I will define pseudoholomorphic curves in symplectic manifolds and describe their local properties.
In this talk I will turn to whole spaces of pseudoholomorphic curves and discuss when their moduli spaces are smooth manifolds of the expected dimension and what we mean by that. I will sketch the standard proof of why these moduli spaces are smooth when curves are simple and the almost complex structure is generic.
Moduli spaces of pseudoholomorphic curves are usually not compact, either because of bubbling phenomena or because the complex structure on the domain can degenerate. Nonetheless, allowing for a slight generalisation of pseudoholomorphic curves, so-called stable maps, we can define compact moduli spaces. I will state and sketch the proof of the famous compactness result due to Gromov.
I will explain why the transversality result described in the second talk does not suffice to define GW invariants in general. Then I will sketch the construction of a global Kuranishi chart for moduli spaces of stable maps in genus zero and show how one can use it to define GW invariants.
The aim of these lectures is to set up the basic material for the study of Gromov-Witten theory. It will be divided into three sections: the construction of moduli spaces, the virtual fundamental cycle, and the localization formula, which is one of the very important tool used in the computation of Gromov-Witten invariants. We will focus on the main ideas, present some key examples, and explain the difficulties in such computations. We finish the course by a discussion on the state-of-the-art and some open problems in this field.
Counting curves of fixed degree and genus passing through a suitable number of points is a natural generalization of the well-known (and easy) problem of counting lines passing through two points. However, although in this more general setting the answer does not depend either on the precise choice of the points, concretely computing the number of such curves remains a challenge. In this talk, we will explain how tropical techniques enable concrete computations of these “enumerative invariants”.
Real Gromov-Witten theory studies the moduli spaces of J-holomorphic curves that are compatible with a fixed anti-symplectic involution of the target and an anti-holomorphic involution of the domain. In this talk, I will introduce these moduli spaces and exhibit their basic features. For instance, as it can be seen on the example of the moduli spaces of stable real curves, they are not necessarily orientable. The positive genus Gromov-Witten invariants of projective spaces - among other targets - can be recovered from the genus 0 invariants through a universal reconstruction formula, known as Givental reconstruction. In order to illustrate the difference between the real and complex Gromov-Witten invariants, I will explain how this formula can be modified to obtain the positive genus real Gromov-Witten invariants of projective spaces.
Tropical refined invariants are polynomials related to the count of curves on toric surfaces. They interpolate between complex and real counts of curves, i.e. between Gromov-Witten and Welschinger invariants. Their general signification is quite mysterious. In this talk I will use floor diagrams to introduce tropical refined invariants, and give few of their properties. The goal is then to state (by hand-waving a bit) a result by Bousseau which relates these polynomials with (relative) Gromov-Witten invariants.