A comparison formula for residue currents

Given two ideals $\mathcal {I}$ and $\mathcal {J}$ of holomorphic functions such that $\mathcal {I} \subseteq \mathcal {J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal {I}$ and $\mathcal {J}$. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes. One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.

Finiteness theorems for holomorphic mapping from products of hyperbolic Riemann surfaces

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds that can be covered by a bounded domain is a finite set.

Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

We study holomorphic maps between C*-algebrasAandB, whenf:BA(0,ϱ)→Bis a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ballU=BA(0,δ). If we assume thatfis orthogonality preserving and orthogonally additive onAsa∩Uandf(U)contains an invertible element inB, then there exist a sequence(hn)inB**and Jordan*-homomorphismsΘ,Θ~:M(A)→B**such thatf(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hnuniformly ina∈U. WhenBis abelian, the hypothesis ofBbeing unital andf(U)∩inv(B)≠∅can be relaxed to get the same statement.

Bloch-to-Hardy composition operators

Let φ be a holomorphic mapping between complex unit balls. We characterize those regular φ for which the composition operators C φ: f ↦ f ○ φ map the Bloch space into the Hardy space.

THE NUMBERS OF PERIODIC ORBITS HIDDEN AT FIXED POINTS OF THREE-DIMENSIONAL HOLOMORPHIC MAPPINGS

Let M be a positive integer and let f be a holomorphic mapping from a ball Δn = {x ∈ ℂn;|x| < δ} into ℂn such that the origin 0 is an isolated fixed point of both f and the M-th iteration fM of f. Then one can define the number [Formula: see text], which can be interpreted to be the number of periodic orbits of f with period M hidden at the fixed point 0. For a 3 × 3 matrix A, of which the eigenvalues are all distinct primitive M-th roots of unity, we will give a sufficient and necessary condition for A such that for any holomorphic mapping f: Δ 3 → ℂ3 with f(0) = 0 and Df(0) = A, if 0 is an isolated fixed point of the M-th iteration fM, then [Formula: see text].

L2 CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

Simple approaches to the proofs of the L2 Castelnuovo–de Franchis theorem and the cup product lemma which give new versions are developed. For example, assume that ω1 and ω2 are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kähler manifold X with ω2 in L2. According to a version of the L2 Castelnuovo–de Franchis theorem obtained in this paper, if ω1 ∧ ω2 ≡ 0, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which ω1 and ω2 are pull-backs. Previous versions required both forms to be in L2.