Classification of 6-dimensional splittable flat solvmanifolds

A flat solvmanifold is a compact quotient Γ∖G where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and Γ is a lattice of G. Any such Lie group can be written as G=Rk ltimes_ϕ Rm with Rm the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice Γ that can be decomposed as Γ=Γ1 ltimes_ϕ Γ2, where Γ1 and Γ2 are lattices of Rk and Rm, respectively. We analyze the relation between these lattices and the conjugacy classes of finite abelian subgroups of GL(n,Z), which is known up to n≤6. From this we obtain the classification of 6-dimensional splittable flat solvmanifolds.Fil: Tolcachier, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry

In this article we study solvmanifolds $\Gamma\backslash G$ equipped with invariant complex structures such that its canonical bundle is trivial. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of a 1-form $\psi$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to recover some known results in the literature as well as to produce new examples of complex solvmanifolds with trivial canonical bundle. Later we consider the non-invariant case and we provide an algebraic obstruction, also in terms of $\psi$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. In addition, this obstruction leads us to a way of building explicit non-invariant sections which we illustrate with some examples. Finally, we apply our results to hypercomplex manifolds in order to provide a negative answer to a question posed by M. Verbitsky.Comment: 32 pages. Corrected some small typos. Comments are welcome

Propiedades geométricas de solvariedades y productos de variedades sasakianas

Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación, 2023.Fil: Tolcachier, Alejandro. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.En esta tesis doctoral estudiamos algunas propiedades geométricas de dos grandes familias de variedades: solvariedades y productos de variedades sasakianas. En primer lugar, las solvariedades son variedades compactas obtenidas como cocientes de grupos de Lie solubles simplemente conexos por subgrupos discretos. Estas variedades son importantes en geometría diferencial por dos razones: por un lado, al estudiar estructuras invariantes varios problemas geométricos se traducen a problemas algebraicos en un álgebra de Lie. Por otro lado, han sido en numerosas ocasiones una fuente de ejemplos (o contraejemplos) a preguntas importantes del área. Comenzamos explorando la G2-geometría de solvariedades de dimensión 7 equipadas con una métrica plana invariante, las cuales pueden ser entendidas desde la teoría de variedades compactas planas. Luego, analizamos el problema de hallar estructuras casi complejas armónicas (puntos críticos de la funcional de energía de Dirichlet) ortogonales respecto de una métrica invariante en solvariedades casi abelianas, para lo cual previamente describimos todas las clases de Gray-Hervella asociadas a álgebras de Lie casi abelianas. Por último nos dedicamos al estudio de solvariedades complejas con fibrado canónico trivial. Aquí mostramos un fenómeno desconocido hasta el momento, que luego aplicamos para dar respuestas a preguntas interesantes en geometría hipercompleja. En segundo lugar, estudiamos métricas distinguidas en el producto de dos variedades sasakianas equipado con una familia a 2 parámetros de estructuras hermitianas. Esta clase generaliza a las variedades de Calabi-Eckmann.In this doctoral thesis, we study some geometric properties of two large families of manifolds: solvmanifolds and products of Sasakian manifolds. Firstly, solvmanifolds are compact manifolds obtained as quotients of simply connected solvable Lie groups by discrete subgroups. These manifolds are important in differential geometry for two reasons: on the one hand, when studying invariant structures, various geometric problems translate into algebraic problems in a Lie algebra. On the other hand, solvmanifolds have often served as a source of examples (or counterexamples) to important questions in the field. We begin by exploring the G2-geometry of 7-dimensional solvmanifolds equipped with an invariant flat metric, which can be understood from the theory of compact flat manifolds. Next, we analyze the problem of finding harmonic almost complex structures (critical points of the Dirichlet energy functional) orthogonal to an invariant metric on almost abelian solvmanifolds. To do this, we first describe all the Gray-Hervella classes associated with almost abelian Lie algebras. Finally, we focus on the study of complex solvmanifolds with trivial canonical bundle. Here, we show a previously unknown phenomenon, which we then apply to provide answers to interesting questions in hypercomplex geometryFil: Tolcachier, Alejandro. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina