# Virtual intersection theories

## TL;DR

This paper develops a unified framework for virtual fundamental classes across various intersection theories, enabling advanced calculations on algebraic stacks with perfect obstruction theories.

## Contribution

It introduces a comprehensive construction of virtual classes in multiple intersection theories and proves key localization and pullback formulas.

## Key findings

- Unified virtual fundamental classes across theories
- Proof of virtual pullback and localization formulas
- Applicability to Deligne-Mumford stacks with obstruction theories

## Abstract

We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.