Introduction to Equivariant Cohomology in Algebraic Geometry (IMPANGA 2010)

TL;DR

This paper provides an introductory survey of equivariant cohomology in algebraic geometry, covering basic concepts, localization techniques, and recent positivity results for Grassmannians, aimed at new learners in the field.

Contribution

It offers an accessible overview of equivariant cohomology, emphasizing localization methods and recent advances in the positivity properties of Grassmannian cohomology rings.

Findings

Localization at fixed points preserves information in equivariant cohomology.

Positivity results for Grassmannian equivariant cohomology rings.

Comprehensive introduction suitable for newcomers to the subject.

Abstract

These are lecture notes from the IMPANGA 2010 Summer School. The lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent positivity results about their equivariant cohomology rings.