Recent zbMATH articles in MSC 32Ghttps://www.zbmath.org/atom/cc/32G2021-04-16T16:22:00+00:00WerkzeugRepresentability theorem in derived analytic geometry.https://www.zbmath.org/1456.140182021-04-16T16:22:00+00:00"Porta, Mauro"https://www.zbmath.org/authors/?q=ai:porta.mauro"Yu, Tony Yue"https://www.zbmath.org/authors/?q=ai:yu.tony-yueIn the paper under review, the authors prove a representability theorem in derived analytic geometry, analogous to
Lurie's generalization of Artin's representablility criteria to derived algebraic geometry.
This is an important, standard type result for the study of moduli problems and
a crucial step towards a solid theory of derived analytic geometry.
More specifically, the authors show that a derived stack for the étale site of derived analytic spaces is a derived analytic stack
if and only if
it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack in the classical
(underived) sense.
The result applies both to complex analytic geometry and non-archimedean analytic geometry.
Central to representability results as in the present paper is deformation theory, which the authors develop here for the derived analytic setup.
The authors define an analytic version of the cotangent complex which controls the deformation theory of the derived stack.
As in the algebraic setting, the cotangent complex represents a functor of derivations.
One key step in order to define the analytic cotangent complex is the elegant description of the \(\infty\)-category of modules over a
derived analytic space \(X\) as the \(\infty\)-category of spectrum objects of a certain \(\infty\)-category associated with \(X\).
Another important construction is the analytification functor which they establish in the derived setting.
To apply derived geometry to classical moduli problems, one may try to enrich classical moduli spaces with derived structures. The paper under review is an important tool in verifying when such enrichments are indeed the correct ones.
Reviewer: Eric Ahlqvist (Stockholm)Hyperelliptic integrals modulo \(p\) and Cartier-Manin matrices.https://www.zbmath.org/1456.140362021-04-16T16:22:00+00:00"Varchenko, Alexander"https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nSummary: The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field \(\mathbb{F}_p\) with a prime number \(p\) of elements were constructed only recently. In this paper we consider an example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus \(g\). It is known that in this case the total \(2g\)-dimensional space of holomorphic (multivalued) solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field \(\mathbb{F}_p\) in this case gives only a \(g\)-dimensional space of solutions, that is, a ``half'' of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field \(\mathbb{F}_p\) can be obtained by reduction modulo \(p\) of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to an example of the elliptic integral considered in the classical paper [\textit{Yu. I. Manin}, Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802)].Morrey type Teichmüller space and higher Bers maps.https://www.zbmath.org/1456.300782021-04-16T16:22:00+00:00"Hu, Guangming"https://www.zbmath.org/authors/?q=ai:hu.guangming"Liu, Yutong"https://www.zbmath.org/authors/?q=ai:liu.yutong"Qi, Yi"https://www.zbmath.org/authors/?q=ai:qi.yi"Shi, Qingtian"https://www.zbmath.org/authors/?q=ai:shi.qingtianSummary: In this paper, we focus on the set of univalent analytic functions \(f\) with \(\log f' \in H_K^2\). Motivated by the study of BMO-Teichmüller spaces and Morrey type spaces, we establish serval equivalent characterizations of Morrey type domains. Furthermore, we show that the higher Bers maps, induced by the higher Schwarzian differential operators, are holomorphic in Morrey type Teichmüller spaces. Finally, one of connected components in the small pre-logarithmic derivative model of the Morrey type Teichmüller space is also obtained.On the non-smoothness of the vector fields for the dynamically invariant Beltrami coefficients.https://www.zbmath.org/1456.300422021-04-16T16:22:00+00:00"Huo, Shengjin"https://www.zbmath.org/authors/?q=ai:huo.shengjin"Guo, Hui"https://www.zbmath.org/authors/?q=ai:guo.huiSummary: For \(\mu \in L^{\infty}(\Delta )\), the vector fields on the unit circle determined by \(\mu \) play an important role in the theory of the universal Teichmüller space. The aim of this paper is to give some characterizations of the vector fields induced by dynamically invariant \(\mu \). We show that those vector fields are not contained in the Sobolev class \(H^{3/2}\). At last, we give some results on dynamically invariant vectors to show that the vector fields, the quasi-symmetric homeomorphisms, and the quasi-circles are closely related.