[Paper Review] On Bochner-Martinelli residue currents and their annihilator ideals

Mattias Jönsson, Elizabeth Wulcan|ArXiv.org|Nov 5, 2008

Commutative Algebra and Its Applications3 citations

TL;DR

This paper provides a geometric characterization of Bochner-Martinelli residue currents associated with holomorphic germs vanishing only at the origin, linking their annihilator ideals to the Rees valuations of the underlying ideal. It proves that the annihilator ideal equals the ideal itself if and only if the ideal is a complete intersection, and shows that non-vanishing components correspond precisely to essential multi-indices tied to Rees divisors in a log-resolution.

ABSTRACT

We study the residue current R^f of Bochner-Martinelli type associated with a tuple f=(f_1,...,f_m) of holomorphic germs at the origin in C^n, whose common zero set equals the origin. Our main results are a geometric description of R^f in terms of the Rees valuations associated with the ideal (f) generated by f and a characterization of when the annihilator ideal of R^f equals (f).

Motivation & Objective

• To understand the annihilator ideal of the Bochner-Martinelli residue current $ R^f $ associated with a tuple $ f = (f_1, \dots, f_m) $ of holomorphic germs vanishing only at the origin.
• To determine when $ \text{ann}\,R^f = (f) $, the ideal generated by the $ f_i $, which is central to the Duality Principle in residue theory.
• To give a geometric description of the non-vanishing components of $ R^f $ in terms of Rees valuations and log-resolutions of the ideal $ (f) $.
• To clarify the dependence of $ R^f $ and its annihilator on the choice of generators of $ (f) $, especially in non-complete intersection cases.
• To establish a precise link between the support of $ R^f $ and the exceptional divisors in a log-resolution that correspond to Rees valuations of $ (f) $.

Proposed method

• Utilizes a log-resolution $ \pi: X \to (\mathbb{C}^n, 0) $ of the ideal $ (f) $, lifting the current $ R^f $ to $ X $ to analyze its structure.
• Defines a multi-index $ \mathcal{I} = \{i_1, \dots, i_n\} $ as essential if the induced map $ [f_{i_1} \circ \pi : \dots : f_{i_n} \circ \pi] $ is surjective on some exceptional divisor $ E \subset \pi^{-1}(0) $, and $ \text{ord}_E(f_{i_k}) \leq \text{ord}_E(f_\ell) $ for all $ k, \ell $.
• Applies Hickel's result relating the ideal $ (f) $ to the Jacobian determinant of $ f $, enabling a connection between algebraic and analytic invariants.
• Uses Andersson's metric-independence result for residue currents to ensure the construction is intrinsic to the ideal $ (f) $, not the choice of Hermitian metric.
• Employs pushforward of currents from the resolution space to $ \mathbb{C}^n $, showing that $ R^f $ is supported on the exceptional divisors corresponding to Rees valuations.
• Analyzes explicit examples via blow-ups (e.g., $ \pi(\sigma, \tau) = (\sigma^2\tau, \sigma) $) to compute $ R^f $ and its annihilator ideals, revealing dependence on generator choices and resolution structure.

Experimental results

Research questions

• RQ1 When does the annihilator ideal of the Bochner-Martinelli residue current $ R^f $ equal the ideal $ (f) $ generated by the holomorphic functions?
• RQ2 What geometric condition on a multi-index $ \mathcal{I} $ ensures that the component $ R^f_{\mathcal{I}} $ is non-zero?
• RQ3 How do the Rees valuations of the ideal $ (f) $ relate to the structure and support of the residue current $ R^f $?
• RQ4 To what extent does the annihilator ideal $ \text{ann}\,R^f $ depend on the choice of generators of $ (f) $, especially when $ (f) $ is not a complete intersection?
• RQ5 Can the annihilator ideal $ \text{ann}\,R^f $ be computed from the contributions of individual Rees divisors in a log-resolution?

Key findings

• The annihilator ideal of $ R^f $ satisfies $ \overline{(f)^n} \subseteq \text{ann}\,R^f \subseteq (f) $, with the left inclusion strict for $ n \geq 2 $, and the right inclusion an equality if and only if $ (f) $ is a complete intersection ideal.
• The component $ R^f_{\mathcal{I}} $ is non-zero if and only if the multi-index $ \mathcal{I} $ is essential, meaning it corresponds to a Rees valuation via a surjective map on an exceptional divisor in a log-resolution.
• The annihilator ideal $ \text{ann}\,R^f $ is independent of the choice of generators $ f_i $ as long as the ideal $ (f) $ remains fixed, even though individual components $ R^f_{\mathcal{I}} $ may depend on the generators.
• In the case of a complete intersection ideal, $ \text{ann}\,R^f = (f) $, confirming the Duality Principle, and $ R^f $ is the pushforward of a current supported on Rees divisors.
• Explicit computations show that $ \text{ann}\,R^f $ can be strictly smaller than the intersection of annihilators of individual components $ R^f_{\mathcal{I}} $, indicating non-additive contributions from different Rees divisors.
• For example, in a resolution with two Rees divisors $ E_1, E_2 $, $ \text{ann}\,R^{E_1} \cap \text{ann}\,R^{E_2} \subsetneq \text{ann}\,R^f $, showing that the full current captures more information than its components.