Relative extremal functions and characterization of pluripolar sets in complex manifolds

Abstract

Abstract. We study a disc formula for the relative extremal function for a subset of a complex manifold and apply it to give a description of pluripolar sets and polynomial hulls.

Characterizations of boundary pluripolar hulls

“…The characterizations of Sadullaev [10], Levenberg-Poletsky [7], also cf. [3], of pluripolar hulls and their proof also hold for b-pluripolar sets. We will include this result with its very similar proof for convenience of the reader in Proposition 3.5.…”

“…It follows that the series v = ∞ j=1 v j is also uniformly convergent on compact sets in D \ {z}, hence it represents a plurisubharmonic function that is continuous up to ∂D \ {z} with boundary values ∞ j=1 F j on ∂D \ {z}. Then by (4) and (5) we have Zeriahi, [16] gave conditions under which a pluripolar set is completely pluripolar. Here we adapt Zeriahi's result to boundary pluripolar sets.…”

“…Edigarian and Sigurdsson, [4], define a domain D ⊂ C n to be weakly regular if for every relatively open subset U of ∂D we have…”

“…For z ∈ D one defines, cf. [10,8,3] (note that [3] appeared as [4], but in this latter paper there is little reference left to the boundary extremal function), ω(z, A, D) = sup{u(z) : u ∈ PSH(D), u ≤ 0, u * ≤ −1 on A}.…”

Vyacheslav Zakharyuta’s complex analysis

“…The disk formula for (locally) pluriregular sets was generalized to all manifolds by Edigarian and Sigurdsson (2006).…”