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.
InthesecondpartofthethesisIdiscussajointworkwithLucaBattistellaandCristina
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.