Siu's semicontinuity theorem

In complex analysis, the Siu semicontinuity theorem implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong number is at least some constant form a complex subvariety. This was conjectured by Harvey & King (1972) and proved by Siu (1973, 1974). Demailly (1987) generalized Siu's theorem to more general versions of the Lelong number.

References

• Demailly, Jean-Pierre (1987), "Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité", Acta Mathematica 159 (3): 153–169, doi:10.1007/BF02392558, ISSN 0001-5962, MR 908144 
• Harvey, F. Reese; King, James R. (1972), "On the structure of positive currents", Inventiones Mathematicae 15: 47–52, doi:10.1007/BF01418641, ISSN 0020-9910, MR 0296348 
• Siu, Yum Tong (1973), "Analyticity of sets associated to Lelong numbers and the extension of meromorphic maps", Bulletin of the American Mathematical Society 79: 1200–1205, doi:10.1090/S0002-9904-1973-13378-6, ISSN 0002-9904, MR 0330505 
• Siu, Yum Tong (1974), "Analyticity of sets associated to Lelong numbers and the extension of closed positive currents", Inventiones Mathematicae 27: 53–156, doi:10.1007/BF01389965, ISSN 0020-9910, MR 0352516