Website of Oliver Leigh
Site d'Oliver Leigh
Oliver Leighs hemsida

Welcome to my website.

I am currently a Postdoctoral Research Fellow in Algebraic Geometry at The University of Oslo.

Previously I was a Postdoctoral Research Fellow at Uppsala University (2021-2023), and Stockholm University (2019-2021).

In 2019 I completed a jointly awarded PhD at the University of Melbourne and the University of British Columbia.
Bienvenue sur mon site web.

Je suis chercheur postdoctoral en géométrie algébrique à l'Université d'Oslo.

Auparavant, j'étais chercheur postdoctoral en géométrie algébrique à l'Université de Uppsala (2021-2023) et à l'Université de Stockholm (2019-2021).

En 2019, j'ai completé un doctorat conjointement à l'Université de Melbourne et à l'Université de la Colombie-Britannique.
Välkommen till min hemsida.

Jag är postdoktor i algebraisk geometri vid Universitetet i Oslo.

Tidigare var jag en postdoktor i algebraisk geometri vid Uppsala universitet (2021-2023) och Stockholms universitet (2019-2021).

2019 slutförde jag en gemensamt tilldelad doktorsexamen vid University of Melbourne och University of British Columbia.

Research Interests
Intérêts de recherche
Forskningsintressen

I am interested in enumerative problems in geometry that are motivated by physics. I primarily study these from an algebro-geometric viewpoint. This includes both Gromov-Witten and Donaldson-Thomas theory.

In Donaldson-Thomas theory we can count subschemes of Calabi-Yau threefolds by examining the local picture. Using this method we can count subschemes by considering 3D partitions and using the topological vertex. Below is a tool for visualising these.
Je m'intéresse aux problèmes énumératifs en géométrie motivés par la physique. J'étudie principalement ceux-ci en utilisant un point de vue géométrique algébrique. Cela inclut à la fois la théorie de Gromov-Witten et celle de Donaldson-Thomas.

Dans la théorie de Donaldson-Thomas, nous pouvons compter les sous-schémas d'une variété de Calabi-Yau de dimension trois. En utilisant cette méthode, nous pouvons compter le sous-schéma en considérant les partitions 3D et en utilisant le sommet topologique. Vous trouverez ci-dessous un outil permettant de les visualiser.
Jag är intresserad av enumerativa problem i geometri som motiveras av fysik. Jag studerar främst dessa från en algebra geometrisk synvinkel. Detta inkluderar både Gromov-Witten och Donaldson-Thomas teori.

I Donaldson-Thomas-teorin kan vi räkna delscheman av tredimensionella Calabi-Yau mångfalder genom att undersöka den lokala bilden. Med hjälp av denna metod kan vi räkna delscheman genom att betrakta 3D-partitioner och använda det topologiska hörnet. Nedan finns ett verktyg för att visualisera dessa.


Research Articles
Articles de recherche
Forskningsartiklar

Abstract
Résumé (en anglais)
Abstrakt (på engelska)
We prove a blow-up formula for the generating series of virtual χ
y
-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological χ
y
-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on ℙ². The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on ℙ², which has recently been extend to the setting of Gieseker
H
-stable sheaves on
H
-polarised surfaces by Kuhn-Tanaka.
arXiv:2205.12953 [math.AG]
Abstract
Résumé (en anglais)
Abstrakt (på engelska)
In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.
DOI: 10.4171/PM/2115 arXiv:2107.00988 [math.AG]
Abstract
Résumé (en anglais)
Abstrakt (på engelska)
The moduli space of stable maps with divisible ramification uses
r
-th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer
r
. In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the
r
-ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.
arXiv:2004.06739 [math.AG]
Abstract
Résumé (en anglais)
Abstrakt (på engelska)
We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied by Bryan in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ``banana configuration''. In [Bryan'19] the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande-Thomas theory for a rational elliptic surface and present new Gopakumar-Vafa invariants for the banana threefold.
DOI: 10.1093/qmathj/haaa007 arXiv:1907.01054 [math.AG]
Abstract
Résumé (en anglais)
Abstrakt (på engelska)
We develop a theory for stable maps to curves with divisible ramification. For a fixed integer
r>0
, we show that the condition of every ramification locus being divisible by
r
is equivalent to the existence of an
r
-th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to r-spin Hurwitz theory. In particular it is expected to provide a proof of the
r
-spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation.
arXiv:1812.06933 [math.AG]
Abstract
Résumé (en anglais)
Abstrakt (på engelska)
We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfies the topological recursion of Eynard and Orantin. This generalises the Bouchard-Mariño conjecture and places Hurwitz-Hodge integrals, which arise in the Gromov-Witten theory of target curves with orbifold structure, in the context of the Eynard-Orantin topological recursion.
DOI: 10.4310/MRL.2016.v23.n5.a3 arXiv:1212.6850 [math.AG]

Contact Information
Informations de Contact
Kontaktinformation

Email:
Email:
E-post:


Visiting address:
Adresse de visite:
Beöksadress:
Room
Salle
Rum
1114,
Niels Henrik Abels hus,
Moltke Moes vei 35,
0851 Oslo,
Norway
Norvège
Norge


Postal address:
Adresse postale:
Postadress:
Oliver Leigh,
Postboks 1053 Blindern,
0316 Oslo,
Norway