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Mumford Relations and the Cohomology...

Pi, Weite,

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Mumford Relations and the Cohomology of Le Potier Moduli Spaces /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Mumford Relations and the Cohomology of Le Potier Moduli Spaces // Weite Pi.
Author:	Pi, Weite,
Description:	1 electronic resource (112 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31844832
ISBN:	9798286445240

Mumford Relations and the Cohomology of Le Potier Moduli Spaces /
Pi, Weite,

Mumford Relations and the Cohomology of Le Potier Moduli Spaces /Weite Pi. - 1 electronic resource (112 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

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 χ-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 [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.

English

ISBN: 9798286445240Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Mumford relations

Mumford Relations and the Cohomology of Le Potier Moduli Spaces /
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245 1 0 $a Mumford Relations and the Cohomology of Le Potier Moduli Spaces / $c Weite Pi.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2025
300 $a 1 electronic resource (112 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
500 $a Advisors: Shen, Junliang Committee members: Loseu, Ivan; Maulik, Davesh.
502 $b Ph.D. $c Yale University $d 2025.
520 $a 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 χ-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 [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.
546 $a English
590 $a School code: 0265
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Mumford relations
653 $a Cohomology
653 $a Geometric relations
653 $a Virasoro operators
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a Yale University. $b Mathematics. $e degree granting institution. $3 3822989
720 1 $a Shen, Junliang $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-12B.
790 $a 0265
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31844832