Project Executive SummaryLocally conformally Kähler (LCK) manifolds are complex manifolds of complex dimension at least two admitting a Kähler covering with deck transformations acting by holomorphic homotheties with respect to the Kähler metric. They belong to complex differential geometry and can be treated using methods pertaining to complex geometry, Riemannian and conformal geometry, algebraic geometry and topology. The very need of combining all these methods constitutes an intrinsic difficulty of the subject. Our project concerns complex and Riemannian properties of LCK manifolds. In complex geometry, we intend to: count the elliptic curves on compact Vaisman manifolds; study the analytic invariants of the recently found new class of LCK manifolds with global spherical shell (which are not Vaisman); determine the properties of holomorphic submersions between LCK manifolds, aiming to prove the non-existence of LCK products; determine the subspace of Lee forms in the 1-cohomology of a given compact LCK manifold; classify 3-dimensional LCK manifolds according to their algebraic dimension; extend the theory to singular analytic spaces; study the possibility of coexistence of an LCK metric with other non-Kähler metrics (e.g. balanced, astheno-Kähler etc.) on LCK manifolds, in particular on solvmanifolds; study the existence problem of LCK metrics on Oeljeklaus-Toma (OT) manifolds. In Riemannian geometry, we shall concentrate on variational properties (harmonic maps and morphism, Yang-Mills fields and generalizations) and deformations of the canonical foliation of Vaisman manifolds. Our methods will combine real and complex differential geometry techniques with algebraic geometry techniques and, for OT manifolds, number theoretic ones.
Financial and Scientific Reports
Research funded by this grant:A. Articles published in ISI ranked journals
B. Papers submitted and preprints
C. Chapters in books published in foreign editorial houses