The project aims to study three problems of the Hermitian geometry of the compact non-Kaehler complex manifolds: The geometry of Hermitian manifolds admitting totally non-degenerate torus action, Hermitian structures on Riemannian manifolds admitting many compatible complex structures and geometry of hypercomplex manifolds and connections with totally antisymmetric torsion. The methods of investigation include: Kodaira-Spencer-Kuranishi deformation theory, Lichnerowicz formulas for Dirac operators and vanishing theorems, twistor correspondence, Kaehler orbifolds, nilpotent Lie groups, moment maps for various spaces with geometric structures, harmonic maps and others. If the project is successful, it is expected a new examples of Einstein Hermitian manifolds to be obtained, the existence problem of HKT structure on arbitrary hypercomplex manifold to be solved (negatively) and some characterization or rigidity type results for manifolds with many complex structures to be found.
The role of the abstract spaces constructed from the complex numbers in the theoretical foundations of quantum mechanics is significant. It increases in the development of quantum mechanic to the theory of quantum fields. A more recent investigations in the theoretical physics are directed toward a unification of the quantum mechanics and general relativity. This, yet hypothetical theory, is called string theory and for a long time the Kaehler geometry of the complex spaces has served as its mathematical foundation. In mathematics little is known about the structure of a general non-Kaehler complex spaces. However, in the fast-growing development of the string theory an abstract models based on the non-Kaehler geometry appeared in the last years. Such spaces have also additional geometrical properties which makes the study of their mathematical structure more tractable. The investigation on the project will contribute to the understanding of some of the mathematical problems related to these physical models.
