Goals of Research Area D
Jointly with Research Area E we face the following subject matters:
Motivic cohomology and diophantine equations.
Kim has introduced a new method into Diophantine approximation and Faltings’ student Hadian-Jazi has shown in his PhD thesis how to use motivic cohomology in this context. Faltings’ paper [D:Fal07] explains the basic facts and constructs a motivic logarithm for arbitrary curves. An open question is the construction of such an object in algebraic K-theory, as well as a good definition of torsors over it.
Although it is known by work of Wiles, Taylor, et al that any elliptic curve over the rational numbers can be uniformised by modular curves, we still have no good bound for the degree of such a uniformisation. But such a bound is essential for all diophantine applications. Recently Faltings has developed some new ideas using Arakelov theory which shall be explored further.
Higgs-bundles and fundamental groups.
The Hodge-Tate theory has been worked out a long time ago. Recently Faltings found a crystalline theory for ‘small’ representations. The general case leads to descent problems on which he plans to work. What is needed is an exponential function for non-abelian cohomology which lifts the exponential for Higgs bundles.
We will analyze and explicate Arthur’s trace formula and apply it to the study of the distribution and structure of the automorphic spectrum of a reductive group. The first goal is to extend Arthur’s trace formula to a larger class of test function rewriting and explicating the geometric side. This is continuation of joint work of W. Müller with Finis and Lapid on the spectral side. It is also the starting point of a larger program. One of the further goals is to extend the Weyl law for the cuspidal spectrum to give the leading term of the traces of Hecke operators on families of automorphic forms for GL(n). This will have applications to the study of the distribution of low lying zeros of automorphic L-functions in families as predicted by Katz and Sarnak.
Shimura varieties. The reduction of Shimura varieties with their natural stratifications, especially in the presence of singularities, will be investigated further. In the parahoric case the analysis of singularities reduces to a study of local models; their special fibers can be interpreted for classical groups as subvarieties of (Grassmannian) quiver varieties defined by linear algebra conditions, but also as closed subvarieties of affine flag varieties, and are also related to subvarieties of group compactifications. It is intended to build a bridge to these other theories to use them to prove a series of open conjectures on local models. Related is the study of degenerate intersections of arithmetic divisors and similar algebraic cycles on Shimura varieties. We aim at proving the Haines-Kottwitz conjecture in cases which are substantially different from the Drinfeld case. We want to study the structure of coefficient spaces and their applications to the theory of Galois representations of p-adic local fields.
The computation and understanding of Euler characteristics of quiver Grassmannians are a key feature in all attempts to categorify cluster algebras. In joint work with Geiß and Leclerc, Schröer developed a conjecture which says that certain generating functions of Euler characteristics of quiver Grassmannians form a basis of any skew-symmetric cluster algebra. The conjecture has been proved for some special cases, but in general it is wide open. The next step in this program is to extend the pool of examples and to develop further inductive methods for the computation of Euler characteristics. In this context, Calabi-Yau categories of dimension two and three arise in a natural way. We would like to construct all stable objects with respect to all discrete stability conditions on the Calabi-Yau categories associated to elements of the Weyl group of a Kac-Moody Lie algebra. By results of Reineke and Keller, this should lead to a better understanding of various quantum dilogarithm identities. We also plan to extend our results on semicanonical bases for commutative cluster algebras to quantum cluster algebras.
Moduli spaces of sheaves.
In a joint project with Göttsche, we will investigate the K-theoretic Nekrasov partition function, which has been used by Göttsche, Nakajima and Yoshioka to prove a wall crossing formula for K-theoretic Donaldson invariants (i.e. holomorphic Euler characteristics of determinant bundles on moduli spaces of sheaves) on rational algebraic surfaces. Göttsche, visiting Bonn as the Hirzebruch professor in 2011/12, and Zagier will use indefinite theta series and mock modular forms to compute the generating functions of K-theoretic Donaldson invariants of rational surfaces as explicit rational functions, and to prove important cases of Le Potier’s strange duality conjecture, which relates holomorphic Euler characteristics of determinant bundles on different moduli spaces on the same surface.
Le Potier’s strange duality conjecture has recently been dealt with successfully by Alina Marian and Oprea in the case of K3 surfaces. For primitive Mukai vectors, moduli spaces of stable sheaves (or complexes of such) on K3 surfaces are compact hyperkähler manifolds whose deformation type has been determined by Huybrechts and Yoshioka. However, their motive still eludes us and for non-primitive Mukai vectors the moduli stacks have hardly been studied at all until recently. In the context of Donaldson-Thomas invariant of open Calabi-Yau manifolds, their Euler numbers have conjecturally been expressed by a multicover formula. We aim at a deeper understanding of motivic and numerical invariants of the various moduli spaces using wall crossing phenomena in the space of Bridgeland stability conditions on the Calabi-Yau category provided by the derived category of coherent sheaves on a K3 surface. The complete description of the space of stability conditions in these cases is a challenging open problem.
Although the main focus of the HIM trimester program on ‘Arithmetic and Geometry’ in 2013 will be different from the one of the Research Area D, we expect a stimulating interaction in particular on topics related to representations of the absolute Galois group (Serre’s conjecture).