ETD

• Analytic sets
• Complex geometry
• Currents
• Lelong Numbers
• Multiplicity
• Pluripotential Theory
• Plurisubharmonic Functions
• Positive
• Siu's Semicontinuity Theorem
• Slicing theory

This thesis lies in the field of complex geometry and pluripotential theory. Its main focus is the study of Lelong numbers, which provide a natural generalization of the notion of multiplicity of analytic subsets to the broader setting of positive closed currents.

Lelong numbers were originally introduced to describe the local singular behavior of plurisubharmonic functions and, more generally, of positive currents. In this framework, analytic sets appear as a special case through their associated currents of integration, while Lelong numbers extend the classical concept of multiplicity to a more flexible analytic setting.

After introducing the basic notions of analytic sets, currents, positivity, and Lelong numbers, we prove Siu’s semicontinuity theorem. This fundamental result asserts that, given a positive closed (p,p)-current
on a complex manifold, the upper level sets of its Lelong numbers are analytic subsets of the manifold.

We first focus on the proof of the theorem in the special case of (1,1)-currents. This case plays a central role in Siu’s original argument, since positive closed (1,1)-currents admit a local representation in terms of plurisubharmonic functions. Such a representation makes the geometric meaning of Lelong numbers more transparent and allows for a more direct and intuitive proof.

We then outline the strategy used to extend the result to general (p,p)-currents. In particular, we describe the role of projection techniques and Federer’s slicing theory, which make it possible to reduce the general case to the (1,1)-case and thus complete the proof of Siu’s theorem.