Recent zbMATH articles in MSC 32G https://zbmath.org/atom/cc/32G 2021-06-15T18:09:00+00:00 Werkzeug Polynomials defining Teichmüller curves and their factorizations mod $$p$$. https://zbmath.org/1460.14063 2021-06-15T18:09:00+00:00 "Mukamel, Ronen Eliahu" https://zbmath.org/authors/?q=ai:mukamel.ronen-e Soit $$\mathcal{M}_{g}$$ l'espace des modules des courbes de genre $$g$$. Pour tout $$d\geq3$$, on définit la courbe de Weierstrass $$W(d) \subset \mathcal{M}_{1} \times \mathcal{M}_{1}$$ de la sorte. C'est le lieu $$(E_{1},E_{2})$$ des paires de courbes elliptiques telles qu'il existe une courbe $$Y \in \mathcal{M}_{2}$$ et une application $$f\colon Y \to E_{1}$$ qui possède un point critique de Weierstrass et telle que le noyau de l'application induite par $$f$$ au niveau des jacobiennes est $$E_{2}$$. L'intérêt de ces notions remonte aux travaux [\textit{C. T. McMullen}, Math. Ann. 333, No. 1, 87--130 (2005; Zbl 1086.14024)] où les courbes de Weierstrass $$W_{d^{2}}$$ sur les surfaces modulaires d'Hilbert sont étudiées en lien avec les courbes de Teichmüller de $$\mathcal{M}_{2}$$. L'un des résultats du présent article est que $$W(d)$$ est birationellement équivalente à $$W_{d^{2}}$$. On peut réaliser la courbe de Weierstrass $$W(d)$$ comme une courbe plane en identifiant $$\mathcal{M}_{1}$$ à $$\mathbb{C}$$ via l'invariant $$j$$. L'auteur montre qu'il existe un polynôme $$\Psi_{d}$$, unique au signe prêt, sans carré en les invariants $$j$$ de $$E_1$$ et $$E_2$$ dont les coefficients sont premiers entre eux tel que $$W(d)$$ est l'ensemble des racines de $$\Psi_{d}$$. Il en déduit que $$W(d)$$ est connexe si $$d=3$$ ou $$d$$ est pair et possède deux composantes connexes si $$d$$ est impair et supérieur ou égal à $$5$$. De plus, si on note $$\chi$$ la caractéristique d'Euler (orbifold) de la courbe $$W_{d^{2}}$$ (calculée par exemple dans [\textit{M. Bainbridge}, Geom. Topol. 11, 1887--2073 (2007; Zbl 1131.32007)]), alors le bidegré du polynôme $$\Psi_{d}$$ est égale à $$(6|\chi|,2|\chi|)$$. Enfin, l'auteur fait quatre conjectures sur la factorisation du polynôme $$\Psi_{d}$$ en caractéristique positive et prouve ces conjectures dans certains cas par un calcul explicite assisté par ordinateur. Reviewer: Quentin Gendron (Guanajuato) Two-dimensional neighborhoods of elliptic curves: formal classification and foliations. https://zbmath.org/1460.32015 2021-06-15T18:09:00+00:00 "Loray, Frank" https://zbmath.org/authors/?q=ai:loray.frank "Thom, Olivier" https://zbmath.org/authors/?q=ai:thom.olivier "Touzet, Frédéric" https://zbmath.org/authors/?q=ai:touzet.frederic Summary: We classify two-dimensional neighborhoods of an elliptic curve $$C$$ with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having $$C$$ as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discuss analytic equivalence and for each formal model, we show that the corresponding moduli space is infinite dimensional. Reviewer: Reviewer (Berlin) A global Torelli theorem for singular symplectic varieties. https://zbmath.org/1460.32014 2021-06-15T18:09:00+00:00 "Bakker, Benjamin" https://zbmath.org/authors/?q=ai:bakker.benjamin "Lehn, Christian" https://zbmath.org/authors/?q=ai:lehn.christian Summary: We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of $$K3^{[n]}$$-type varieties. Reviewer: Reviewer (Berlin) The dimension of the moduli spaces of curves defined by topologically non quasi-homogeneous functions. https://zbmath.org/1460.32067 2021-06-15T18:09:00+00:00 "Loubani, Jinan" https://zbmath.org/authors/?q=ai:loubani.jinan Summary: We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case [\textit{Y. Genzmer} and \textit{E. Paul}, Mosc. Math. J. 11, No. 1, 41--72 (2011; Zbl 1222.32056)] and the quasi homogeneous case [\textit{Y. Genzmer} and \textit{E. Paul}, J. Singul. 14, 3--33 (2016; Zbl 1338.32028)], we describe an algorithm which computes the dimension of the generic strata of the local moduli space of curves. Reviewer: Reviewer (Berlin) Weierstrass Prym eigenforms in genus four. https://zbmath.org/1460.37027 2021-06-15T18:09:00+00:00 "Lanneau, Erwan" https://zbmath.org/authors/?q=ai:lanneau.erwan "Nguyen, Duc-Manh" https://zbmath.org/authors/?q=ai:nguyen.duc-manh Summary: We prove the connectedness of the Prym eigenforms loci in genus four (for real multiplication by some order of discriminant $$D$$), for any $$D$$. These loci were discovered by \textit{C. T. McMullen} [Duke Math. J. 133, No. 3, 569--590 (2006; Zbl 1099.14018)]. Reviewer: Reviewer (Berlin) Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces. https://zbmath.org/1460.14120 2021-06-15T18:09:00+00:00 "Allegretti, Dylan G. L." https://zbmath.org/authors/?q=ai:allegretti.dylan-g-l The author considers two spaces associated to a quiver with potential: a complex manifold parametrizing Bridgeland stability conditions on a certain $$3$$-Calabi-Yau triangulated category, and a cluster variety. They can be interpreted as moduli spaces of geometric structures on surfaces. \textit{T. Bridgeland} and \textit{I. Smith} [Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)] have shown that the space of stability conditions is isomorphic to a moduli space of meromorphic quadratic differentials, while \textit{V. V. Fock} and \textit{A. B. Goncharov} [Invent. Math. 175, No. 2, 223--286 (2009; Zbl 1183.14037)] have proved that the cluster variety is birational to a moduli space of local systems equipped with additional framing data. The author shows that if the quiver with potential arises from an ideal triangulation of a marked bordered surface, then one can construct a natural map from a dense subset of the space of stability conditions to the cluster variety. Using this construction, he gives solutions to a family of Riemann-Hilbert problems arising in Donaldson-Thomas theory. Reviewer: Vladimir P. Kostov (Nice) Characteristic classes of moduli spaces -- Riemann surface, graph, homology cobordism. https://zbmath.org/1460.55018 2021-06-15T18:09:00+00:00 "Morita, Shigeyuki" https://zbmath.org/authors/?q=ai:morita.shigeyuki The author surveys the theory of characteristic classes associated with mapping class groups of surfaces and other groups that have been studied using parallel techniques, like the arithmetic groups $$\mathrm{Sp}(2g,\mathbb{Z})$$ and $$\mathrm{GL}(2g,\mathbb{Z})$$, and the related moduli spaces (moduli spaces of surfaces, moduli spaces of graphs, etc.) The author's aim is to build a unifying theory of these objects studied. The stress is on a collection of Lie algebras which are naturally associated with these groups. One of them is the Lie algebra which played an important role in the theory of Johnson homomorphism, namely, the one consisting of the symplectic derivations of the free Lie algebra generated by the homology groups of surfaces. It turned out that this Lie algebra has connections with number theory, outer automorphisms of free groups, homology cylinders in 3-manifolds and homology cobordisms, and other topics. The author studies in particular a generating system for this Lie algebra. He reviews his joint work with Takuya Sakasai and Masaaki Suzuki, on this topic, as well as works of Nariya Kawazumi, Yusuke Kuno, Takao Satoh and others. Reviewer: Athanase Papadopoulos (Strasbourg) Cubic curves and totally geodesic subvarieties of moduli space. https://zbmath.org/1460.14062 2021-06-15T18:09:00+00:00 "McMullen, Curtis T." https://zbmath.org/authors/?q=ai:mcmullen.curtis-t "Mukamel, Ronen E." https://zbmath.org/authors/?q=ai:mukamel.ronen-e "Wright, Alex" https://zbmath.org/authors/?q=ai:wright.alex The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. In this paper the authors show that the flex locus provides the first example of a primitive totally geodesic subvariety of moduli space and the gothic locus provides new $$\mathrm{SL}_2(\mathbb R)$$-invariant varieties in Teichmüller dynamics. A number of interesting properties of these loci from the viewpoints of projective geometry and flat surfaces are discussed. Reviewer: Dawei Chen (Chestnut Hill) Weil-Petersson Teichmüller space revisited. https://zbmath.org/1460.30013 2021-06-15T18:09:00+00:00 "Wu, Li" https://zbmath.org/authors/?q=ai:wu.li "Hu, Yun" https://zbmath.org/authors/?q=ai:hu.yun "Shen, Yuliang" https://zbmath.org/authors/?q=ai:shen.yuliang This paper is concerned with Weil-Petersson geometry. A sense-preserving homeomorphism of the unit circle is said to belong to the Weil-Petersson class if it has a quasiconformal extension to the unit disk whose Beltrami coefficient is square integrable in the Poincaré metric. \textit{L. A. Takhtajian} and \textit{L.-P. Teo} showed in the paper [Weil-Petersson metric on the universal Teichmüller space. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1243.32010)] that the space $$\mathrm{WP}(S^1)$$ of Weil-Petersson classes of homeomorphisms of the unit circle is endowed with an Hilbert manifold structure. In the paper under review, the authors show that the smooth Hilbert manifold structure on $$\mathrm{WP}(S^1)$$ inherited froom $$H^{\frac{1}{2}}$$ by the pullback $$g\mapsto \log \vert g'\vert$$ is compatible with the Hilbert manifold structure introduced by Takhtajian and Teo. As an application the authors obtain new proofs of results in the papers [the third author, Am. J. Math. 140, No. 4, 1041--1074 (2018; Zbl 1421.30059); the third author and \textit{S. Tang}, Adv. Math. 359, Article ID 106891, 25 p. (2020; Zbl 1436.30015)]. Reviewer: Athanase Papadopoulos (Strasbourg) The Kodaira problem for Kähler spaces with vanishing first Chern class. https://zbmath.org/1460.32026 2021-06-15T18:09:00+00:00 "Graf, Patrick" https://zbmath.org/authors/?q=ai:graf.patrick "Schwald, Martin" https://zbmath.org/authors/?q=ai:schwald.martin Summary: Let $$X$$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $$X$$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $$X$$ has toroidal singularities, if $$X$$ has finite quotient singularities and if the cohomology group $$\text{H}^2 \!(X, \mathscr{T}_X)$$ vanishes. Reviewer: Reviewer (Berlin)