arXiv subjects

Daniele Grandini

Publications and source records attributed to Daniele Grandini.

13 recordsLinked to original sources

arXiv2026

An Efficiently Computable Lower Bound for the Independence Number of Hypergraphs

Let $k\ge2$ be fixed. We study the integer lower bound $\ell(G)$ obtained by inverting the classical counting inequality for Turán systems. For a $k$-uniform hypergraph with $n$ vertices and $m$ edges, the bound can be evaluated exactly by binary search in time polynomial in the binary lengths of $n$ and $m$. We exhibit separations from the Turán-Spencer and Caro-Tuza bounds for every fixed $k\ge3$, and from the Csaba-Plick--hokoufandeh bound in the $3$-uniform case. For the Caro-Tuza comparison, the separation grows linearly in $k$ on infinitely many regular $k$-uniform hypergraphs.

math.CO↗

arXiv2026

Independence Polynomials of 2-step Nilpotent Lie Algebras

Motivated by the Dani-Mainkar construction, we extend the notion of independence polynomial of graphs to arbitrary 2-step nilpotent Lie algebras. After establishing efficiently computable upper and lower bounds for the independence number, we discuss a metric-dependent generalization motivated by a quantum mechanical interpretation of our construction. As an application, we derive elementary bounds for the dimension of abelian subalgebras of 2-step nilpotent Lie algebras.

math.RA↗

arXiv2026

Tensorial Constraints for Commuting Endomorphisms of the Generalized Tangent Bundle

In this paper we consider families of mutually commuting endomorphisms of the generalized tangent bundle. We identify natural tensorial constraints extending the notion of a generalized Kähler structure to endomorphisms that are not necessarily generalized almost complex structures. These tensors form ideals whose generators we explicitly construct and study using Gröbner basis techniques.

math.DG↗

arXiv2024

On the cohomology of Lie algebras associated with graphs

We describe a canonical decomposition of the cohomology of the Dani-Mainkar metabelian Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the cohomology in all degrees of Lie algebras associated with arbitrary star graphs. We also describe a procedure to reduce the calculation of the cohomology of solvable Lie algebras associated with graphs through the Grantcharov-Grantcharov-Iliev construction to the cohomology of Dani-Mainkar Lie algebras.

math.RA↗

arXiv2023

A note on the shifted Courant-Nijenhuis torsion

We characterize the vanishing of the shifted Courant-Nijenhuis torsion as the strongest tensorial integrability condition that can be imposed on a skew-symmetric endomorphism of the generalized tangent bundle.

math.DG↗

arXiv2022

Polynomial Structures in Generalized Geometry

On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham operator and the Courant-Dorfman bracket. In particular we isolate several conditions that when restricted to the motivating example of generalized almost complex structure are equivalent to the notion of integrability.

math.DG↗

arXiv2017

Generalized Almost Product Structures and Generalized CRF-structures

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham operator. As applications we give a characterization of generalized complex manifolds that are locally the product of generalized complex factors and discuss infinitesimal deformations of generalized CRF-structures.

math.DG↗

arXiv2015

An Abstract Morimoto Theorem for Generalized $F$-structures

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$-structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized complex structures on product manifolds in which one factor is a Lie group and we generalize a theorem of Blair, Ludden and Yano on Hermitian bicontact manifolds.

math.DG↗

arXiv2014

Generalized Contact Geometry and T-Duality

We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant-Dorfman bracket, including T-duality. We define a notion of geometric type for generalized almost contact structures, and study its behavior under T-duality.

math.DG↗

arXiv2014

Holomorphic Poisson Cohomology

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or complex parallelizable manifolds.

math.DG↗

arXiv2011

Borel and Continuous Systems of Measures

We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.

math.FA↗

arXiv2011

Weak Pullbacks of Topological Groupoids

We introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of topological groupoids, and subsequently in HG. The category HG is the setting for topological groupoidification, which we present in separate papers, and in which the weak pullback is a key ingredient.

math.FA↗

arXiv2006

Quaternionic Kähler Reductions of Wolf Spaces

The main purpose of the following article is to introduce a \emph{Lie theoretical} approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called \emph{generalized Wolf spaces}.

math.DG↗