Bert van Geemen

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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)
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)
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)
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)
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)