and poles of \(\Psi\) constitute the set of critical points.
For a differential of order \(k\) given locally by \(f(z)dz^{k}\) in a neighborhood of a non-critical point, there are \(k\) locally distinct directions, called \textit{horizontal}, which are given by the conditions that \(f(z)dz^{k}\) is real and positive.
Recall that a quadratic differential is called \textit{Strebel} if almost all horizontal trajectories are closed. Such a phenomenon can never happen for a \(k\)-differential of order \(k\geq 3\) unless it is a power of a \(1\)-form or a quadratic differential. The authors introduce thus the notion of quasi-Strebel structure and give sufficient conditions for a meromorphic \(k\)-differential to be quasi-Strebel. The main result is the following:
\textbf{Theorem 1.} Let \(\Psi\) be a meromorphic \(k\)-differential such that
\begin{itemize}
\item no poles have of order smaller than \(-k\);
\item at a pole of order \(-k\), the residue belongs to \(i^{k}\mathbb{R}\).
\end{itemize}
Then,
\begin{itemize}
\item[1.] if \(k>2\) is even, then \(\Psi\) has a quasi-Strebel structure;
\item[2.] if \(k>2\) is odd, and, up to a common factor, the period of \(\sqrt[k]{\Psi}\) along every path connecting any two singularities of \(\Psi\) belongs to \(\mathbb{Q}[e^{\frac{2i\pi}{k}}]\), then \(\Psi\) has a quasi-Strebel structure.
\end{itemize}
Reviewer: Andrea Tamburelli (Houston)