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Title: Gopakumar-Vafa invariants in genus 0 and 1
Author: Carocci, Francesca
ISNI:       0000 0004 7658 7047
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2018
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In this thesis we explore two possible approaches to the study of Gopakumar-Vafa invari- ants in genus 0 and 1, mostly concentrating on their relation with certain moduli spaces of morphisms. The work is divided in two parts. In the first part we study via a new technique inspired by Hitchin's spectral curve construction, a version of Donaldson-Thomas invari- ants which Katz has proposed as a mathematical definition of genus 0 GV invariants. The spectral curve point of view leads us, in the reduced class case, to express the local invariants of a fixed curve in terms of maps from curves homeomorphic to genus 0 ones. At the end of the first part we look at some examples in the non reduced class case and explore the spectral construction techniques in this set up. In the second part of the thesis I discuss a joint work with Luca Battistella and Cristina Manolache. We first recall Li-Zinger definition of reduced genus 1 invariants, which coincide with genus 1 GV for the quintic 3-folds for the reduced class case, and prove that such invariants coincide with those arising from the moduli space of 1-stable maps. In such moduli space elliptic tails are destabilised and replaced by non degenerate maps from cuspidal curves. We think of this as a first step in a longer project, with the aim is that to define alternate compactifications of the moduli space of maps in higher genus whose associated invariants do not take into account degenerate contributions.
Supervisor: Thomas, Richard Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral