Download this article
 Download this article For screen
For printing
Recent Issues

Volume 29, 1 issue

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Virtual classes via vanishing cycles

Tasuki Kinjo

Geometry & Topology 29 (2025) 349–397
Abstract

We develop a new method to construct the virtual fundamental classes for quasismooth derived schemes (and more generally, derived 1-Artin stacks) using the perverse sheaves of vanishing cycles on their 1-shifted cotangent spaces. It is based on the author’s previous work that can be regarded as a version of the Thom isomorphism for 1-shifted cotangent spaces. We use the Fourier–Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend–Fantechi and Li–Tian, under the schematic and quasiprojectivity assumption. We also discuss an approach to construct DT4 virtual classes for 2-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

Keywords
virtual fundamental classes, vanishing cycles, derived algebraic geometry, shifted symplectic geometry, cohomological Donaldson–Thomas theory
Mathematical Subject Classification
Primary: 14A30
Secondary: 14N35
References
Publication
Received: 3 October 2022
Revised: 31 December 2023
Accepted: 17 February 2024
Published: 22 January 2025
Proposed: Gang Tian
Seconded: Richard P Thomas, Jim Bryan
Authors
Tasuki Kinjo
Research Institute for Mathematical Sciences
Kyoto University
Kyoto
Japan

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2025, the article will become Open Access in early 2025. Otherwise, this article (and all 2025 articles) will be available only to paid subscribers.