Moduli spaces of multi-polarised projective varieties

Moduli spaces arise naturally in classification problems in algebraic and differential geometry, and play important roles in many different areas. 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 discrete invariants and try to construct a moduli space; that is, an algebraic variety whose points correspond in a natural way to the equivalence classes of the objects to be classified. This works nicely for nonsingular curves, though to include singular curves 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.

The classification of complex projective curves is part of one of the most fundamental classification problems in algebraic geometry: that of classifying complex projective varieties (of fixed dimension). It is usual to work with polarised projective varieties (X,L) where L is an ample line bundle over the projective variety X, and to try to impose suitable stability conditions so that moduli spaces of (semi)stable polarised complex projective varieties can be constructed. (In the case when X is a nonsingular complex projective curve of genus at least two then we can choose a suitable power of the canonical line bundle as the polarisation). Very significant advances in this direction have been made in recent years, by combining methods from algebraic, differential and symplectic geometry, relating the so-called K-stability of (X,L) to the existence of special Kahler metrics on X.

The aim of this research project is to study moduli spaces of complex projective varieties X equipped not just with one ample line bundle L, but instead with finitely many ample line bundles representing a (subset of a) basis of the Neron-Severi group of X. Given one ample line bundle L on X, we can use the sections of a sufficiently large power of L to embed X in a projective space. Then one can hope to apply ideas coming from Mumford's geometric invariant theory (GIT), developed in the 1960s to construct and study quotients of algebraic varieties by reductive group actions, to define notions of (semi)stability for the action of the associated special linear group on the Hilbert scheme representing projective subschemes of this projective space with the same Hilbert polynomial as X. However these depend on the power of L chosen, and do not have obvious geometric interpretation; the motivation behind the definition of K-(semi)stability is to provide some sort of asymptotic version of this GIT (semi)stability as the power of the line bundle tends to infinity.

Given several different ample line bundles on X we can take sections of tensor products of powers of these line bundles to embed X in projective toric varieties. We can then study the corresponding group actions on the corresponding toric varieties, and analogues of K-stability in these situations. The project aims to investigate this in the case when dimX=2, which is the lowest dimension which is not already covered by the traditional situation with just one ample line bundle.


This project falls within the EPSRC Geometry and Topology research area. No companies or collaborators are involved.