arXiv research

Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.

Study on Hermitian metrics on Lie algebras with specific ideals.

New algebraic structures for Hermitian geometry cohomologies.

Study on balanced Hermitian structures on Lie algebras twisted by representations.

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.

We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian…

Flat Hermitian Lie algebras are always Kähler.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

Characterizes a class of almost Hermitian 4-manifolds using integral identities.

Develops a universal Hermitian projective calculus for complex hyperbolic two-space

Introduces symplectic groups over noncommutative algebras and their geometric actions.

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structur…

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form $ω$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfr…

A hermitian algebra is a unital associative ${\mathbb C}$-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ${\mathbb R}$. In the case of an algebra ${\mathcal A}$ endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open an…

Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.

We start by analysing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a…

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special funct…

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure…

Deforms orbits in Lie algebras to Lagrangian submanifolds.

The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on two parameters. Some geometric characteris…

Study of curvature flow on complex Lie groups, leading to soliton convergence.

Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.

Study on 4D Lie groups and related almost hypercomplex manifolds.

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehl…

Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this posit…

We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri-Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen …

Characterizes complex structures on specific Lie groups.

Characterizes tangent cones for specific connections on reflexive sheaves.

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …

Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dim…

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by M.L. Barberis, are constructed here. The different cases regarding the signature of the basic pseudo…

Study of Hermitian metrics on Lie algebroids over complex spaces.

The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.

Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.

Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians $L_m$ are defined for each positive integer $m$, which generalize the ordinary Laplace-Beltrami operator. We show …

Study of $H$-eigenvalues for complex tensors and their applications in differential geometry.

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…

A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complet…

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebrai…

Study Lie algebras with complex structures, focusing on degenerations and deformations.

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric propertie…

Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.

The Streets-Tian conjecture is confirmed for specific types of Hermitian manifolds.