Skoda–El Mir theorem

The Skoda–El Mir theorem is a theorem of complex geometry, stated as follows:

Theorem (Skoda,^[1] El Mir,^[2] Sibony ^[3]). Let X be a complex manifold, and E a closed complete pluripolar set in X. Consider a closed positive current $\Theta$ on $X \backslash E$ which is locally integrable around E. Then the trivial extension of $\Theta$ to X is closed on X.

Notes

1. H. Skoda. Prolongement des courants positifs fermes de masse finie, Invent. Math., 66 (1982), 361–376.
2. H. El Mir. Sur le prolongement des courants positifs fermes, Acta Math., 153 (1984), 1–45.
3. N. Sibony, Quelques problemes de prolongement de courants en analyse complexe, Duke Math. J., 52 (1985), 157–197

References

• J.-P. Demailly, L² vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)