Residue currents associated with weakly holomorphic functions

Abstract

We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as the transformation law, the Poincar\'e-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli type residue current associated with $f$ when $f$ defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In particular, corrected and clarified some things in Section 5 and 6 regarding products of weakly holomorphic functions and currents, and the definition of the Bochner-Martinelli type current

Sparse effective membership problems via residue currents

“…One can define pseudomeromorphic currents also on singular varieties, so that the properties above hold true, see [24].…”

“…One can show that π * (∂|f | 2λ ∧ u), where u is defined by (2.4), has an analytic continuation as a current on X + to where Re λ > −ǫ, such that π * R + = R f , where R + = π * (∂|f | 2λ ∧ u)| λ=0 , see [24]. In X + , π * f = f 0 f ′ , where f 0 is holomorphic and f ′ is a nonvanishing tuple.…”

“…The Bochner-Martinelli residue current can actually be defined also on singular varieties; it will however not have as nice properties as in the smooth case, cf. [7,24]. It would be interesting to investigate the general situation more carefully.…”

Direct images of semi-meromorphic currents

“…see Section 2.2. The notion of pseudomeromorphic currents plays a decisive role in, for instance, [10,11,8,18,19,7,28,29,25,21,27].…”

Pseudomeromorphic currents on subvarieties

“…These basic properties are very useful, or even indispensable, tools in, for instance [1,2,[8][9][10][11][12][13][14][15]. If μ is pseudomeromorphic and has support on a pure-dimensional subvariety V ⊂ X we say that μ has the standard extension property (SEP), with respect to V , if 1 A μ = 0 for each germ of a subvariety A ⊂ V of positive codimension, at any point of V .…”