Shifted symplectic structures and derived moduli spaces

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Adhikari, Nachiketa

Contributors dc:contributor

• Katz, Sheldon
• Heller, Jeremiah
• Dodd, Christopher
• Janda, Felix

• Algebraic Geometry
• Moduli Spaces
• Derived Stacks
• Shifted Symplectic Structures
• Quot Scheme
• Hilbert Scheme
• Coherent Sheaves
• Calabi-yau Manifolds

Statement dc:rights

• Copyright 2024 Nachiketa Adhikari

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F125537