Bert van Geemen

Kuga-Satake Varieties and the Hodge Conjecture

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of KugaSatake varieties. The Hodge conjecture is discussed in section2. An excellent survey of the Hodge conjecture for abelian varieties is [G].

On Hitchin’s connection

The aim of this paper is to give an explicit expression for Hitchins connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We recall the definition of this connection (which arose in Quantum Field Theory) and characterize it for families of genus two curves. We then consider a particular family (over a configuration space), the corresponding vector bundles are (almost) trivial. The Heat equations which give the connection these bundles are related to the Lie algebra so(6). We compute some local monodromy representations. As a byproduct we obtain, for any representation space V of so(2g+2), a flat connection on the trivial bundle with fiber V over a configuration space and thus a monodromy representation of a pure braid group.

Genus Four Superstring Measures

A main issue in superstring theory are the superstring measures. D’Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are the product of modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently, S. Grushevsky published this result as part of a more general approach.

On the Hitchin system

1.1 What is known as the Hitchin system is a completely integrable hamiltonian system (CIHS) involving vector bundles over algebraic curves, identified by Hitchin in ([H1], [H2]). It was recently generalized by Faltings [F]. In this paper we only consider the case of ranktwo vector bundles with trivial determinant. In that case the Hitchin system corresponding to a curve C of genus g is obtained as follows. Let

From qubits to E7

There is an intriguing relation between quantum information theory and super gravity, discovered by M. J. Duff and S. Ferrara. It relates entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant on the 56-dimensional representation of the Lie group E7. In this paper we recall the relatively straightforward manner in which three-qubits lead to E7, or at least to the Weyl group of E7. We also show how the Fano plane emerges in this context.

An Introduction to the Hodge Conjecture for Abelian Varieties

In this lecture we give a brief introduction to the Hodge conjecture for abelian varieties. We describe in some detail the abelian varieties of Weil-type. These are examples due to A. Weil of abelian varieties for which the Hodge conjecture is still open in general.

Heeke Eigenforms in the Cohomology of Congruence Subgroups of SL (3, Z)

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of Sl(3; Z). To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra–Lenstra–Lovasz algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Green and by Top and van Geemen in previous work. Comparing our tables with results from computations of Galois representations, we find some new numerical evidence for the conjectured relation between modular forms and Galois representations.

A LINEAR SYSTEM ON NARUKI'S MODULI SPACE OF MARKED CUBIC SURFACES

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Narukis toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E6.

Mirror quintics, discrete symmetries and Shioda maps

In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard Fuchs equation associated to the holomorphic 3-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one parameter families to the family of Mirror Quintics. Our constructions generalize to degree n Calabi Yau varieties in (n-1)-dimensional projective space.

An isogeny of K3 surfaces

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.