Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-29T05:38:36.110Z Has data issue: false hasContentIssue false

Introduction

Published online by Cambridge University Press:  05 October 2009

Alejandro Adem
Affiliation:
University of British Columbia, Vancouver
Johann Leida
Affiliation:
University of Wisconsin, Madison
Yongbin Ruan
Affiliation:
University of Michigan, Ann Arbor
Get access

Summary

Orbifolds lie at the intersection of many different areas of mathematics, including algebraic and differential geometry, topology, algebra, and string theory, among others. What is more, although the word “orbifold” was coined relatively recently, orbifolds actually have a much longer history. In algebraic geometry, for instance, their study goes back at least to the Italian school under the guise of varieties with quotient singularities. Indeed, surface quotient singularities have been studied in algebraic geometry for more than a hundred years, and remain an interesting topic today. As with any other singular variety, an algebraic geometer aims to remove the singularities from an orbifold by either deformation or resolution. A deformation changes the defining equation of the singularities, whereas a resolution removes a singularity by blowing it up. Using combinations of these two techniques, one can associate many smooth varieties to a given singular one. In complex dimension two, there is a natural notion of a minimal resolution, but in general it is more difficult to understand the relationships between all the different desingularizations.

Orbifolds made an appearance in more recent advances towards Mori's birational geometric program in the 1980s. For Gorenstein singularities, the higher-dimensional analog of the minimal condition is the famous crepant resolution, which is minimal with respect to the canonical classes.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Introduction
  • Alejandro Adem, University of British Columbia, Vancouver, Johann Leida, University of Wisconsin, Madison, Yongbin Ruan, University of Michigan, Ann Arbor
  • Book: Orbifolds and Stringy Topology
  • Online publication: 05 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543081.001
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Introduction
  • Alejandro Adem, University of British Columbia, Vancouver, Johann Leida, University of Wisconsin, Madison, Yongbin Ruan, University of Michigan, Ann Arbor
  • Book: Orbifolds and Stringy Topology
  • Online publication: 05 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543081.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Introduction
  • Alejandro Adem, University of British Columbia, Vancouver, Johann Leida, University of Wisconsin, Madison, Yongbin Ruan, University of Michigan, Ann Arbor
  • Book: Orbifolds and Stringy Topology
  • Online publication: 05 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543081.001
Available formats
×