Residue currents of holomorphic morphisms

Abstract

Abstract. Given a generically surjective holomorphic vector bundle morphism f : E → Q, E and Q Hermitian bundles, we construct a current R f with values in Hom (Q, H), where H is a certain derived bundle, and with support on the set Z where f is not surjective. The main property is that if φ is a holomorphic section of Q, and R f φ = 0, then locally f ψ = φ has a holomorphic solution ψ. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M Noether's theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to f ψ = φ, and as an example we give new results related to the H p -corona problem.

Explicit representation of membership in polynomial ideals

“…The proper analogue of (1.7) is, cf. [4], that ≤ C| det F| min(n,m−r +1) (1.11) holds locally, where is a somewhat stronger norm than the natural norm | |; i.e., ≤ | |. We have the following generalization of the previous theorems.…”

Integral representation with weights II, division and interpolation

“…Then for each r-tuple φ in H p (D), p < ∞, one can find an m-tuple ψ in H p such that F ψ = φ. This was proved in [25] (and with a sharper estimate in [4], see [4] also for a further discussion), by reducing it to the case r = 1 via the Fuhrmann trick, [21], and the case r = 1 is known since long ago, see [7] and the references given there. An explicit solution formula in case r = 1 is given in [6], and copying the arguments there, and using the special choice of Hefer forms defined in Section 4, (most likely)…”

“…If we extend u across Z as U = | det f | 2λ u| λ=0 , cf., Example 3 and see [4] for details, and let R =∂| det f | 2λ ∧u| λ=0 , then we get currents satisfying (5.2). Moreover, U is smooth outside Z and R has support on Z.…”

“…When r = 1, and codim Z = m this is precisely the duality theorem for a complete intersection ( [19] and [28]). In particular we get an explicit proof of the following statement from [4], generalizing the Briançon-Skoda theorem, [15] (for an explicit proof in the case r = 1, see [10] and [22]): Suppose that f is an r × m matrix of holomorphic functions that is generically surjective, and that φ is an r-column of holomorphic functions. If φ 2 ≤ C(det f f * ) min(n,m−r+1) , then Rφ = 0 and hence φ = f ψ locally.…”

Residue current approach to Ehrenpreis–Malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags

“…Recall that for a given section s of a hermitian vector bundle E, one can define a section s * of E * by s * = (·, s); it is called the dual section of s. The following lemma is the key tool for obtaining a good estimate of residue currents as we will see in Section 5. Though we do not include the proof of Lemma 4.1, we recall the explicit formula for s which can be found in [3]. Take any smooth local frame {ǫ j } r j=1 (r = rank Q) of Q.…”