Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Shen, Junliang

Abstract

The goal of this thesis is to study the cohomological aspect of Le Potier's moduli space of $1$-dimensional sheaves on the projective plane. Our main technical tools are Mumford-type geometric relations, formulated originally over moduli of bundles on curves and generalized to sheaves over del Pezzo surfaces in this thesis. We prove that these relations can be obtained effectively from certain primitive ones via the Virasoro operators. As applications, we prove a minimal generation result, a cohomological $\chi$-dependence result, and fully compute the cohomology rings in low degrees. Finally, we propose a ``Perverse $=$ Chern'' conjecture relating two filtrations of highly different nature on the cohomology. This can be viewed as a compact and Fano analogue of the $P=W$ conjecture of de Cataldo--Hausel--Migliorini. The main body of this thesis is a synthesis of three research papers \cite{PS23, LMP24a, KLMP24}. The first paper is joint work with Junliang Shen, the second is joint with Woonam Lim and Miguel Moreira, and the third is joint with Yakov Kononov, Woonam Lim, and Miguel Moreira.

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