University of Illinois Urbana-Champaign

Shifted symplectic structures and derived moduli spaces

Adhikari, Nachiketa

Content Files
ADHIKARI-DISSERTATION-2024.pdf

Permalink

https://hdl.handle.net/2142/125537

Description

Title
Shifted symplectic structures and derived moduli spaces
Author(s)
Adhikari, Nachiketa
Issue Date
2024-06-26
Director of Research (if dissertation) or Advisor (if thesis)
Katz, Sheldon
Doctoral Committee Chair(s)
Heller, Jeremiah
Committee Member(s)
  • Dodd, Christopher
  • Janda, Felix
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • algebraic geometry
  • moduli spaces
  • derived stacks
  • shifted symplectic structures
  • quot scheme
  • hilbert scheme
  • coherent sheaves
  • calabi-yau manifolds
Abstract
Derived algebraic geometry is a vast generalization of algebraic geometry that involves higher categories in a fundamental way. Constructions of derived moduli spaces and their intrinsic features as derived objects (for example, the cotangent complex, higher morphism spaces) have proved instrumental in understanding the behavior of many ordinary moduli spaces. This in turn has facilitated advancements in enumerative geometry (among other fields), especially that of Calabi-Yau threefolds and fourfolds. This thesis, after providing a brief introduction to the derived algebraic framework developed by Toën-Vezzosi and Lurie, builds on existing work of Toën-Vaquié to prove the existence and representability of the derived quot scheme. This derived moduli space is in a natural way a generalization of the ordinary quot scheme, which, while crucial for constructing several other interesting moduli spaces, is also interesting in its own right. Shifted symplectic structures, introduced by Pantev-Toën-Vaquié-Vezzosi, are extensions of ordinary symplectic structures to the derived setting. They capture several phenomena seen at the classical level, such as symmetric obstruction theories, while also shedding light on new phenomena. This thesis (including joint work with Yun Shi described in Sections 4.3 and 4.4) builds on work of Brav-Bussi-Joyce to show that certain shifted symplectic derived spaces (including moduli spaces of coherent sheaves on fourfolds) can be locally obtained as derived critical loci.
Graduation Semester
2024-08
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/125537
Copyright and License Information
Copyright 2024 Nachiketa Adhikari

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois