Shifted symplectic structures and derived moduli spaces

Adhikari, Nachiketa

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

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

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