Moduli spaces of stable and unstable maps to curves and surfaces

Moduli spaces arise naturally in classification problems in geometry, and play important roles in many different areas, in particular algebraic, differential and symplectic geometry. The aim of this research project is to use ideas and methods from algebraic geometry, differential geometry and symplectic topology, exploiting the complementary areas of expertise of the three supervisors, to study moduli spaces of holomorphic maps from complex projective curves to low-dimensional complex projective varieties, allowing singularities in their domains and targets.

A moduli problem, for example the classification of nonsingular complex projective curves up to isomorphism, or equivalently compact Riemann surfaces up to biholomorphism, can usually be resolved into some basic steps. The first step is to find as many discrete invariants of the objects to be classified as possible (in the case of nonsingular complex projective curves the genus is the only discrete invariant). The second step is to fix the values of all the discrete invariants and try to construct a moduli space; that is, an algebraic variety (or some more general geometric object) whose points correspond in a natural way to the equivalence classes of the objects to be classified. This works nicely for nonsingular curves (at least if we are willing to work with 'coarse' moduli spaces), but if we want to include singular curves then much more care is needed. Complex projective curves with very mild singularities (so-called stable curves) can be included without difficulty; the moduli spaces of stable curves of different genera are themselves projective varieties whose enumerative geometry has been intensively studied over the last decades. However if we include curves with more serious singularities, then the existence of a moduli space is too much to hope for, unless we are willing to replace spaces by much more sophisticated and less tractable geometric objects called stacks. A moduli stack may have an associated moduli space, but it is often the case that not even a coarse moduli space will exist.

Traditionally, 'unstable' objects are left out in order to construct a moduli space, so the final step in the classification problem is to understand which objects are (semi)stable and will appear in the moduli space. However in recent years new methods (using 'non-reductive geometric invariant theory') have been developed which allow the construction of moduli spaces of unstable objects ... more precisely, the final step in the classification problem becomes the assignment of a refined version of 'Harder-Narasimhan type' to each unstable object, and the construction of moduli spaces of fixed Harder-Narasimhan type ... and Frances Kirwan's student Joshua Jackson's 2019 thesis studied moduli spaces of unstable complex projective curves.

Instead of just considering moduli of curves, a hugely fruitful area of research over the last three decades has been to consider maps of curves to a fixed target projective variety X, usually taken to be nonsingular. Again if we restrict to 'stable maps' where the domain curve has only very mild singularities, then we can hope to construct coarse moduli spaces and define and calculate enumerative invariants of X: the 'Gromov-Witten invariants' which have played and continue to play such important roles in algebraic geometry and symplectic topology (as well as in string theory). When X is a single point we recover the moduli spaces of stable curves. This project aims to extend Joshua Jackson's work from the case when X has dimension 0 to dimensions 1 and 2, constructing and studying moduli spaces of unstable maps to low-dimensional projective varieties X (where X itself may also vary), and to bring in symplectic and differential methods as well as algebraic ones to do this.

This project falls within the EPSRC Mathematical Sciences research area