I am currently working in the field of algebraic geometry. More specifically, I am interested in the geometry of certain types of surfaces and threefolds, known respectively as K3 surfaces and Calabi-Yau threefolds. Surfaces and threefolds of this type have attracted significant interest in recent years, fuelled largely by open questions in several areas of mathematics and physics.
Classification of K3 surfaces
One topic that I am currently working on is broadly concerned with the classification of K3 surfaces. One of the main tools used to perform such classifications in algebraic geometry is the theory of moduli spaces. In a sense, a moduli space can be thought of as a space that parametrises algebro-geometric objects of a certain type. Much of the power of this approach lies in the fact that the moduli spaces themselves are algebro-geometric objects in their own right, so can be studied using the tools of the subject.
The problem that interests me is that of compactifying the moduli space of a certain type of K3 surface; specifically, those K3 surfaces which admit a polarisation of degree two. This problem deals with what happens at the boundary of this moduli space. It is known that as one moves towards this boundary the K3 surfaces parametrised by the moduli space become singular: the goal is to be able to explicitly describe the points on this boundary, along with the singular K3 surfaces that they correspond to.
In some of my previous work, I used a minimal model programme approach to classify the geometric possibilities for the singular K3 surfaces occurring at this boundary. In joint work with V. Alexeev, I am currently working on using these results to understand the boundary of the moduli space of K3 surfaces of degree two. Our aim is to solve the problem of compactifying this moduli space, which has remained open for over 30 years.
Threefolds admitting K3 surface fibrations
Another topic that I am currently working on involves the explicit construction and study of threefolds fibred by K3 surfaces. In some of my previous work, I generalised a lower-dimensional construction by Catanese and Pignatelli to find an explicit construction for threefolds fibred by K3 surfaces of degree two. However, it has so far proven difficult to obtain detailed information about the threefolds produced by this construction, largely due to gaps in our understanding of the moduli space of K3 surfaces of degree two (see above).
To avoid this issue, I have recently been collaborating with C. Doran, A. Harder, and A. Novoseltsev to study threefolds that are fibred by another type of K3 surface; specifically, those that admit polarisations by lattices of high rank. These high rank lattice polarised K3 surfaces have relatively simple moduli spaces, which makes the explicit study of threefolds fibred by them much simpler.
In fact, it turns out that there are close parallels between the theory of such threefolds and the theory of elliptic surfaces, as originally studied by Kodaira and Tate. In particular, much of the geometry is controlled by two invariants: the generalised functional invariant, which controls the geometry of a general fibre, and the generalised homological invariant, which encodes information about the singular fibres. We hope that by understanding these invariants better, we can formulate a complete theory of threefolds fibred by high rank lattice polarised K3 surfaces.
Explicit construction of Calabi-Yau threefolds
As an application of this theory, I have recently been collaborating with A. Clingher, C. Doran, A. Harder, J. Lewis, and A. Novoseltsev to explicitly construct and study new examples of Calabi-Yau threefolds. Calabi-Yau threefolds provide important examples of phenomena arising in many fields, including algebraic and differential geometry, mirror symmetry and string theory, but at present many of their properties are quite poorly understood. Due to this, a substantial amount of recent research has focussed on the construction and study of explicit examples, in an attempt to better understand them.
The geometry of the Calabi-Yau threefolds arising from our K3 fibration construction has so far proved to be remarkably rich. For particular choices of polarising lattice, we have found that such threefolds (including the well-known mirror quintic threefold) are completely determined by their generalised functional invariants, which in this case take the form of a map between two curves. Due to this, many of their fundamental geometric properties depend only upon the ramification behaviour of this map. This fact can be used to explicitly study the moduli spaces of such Calabi-Yau threefolds, by relating them to widely-studied Hurwitz spaces, which characterise ramified covers between curves. We hope that this will lead to answers to several important fundamental questions about the moduli space of Calabi-Yau threefolds.
At its most fundamental level, mirror symmetry is the observation that (almost) every Calabi-Yau threefold has a mirror partner, which is another Calabi-Yau threefold with closely related properties. This correspondence has been the subject of intense study over the last two decades, but there are still a great number of open questions about it.
Our K3 surface fibred Calabi-Yau threefolds provide a fascinating arena in which to study aspects of mirror symmetry. In this setting, mirror symmetry seems to be closely related to the so-called Fano-LG correspondence, which relates objects called Fano varieties to Landau-Ginzburg models. In particular, it seems that our K3 surface fibred Calabi-Yau threefolds may be realised, in a certain sense, as pairs of Landau-Ginzburg models glued together, and their mirror partners may be formed by gluing the corresponding Fano varieties. This provides a new link between mirror symmetry for Calabi-Yau threefolds and the Fano-LG correspondence. A significant portion of my current research is aimed at trying to understand how this picture interacts with the various established theories of mirror symmetry appearing in the literature.
In addition to this, our methods seem to provide a good way to construct new examples of, and occasionally counterexamples to, conjectured phenomena in mirror symmetry. In particular, so far we have explicitly constructed a family of Calabi-Yau threefolds whose existence had previously only been inferred, using techniques from mirror symmetry, by Doran and Morgan, and produced an explicit counterexample to a well-known mirror symmetric conjecture of Morrison.
Broadly speaking, much of this work is tied together by the problem of classifying and explicitly studying general Calabi-Yau threefolds that admit fibrations by polarised K3 surfaces. In order to attack this problem, one needs to understand how to classify polarised K3 surfaces, how to construct threefolds from them, and how to extract explicit geometric information about the constructed threefolds. In the long term, I hope that my current work will provide a starting point for such an attack.
A more detailed account of much of this work may be found in the papers listed in the publications, preprints and other writing section of this website.
This page is maintained by Alan Thompson and was last updated on 04/09/17. Please email comments and corrections to amt69 (at) cam.ac.uk.