Some aspects of positivity in complex geometry

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Manifolds with semi-positive Ricci curvature are central objects in complex geometry. In many aspects, this class however is too small. Instead, one considers varieties admitting a singular metric with semi-positive curvature current. From an algebraic point of view, this is a property which need only to be checked along curves. Technically, we speak of varieties with nef anticanonical classes. This varieties are in the focus of the project. In particular, we aim to study the global structure of those manifolds (Albanese, birational geometry, algebraic reduction). We further want to investigate, whether a manifold with nef anticanonical class can be deformed into varieties with semi-positive Ricci curvature. These subclass is much better understood, using differential-geometric methods. Moreover, we plan to investigate whether general non-algebraic Kähler manifolds with nef anticanonical classes can be approximated by algebraic ones. This would allow to use algebraic methods to study these Kähler varieties. Finally, general positivity notions for line bundles and their connections to the geometry of curves with positive normal bundles with be studied in the frame of a duality theory.