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One of the most relevant topics in web theory is linearization. A particular class of linearizable webs is the Grassmannizable web. Akivis gave a characterization of such a web, showing that Grassmannizable webs are equivalent to isoclinic and transversally geodesic webs. The obstructions given by Akivis that characterize isoclinic and transversally geodesic webs are computed locally, and it is difficult to give them an interpretation in relation to torsion or curvature of the unique Chern connection associated with a web. In this paper, using Nagy's web formalism, Frölisher—Nejenhuis theory for derivation associated with vector differential forms, and Grifone's connection theory for tensorial algebra on the tangent bundle, we find invariants associated with almost-Grassmann structures expressed in terms of torsion, curvature, and Nagy's tensors, and we provide an interpretation in terms of these invariants for the isoclinic, transversally geodesic, Grassmannizable, and parallelizable webs. [ABSTRACT FROM AUTHOR]

Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities. [ABSTRACT FROM AUTHOR]

One of the most relevant topics in web theory is linearization. A particular class of linearizable webs is the Grassmannizable web. Akivis gave a characterization of such a web, showing that Grassmannizable webs are equivalent to isoclinic and transversally geodesic webs. The obstructions given by Akivis that characterize isoclinic and transversally geodesic webs are computed locally, and it is difficult to give them an interpretation in relation to torsion or curvature of the unique Chern connection associated with a web. In this paper, using Nagy’s web formalism, Frölisher—Nejenhuis theory for derivation associated with vector differential forms, and Grifone’s connection theory for tensorial algebra on the tangent bundle, we find invariants associated with almost-Grassmann structures expressed in terms of torsion, curvature, and Nagy’s tensors, and we provide an interpretation in terms of these invariants for the isoclinic, transversally geodesic, Grassmannizable, and parallelizable webs.

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

We prove that the Landau–Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperKähler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine structure of the special Lagrangian fibrations constructed by Collins–Jacob–Lin in P 1 × P 1 [16] and compare it with the integral affine structures used in the work of Carl–Pumperla–Siebert [9]. We also construct the Floer-theoretical Landau–Ginzburg mirrors of smoothing of A n -singularities and monotone del Pezzo surfaces, by using the gluing method of Cho–Hong–Lau [13] and Hong–Kim–Lau [41]. They agree with the result of limit of hyperKähler rotations. [ABSTRACT FROM AUTHOR]

A complex vector space V is a prehomogeneous G -module if G acts rationally on V with a Zariski-open orbit. The module is called étale if dim ⁡ V = dim ⁡ G . We study étale modules for reductive algebraic groups G with one-dimensional center. For such G , even though every étale module is a regular prehomogeneous module, its irreducible submodules have to be non-regular. For these non-regular prehomogeneous modules, we determine some strong constraints on the ranks of their simple factors. This allows us to show that there do not exist étale modules for G = GL 1 × S × ⋯ × S , with S simple. [ABSTRACT FROM AUTHOR]

Recently, we proved the Griffiths conjecture on the boundedness of the period maps, and then used it to prove global Torelli theorem on the Torelli space under certain natural conditions. This paper serves as a review of these results and an introduction to the main ideas behind the proofs. [ABSTRACT FROM AUTHOR]

Abstract: The generalized curvature tensor and Christoffel symbols are determined in background by a modified ansatz of the de Wit–Freedman type by imposing gauge invariance. The resulting set of recurrence relations and difference equations is solved. The Riemann curvature tensor is derived by antisymmetrization. All results are presented as finite power series in the inverse AdS radius and are unique. The fourth order, which is complete for fields up to spin five, is calculated explicitly. Higher orders can be obtained with the same method. [Copyright &y& Elsevier]

Abstract: In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection. [Copyright &y& Elsevier]

The Nagano–Yagi–Goldmann theorem states that on the torus T2, every affine (or projective) structure is invariant or is constructed on the basis of some Goldmann rings [Nagano, T., Yagi, K., Osaka J. Math. 11 (1974) 181]. It shows the interest to study the invariant affine structure on the torus T2 or on abelian Lie groups. Recently, the works of Kim [J. Differential Geom. 24 (1986) 373] and Dekimpe–Ongenae [P.A.M.S., 128 (11) (2000) 3191] precise the number of non-equivalent invariant affine structures on an abelian Lie group in the case these structures are complete. In this paper, we propose a study of complete and non-complete affine structures on abelian Lie groups based on the geometry of the algebraic variety of finite dimensional associative algebras. [Copyright &y& Elsevier]