Complex Analytic Geometry

The Hartogs extension theorem on (n – 1)-complete complex spaces

“…Firstly, connectedness of X is not a restriction, since otherwise, Ω would be contained in a single component of X. Secondly, as X is (n − 1)-complete, i ∂∂ ρ Xreg has at least 2 positive eigenvalues at every point z ∈ X reg , and consequently, each super-level set z ∈ X : ρ(z) > c , has a pseudoconcave boundary at every smooth point z ∈ X reg with dρ(z) = 0 and in fact, the Levi-form of this boundary has at least one 2 The previous definitions are known to be independent of the choices -covering, embeddings ϕ j , dimensions N j , extensions (•), see [5,7,10]. 3 The first statement yields immediately that every point z ∈ X has a neighborhood basis V k k∈N such that X reg ∩ V k is connected; also, X reg itself is connected.…”

“…N j as the zero-set {g j,ν = 0} of finitely many functions g j,ν holomorphic in a neighborhood of the closure of B j , the local ambient v j : B j → {−∞} ∪ R is essentially of the form 5 :…”

“…by definition, a function which is locally the sum of a psh function and of a C ∞ function, or equivalently, a function v whose complex Hessian i ∂∂ v has bounded negative part. 5 In addition, a regularized maximum function ( [3]) is used to smoothly glue these different definitions on all finite intersections A j1 ∩ · · · ∩ A jm and the formula given here is exact on a sub-ball C j ⊂ B j .…”

“…Then X sing is of codimension 2 at every point of X ( [5,10]) and for every open set U ⊂ X, both restriction maps…”

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“…In addition, one also has the Projection Formula f%{@® 0(f*B)) = {fl@) ® 6(B) for any bundle B on Y, a fact frequently used in Part II. The functor f% does not necessarily preserve local freeness, but if / is both flat and proper and Y is reduced, then for a locally free sheaf ^on X such that Hq(f~1(y), £% ® 6f-\(Y)) is °f constant dimension for y^ e Y, f%^is a locally free sheaf on Y, and the corresponding bundle has fibre H%f~\y), @ <S> §f-i(y))dXy e y (Fischer [11]). …”