[Paper Review] Local intersection numbers and a generalized King formula

Mats Andersson, Elizabeth Wulcan+2 more|arXiv (Cornell University)|Jan 1, 2012

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper establishes that the Lelong numbers of generalized Monge-Ampère products $(dd^c\log|f|^2)^k$ restricted to the zero set of an ideal sheaf $\mathcal{J}$ coincide with the local intersection numbers defined by Tworzewski and Gaffney-Gassler via a St{\'u}ckrad-Vogel procedure. It introduces a generalized King formula accounting for fixed and moving components in Vogel sequences, using a novel calculus for Bochner-Martinelli currents as a key tool.

ABSTRACT

Let $\mathcal J$ be an ideal sheaf on a reduced analytic space $X$. Given generators $f_1,...,f_m$ of $\mathcal J$ let $M^f_k$ be restrictions to the zero set of $\J$ of the generalized Monge-Ampere products $(dd^c\log|f|^2)^k$. We prove that the Lelong numbers at $x$ of these currents coincide with the list of locally defined numbers introduced independently by Tworzewski and Gaffney-Gassler using a local St\uckrad-Vogel procedure. We also give a generalization of the classical King formula that takes into account the difference of fixed and moving components of Vogel sequences associated with $\mathcal J$. xA basic tool is a new calculus for products of a certain kind of positive closed currents, so-called Bochner-Martinelli currents.

Motivation & Objective

• To establish a precise link between Lelong numbers of generalized Monge-Ampère currents and local intersection numbers defined via the St{\'u}}ckrad-Vogel procedure.
• To resolve the discrepancy between fixed and moving components in Vogel sequences associated with an ideal sheaf $\mathcal{J}$.
• To develop a new calculus for products of Bochner-Martinelli currents, enabling the analysis of singular positive closed currents.
• To generalize the classical King formula to incorporate local intersection-theoretic data in complex analytic geometry.

Proposed method

• Define generalized Monge-Ampère products $(dd^c\log|f|^2)^k$ for generators $f_1,\dots,f_m$ of an ideal sheaf $\mathcal{J}$, restricted to the zero set of $\mathcal{J}$.
• Introduce Bochner-Martinelli currents as a new class of positive closed currents with controlled singularities, enabling product calculus.
• Use the Bochner-Martinelli calculus to compute Lelong numbers of the generalized Monge-Ampère currents.
• Apply the St{\'u}}ckrad-Vogel procedure locally to define intersection numbers and compare them to the Lelong numbers of the currents.
• Derive a generalized King formula that accounts for fixed and moving components in Vogel cycles associated with $\mathcal{J}$.

Experimental results

Research questions

• RQ1 Do the Lelong numbers of the generalized Monge-Ampère currents $(dd^c\log|f|^2)^k$ restricted to $V(\mathcal{J})$ match the local intersection numbers defined by Tworzewski and Gaffney-Gassler?
• RQ2 How can the classical King formula be extended to include contributions from both fixed and moving components in Vogel sequences?
• RQ3 What is the role of Bochner-Martinelli currents in computing intersection numbers via Lelong numbers?
• RQ4 Can a consistent calculus for products of positive closed currents be developed for this class of singular currents?
• RQ5 What is the precise relationship between the algebraic structure of the ideal $\mathcal{J}$ and the analytic invariants of its associated currents?

Key findings

• The Lelong numbers of the currents $M^f_k = (dd^c\log|f|^2)^k|_{V(\mathcal{J})}$ at a point $x$ are equal to the local intersection numbers defined via the St{\'u}}ckrad-Vogel procedure.
• The generalized King formula accounts for both fixed and moving components in Vogel cycles, refining the classical formula.
• The Bochner-Martinelli current calculus provides a robust framework for computing products of singular positive closed currents in this context.
• The equivalence between analytic Lelong numbers and algebraic local intersection numbers holds in the setting of reduced analytic spaces.
• The method allows for a unified treatment of intersection theory and Lelong number theory for ideal sheaves via current-theoretic techniques.