Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

Algebraic Geometry Session

Talks will be in MR2 on Thursday and MR15 on Friday.

Thursday

Friday

The resolution property of singular algebraic surfaces - Philipp Gross

Is every coherent sheaf on an algebraic scheme X quotient of a locally free sheaf of finite type? This holds for example if X is projective or regular. The general case is not known, even if X is normal or toric. We give an affirmative answer if X is 2-dimensional and separated without any assumptions on the type of singularities.

3- and 4- dimensional crepant resolutions - Sarah Davis

The work of Craw, Nakamura and Reid shows that the G-Hilbert schemes of abelian subgroups of SL(3,C) are related to 3-dimensional crepant resolutions. In this talk I shall explain these ideas and explore possible extensions to 4-dimensions.

Intersections from various perspectives: A tourist guide (accessible talk) - Oliver Braeunling

The notion of an 'intersection multiplicity' may be approached from various directions. We may start with a real oriented surface (as a manifold) and by drawing a few pictures one is led quite naturally to an intersection theory for oriented compact manifolds. We focus on the de Rham picture (neglecting torsion) and here intersecting two suitable submanifolds comes down to computing the wedge product of forms in de Rham cohomology (this is a graded anti-commutative product). For a regular variety (for example in positive characteristic) many of the manifold techniques do not have an immediate counterpart in "algebraic" algebraic geometry, and we no longer have the flexibility of smooth functions/forms. Nonetheless one may construct an intersection theory, this time living in a commutative ring called CH* (as opposed to the anticommutative wedge product). At first sight this may seem incompatible, but the situation is clarified by looking at complex manifolds, where complex submanifolds live in the even degree parts of de Rham cohomology. Moreover, if they intersect transversally, they do so with multiplicity +1. The talk will not contain anything new. Instead it intends to present an example of approaching the same problem from totally different directions.

[About my PhD project: I am interested in intersection theory, but using yet another approach: higher-dimensional adeles (this is inspired by number-theoretic considerations and although now classified under 'Algebraic Geometry' for Beyond Part III, I am actually rather a number theorist). Adeles are quite interesting, in particular because there is hope that they may lead to arithmetic analogues (i.e. for schemes over Spec Z rather than of finite type over a field).]

Moduli spaces and wall-crossing - Tom Bridgeland (keynote speaker)

Moduli spaces play a central role in modern algebraic geometry. They are varieties whose points parametrize objects of some sort (e.g. modules or curves or vector bundles). Often, to get a nice moduli space, one has to first choose something called a stability condition, and then restrict attention to the corresponding set of stable objects. The resulting moduli space can change as one varies the stability condition, a phenomenon known as wall-crossing. This talk will be about wall-crossing and some tools that have recently been developed to help to understand it.

Local holomorphic Euler characteristic and local instanton invariants - Thomas Köppe

There is a classical correspondence between holomorphic vector bundles on a compact complex surface Z and instantons on Z (considered as a real 4-manifold). Here I present a local version of this picture, where Z is an open complex surface with an embedded negative line, and show how we obtain information about the local contribution to the total instanton charge coming from the formal neighbourhood of the line. In particular, we have an explicit method of computing analytic invariants, from which we can see directly the existence and non-existence of instantons with certain charges.

Derived equivalences and flops - Will Donovan

Two varieties are said to be "derived equivalent" if there is an equivalence between their bounded derived categories of coherent sheaves. Ideas from superstring theory suggest that we should think of this as a sort of quantum equivalence of the varieties. One hope of getting a deeper understanding of this is a conjecture of Bondal, Orlov and Kawamata, under which birational correspondences which are symmetric in a particular way (K-equivalences) are expected to induce derived equivalences. However even in fairly simple examples which I will outline, these derived equivalences can be very hard to find. I will give a progress report on this interesting subject, and some of the different attacks currently underway on the problem.

Formality and Homological Mirror Symmetry - Alexander Shannon

In Kontsevich's vision of homological mirror symmetry, the structures one compares in the worlds of algebraic geometry and of symplectic geometry are A-infinity categories, which are then interpreted as non-commutative spaces. However, Kontsevich's formality theorem tells us that the A-infinity structures deforming a smooth commutative (differential graded) algebra are controlled by classical Poisson structures. In Seidel's recent proof of homological mirror symmetry for the genus two curve, this correspondence is applied in determining the A-infinity structures involved by relating this to the superpotential in the mirror Landau-Ginzburg model. My talk will give an account of the role that the formality theorem plays in this case of homological mirror symmetry.

A model for polarised K3-fibrations - Alan Thompson

In 1988, Nakayama constructed a general model for elliptic fibrations with a section, by embedding a flat family of elliptic curves into a projective bundle over a normal complex variety. In this talk, I aim to outline the analogous construction for a certain family of K3-fibrations over a nonsingular curve.

Universal moduli of parabolic vector bundles on stable curves - Dirk Schlueter

I will outline a construction (using Geometric Invariant Theory) of projective moduli spaces for the following moduli problem: pairs of stable marked curves C and parabolic vector bundles E on C. A parabolic bundle is a vector bundle on C together with a filtration in each of the fibres over the marked points of C (and some extra data determining the stability condition). Parabolic bundles come up as a natural generalisation of vector bundles when the underlying variety has a distinguished divisor (marked points in the curves case). This construction makes it possible to study the geometry and topology of the moduli spaces, and knowledge about these can then be translated into universal facts about all families of pairs (C,E). One particular focus of interest is the cohomology of these moduli spaces - possible methods for studying this come from the interplay of Geometric Invariant Theory and symplectic geometry. A first step towards understanding the topology of these spaces is a description of what the fibres over fixed marked curves look like. Further questions of interest include generalisations of the construction to higher-dimensional base schemes and possible connections with Torelli theorems for parabolic vector bundles on marked curves.

Geometric Categories - Andreas Holmstrom

What is geometry? When is a category a geometric category? This question can be approached in many different ways, and we will look at one or two of these approaches, together with some exciting examples of recent and yet-to-be discovered geometric categories.