On proper intersections on a singular analytic space

TL;DR

This paper introduces a new intrinsic method for defining proper intersections of cycles on singular analytic spaces, generalizing classical intersection theory and enabling global formulas.

Contribution

It develops an intrinsic intersection product for nice cycles on singular spaces using residue calculus, extending classical intersection theory to singular settings.

Findings

Defines a class of nice cycles including all effective $Q$-Cartier divisors.

Provides an intrinsic intersection product for equidimensional nice cycles.

Establishes global formulas for intersections on singular analytic spaces.

Abstract

Given a reduced analytic space $Y$ we introduce a class of {\it nice} cycles, including all effective $\mathbb{Q}$-Cartier divisors. Equidimensional nice cycles that intersect properly allow for a natural intersection product. Using $\bar{\partial}$-potentials and residue calculus we provide an intrinsic way of defining this product. The intrinsic definition makes it possible to prove global formulas. In case $Y$ is smooth all cycles are differences of nice cycles, and so we get a new way to define classical proper intersections.