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   0908144
• Harvey, F. Reese; King, James R. (1972), "On the structure of positive currents", Inventiones Mathematicae , 15 (1): 47–52, Bibcode:1972InMat..15...47H, 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 (6): 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 (1–2): 53–156, Bibcode:1974InMat..27...53S, doi:10.1007/BF01389965, ISSN   0020-9910, MR   0352516