Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

On Kähler-like and G-Kähler-like almost Hermitian manifolds

We introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0, then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.

Psychometric analysis of the scales of validity, anancastiness, impulsivity and hypersensitivity of the questionnaire of neurotic disorders

The classical test theory, on the basis of which the majority of diagnostic tools was created, has many defects associated with the lack of measuring principles in it. Today, advanced psychometric technologies based on modern test theory are relevant. These include the Rush model. These technologies surpass the classical theory of tests in scientific validity and applied efficiency, created on the basis of their psychometric scales. The purpose of the study: conduct a psychometric analysis of the scales of validity, anancastiness, impulsivity and hypersensitivity, created on the basis of the statements of the questionnaire of neurotic disorders.Surveyed 296 people. The main statistical method of work is the Rush metric system. Results: new scales have been developed: the scales of validity, anancastiness, impulsivity and hypersensitivity based on the questionnaire of neurotic disorders; statements of the presented scales have adequate constructive validity; measures of difficulty points — mainly within the regulatory range from -2 to +2 logites; the scales are one-dimensional, have a relatively balanced metric structure; reliability index validity scale is 0.88, the anancastiness scale — 0,87, the impulsivity scale — 0.82, the hypersensitivity scale — 0,93; scales of anancastiness and impulsivity are able to differentiate 3 levels of severity of properties, and scales of validity and hypersensitivity — 2.

Analysis of psychometrical properties of the scale of social anxiety of the neurotic disturbances questionnaire

Psychodiagnostic measuring instruments created within the framework of the classical theory of tests are distinguished by the instability of all psychometric parameters. Therefore, there arose the need to use modern scientifically grounded approaches for designing techniques that lack these shortcomings. The purpose of the study: to carry out an analysis of the psychometric properties of the scale of social anxiety of the questionnaire of neurotic disorders. A total of 296 people were tested. The main statistical method of work is the metric Rush system. Results: the approval of the scale of social anxiety possess an adequate constructual validity, measures of difficulty points are in the range from -2 to +2 logits, the scale is one-dimensional, has a relatively balanced metric structure, the reliability index is 0.83, the scale is able to differentiate the three levels of expression of the construct.

Astheno–Kähler and Balanced Structures on Fibrations

We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, strong Kähler with torsion (SKT), and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide a construction of astheno-Kähler structures on torus bundles over Kähler manifolds leading to new examples. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metric, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups SU(3) and G2 admit SKT and astheno-Kähler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space M with invariant volume admits a balanced metric, then its first Chern class c1(M) does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.

BALANCED METRICS ON HOMOGENEOUS VECTOR BUNDLES

Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M, ω) and let E = E1 ⊕ ⋯ ⊕ Em → M be its decomposition into irreducible factors. Suppose that each Ej admits a ω-balanced metric in Donaldson–Wang terminology. In this paper we prove that E admits a unique ω-balanced metric if and only if [Formula: see text] for all j, k = 1,…, m, where rj denotes the rank of Ej and Nj = dim H0(M, Ej). We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety (M, ω) and we show the existence and rigidity of balanced Kähler embedding from (M, ω) into Grassmannians.