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Results of Beauville imply that Zα is finite dimensional (cf. 2.5 below). In case A = J(C), the jacobian of a curve C, Ceresa has shown that the cycle C−C := C−(−1)∗C ∈ ZC is not algebraically equivalent to zero for generic C of genus g ≥ 3, which implies that for such a curve dimQ ZC ≥ 2. In this note we investigate the subspace ZWm of CHm(J(C))Q, with Wm… (More)

Abstract. 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 determined by 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… (More)

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… (More)

- K. Hulek, Jeroen Spandaw, Bert van Geemen, Duco van Straten
- 2000

The moduli space of
1; 3-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth±Nieto quintic. Barth and Nieto have shown that these varieties have Calabi±Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common… (More)

The aim of this paper is to give an explicit expression for Hitchin’s connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We sketch the general situation. Let π : C → S be a family of projective smooth curves. Let p : M → S be the family of moduli spaces of (S-equivalence classes of) rank r bundles with trivial… (More)

- Bert van Geemen
- 2000

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 Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge… (More)

- Bert van Geemen
- 2007

The Langlands philosophy contemplates the relation between auto-morphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3. Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations. Here we announce that we found an… (More)

- Bert van Geemen
- 2007

To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V −n/2 of weight k + n for n positive and, under certain circumstances, also for n negative. We show that these ‘half twists’ come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field.

We study a proposal of D’Hoker and Phong for the chiral superstring measure for genus three. A minor modification of the constraints they impose on certain Siegel modular forms leads to a unique solution. We reduce the problem of finding these modular forms, which depend on an even spin structure, to finding a modular form of weight 8 on a certain subgroup… (More)

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on MorrisonNikulin… (More)