Related papers: Moduli spaces of (co)closed $\mathrm{G}_2$-structu…
Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form…
We determine the moduli space of metric 2-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a 4-dimensional contractible simplicial complex.
We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi…
Inspired by considerations in $M$-theory, we prove the equivalence between the moduli spaces of (suitably complexified) torsion free $G_{2}$-structures on 7-manifolds which are families of hyperK\"ahler ALE 4-manifolds fibered over compact…
A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit…
The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the…
Let $X$ be a smooth projective curve of genus $g \geq 3$, and let $G$ be a nontrivial connected reductive affine algebraic group over $\mathbb{C}$. Examining the moduli spaces of regularly stable $G$-Higgs bundles and holomorphic…
We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is…
This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states…
We study a known filtration of the second cohomology of a finite dimensional nilpotent Lie algebra $\mathfrak{g}$ with coefficients in a finite dimensional nilpotent $\mathfrak{g}$-module $M$, that is based upon a refinement of the…
In this paper we study the moduli space of 4-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds,…
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian…
We review recent results concerning closed G$_2$-structures on seven-dimensional manifolds. In particular, we discuss the construction of examples and some related problems.
This is a short note on generalized $G_2$-structures obtained as a consequence of a $T$-dual construction given in a previous work of the authors together with Leonardo Soriani. Given classical $G_2$-structure on certain seven dimensional…
In this paper, we investigate the existence of an elementary abelian closure in characteristic not $2$ for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for…
We describe the topology of the moduli spaces of flat metrics for all the 3-dimensional closed manifolds. We give an algebraic description of the moduli spaces for the 4-dimensional closed flat manifolds with a single generator in their…
The article presents the structure of the automorphism groups of two types of non-nilpotent Leibniz algebras with a dimension of 3.
In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We…
We show that the moduli space of nonnegatively curved metrics on each member of a large class of 2-connected 7-manifolds, including each smooth manifold homeomorphic to $S^7$, has infinitely many connected components. The components are…
A real Lie algebra is said to be characteristically solvable if its derivation algebra is solvable. We explicitly determine the moduli space of left-invariant metrics, up to isometric automorphism, for $6$-dimensional nilmanifolds whose…