Some invariants for surfaces with an automorphism
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Korde, Atharva Dilip
issued: 2025-10-16T19:06:17Z
issued: 2025
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: In this dissertation, the main objects of study are sheaves with a 'twisted endomorphism' on surfaces with an automorphism. The results are divided into three main chapters: In Chapter 2, we work generally and define these objects on varieties with an automorphism. The moduli stacks of these objects are shown to be algebraic. When the automorphism is of finite order, we show that the category of such objects is equivalent to a category of modules over a certain sheaf of noncommutative algebras. After fixing a stability condition, we study Hilbert schemes of points of this sheaf of noncommutative algebras. In Chapter 3, we specialize to the case of surfaces. The topological Euler characteristics of the Hilbert schemes in Chapter 2 are packaged in a generating series, which we compute. The answer generalizes the situation when the automorphism is the identity. Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) type invariants arising from these objects. We define DT-type invariants when the structure sheaf of the surface has no higher cohomology and the quotient map is a cyclic covering of smooth surfaces. This gives new examples of invariants for automorphisms of some del Pezzo surfaces, which also may be thought of as invariants of noncommutative Fano threefolds.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/92641
note: Some invariants for surfaces with anautomorphismbyAtharva Dilip KordeA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2025© Atharva Dilip Korde 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Some invariants for surfaces with an automorphismsubmitted by Atharva Dilip Korde in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in MathematicsExamining Committee:Kai Behrend, Professor, Mathematics, UBCSupervisorJim Bryan, Professor, Mathematics, UBCSupervisory Committee MemberKalle Karu, Professor, Mathematics, UBCSupervisory Committee MemberSabin Cautis, Professor, Mathematics, UBCUniversity ExaminerMark van Raamsdonk, Professor, Physics and Astronomy, UBCUniversity ExaminerJørgen Vold Rennemo, Professor, Mathematics, University of OsloExternal ExamineriiAbstractIn this dissertation, the main objects of study are sheaves with a ‘twistedendomorphism’ on surfaces with an automorphism. The results are dividedinto three main chapters:In Chapter 2, we work generally and define these objects on varietieswith an automorphism. The moduli stacks of these objects are shown tobe algebraic. When the automorphism is of finite order, we show that thecategory of such objects is equivalent to a category of modules over a certainsheaf of noncommutative algebras. After fixing a stability condition, westudy Hilbert schemes of points of this sheaf of noncommutative algebras.In Chapter 3, we specialize to the case of surfaces. The topologicalEuler characteristics of the Hilbert schemes in Chapter 2 are packaged in agenerating series, which we compute. The answer generalizes the situationwhen the automorphism is the identity.Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) typein…
has type: Thesis/Dissertation
date issued: 2025-11
is shown at: 10.14288/1.0450471
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Behrend, K. (Kai)
type: Text
identifier URI: http://hdl.handle.net/2429/92641