Integral representation with weights II, division and interpolation

Abstract

Abstract. Let f be a r × m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that φ = f ψ, provided that φ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case r = 1. IntroductionThis paper is a continuation of [1] where we introduced a new way to generate weighted representation formulas for holomorphic functions, generalizing [11]. In this paper we focus on division and interpolation and we introduce new formulas for matrices of holomorphic functions. As applications we obtain generalizations of various results previously known for a row matrix.Let f = (f 1 , . . . , f m ) be a tuple of holomorphic functions defined in, say, a neighborhood of the closure of the unit ball D in C n with common zero set Z, and assume that df 1 ∧ . . . ∧df n = 0 on Z. In [12] was constructed a representation formulafor holomorphic functions φ, where both T and S are holomorphic in z, T (·, z) is integrable, and S(·, z) is a current of order zero (i.e., with measure coefficients) with support on Z. If φ vanishes on Z, thus (1.1) provides an explicit representation of φ as an element of the ideal generated by f . Moreover, if φ is just defined on Z, thenis a holomorphic extension, i.e., a holomorphic function in D that interpolates φ on Z. The formula (1.1) was extended to the case where f Date: December 9, 2017. 1991 Mathematics Subject Classification. 32 A 26, 32 A 27.

Explicit representation of membership in polynomial ideals

“…To be precise with the signs one has to introduce a superbundle structure on E = ⊕E k ; then for instance f is mapping of even order since it maps E k → E k−1 (and therefore f anti-commutes with odd order forms) whereas, e.g., H is even since H k is a form of degree k − (mod 2) that takes values in Hom (E , E k ), giving another facor k − (mod 2). See Section 5 in [5] for details. Proof of Proposition 4.2 The first part of the proposition follows in the same way as the corresponding statement (5.4) in [5].…”

“…See Section 5 in [5] for details. Proof of Proposition 4.2 The first part of the proposition follows in the same way as the corresponding statement (5.4) in [5]. However, we will provide an argument for a more general statement, which also implies the more general formula in Remark 6.…”

Weighted Koppelman formulas and the ∂¯-equation on an analytic space

“…If Ω is pseudoconvex and K is a holomorphically convex compact subset, then one can find a weight with respect to some neighborhood Ω of K, depending holomorphically on z, that has compact support (with respect to ζ ) in Ω, see, e.g., [2,Example 2]. Here is an explicit choice when K is the closed ball B and η = ζ − z:…”

“…The starting point is a certain residue current R, introduced in [5], that is associated to a subvariety X ⊂ Ω, and the integral representation formulas from [2]. We discuss the current R, and its associated structure form ω on X, in Section 2, and in Section 3 we recall from [4] the construction of the Koppelman formulas.…”

Weighted integral formulas on manifolds

“…More examples and references can be found in the book [4]. More recently, Andersson [3] introduced a method generalizing [9] and [6] which is even more flexible and also easier to handle. It allows for some recently found representations with residue currents, for applications to division and interpolation problems, and also allows for f to be a matrix of functions.…”