In this paper, we consider the weight i de Rham-Gauss-Manin bundles on a smooth variety arising from a smooth projective morphism f : X U −→ U for i ≥ 0. We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.

Abstract. In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to an r-simplex whose points parametrize flat connections on a smooth manifold X. These invariants lie in degrees (2p − r − 1)-cohomology with C/Z-coefficients, for p > r ≥ 1. This corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in C/Zcohomology of the underlying smooth manifold X.

Abstract. In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow-Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle explicit Chow-Künneth projectors are obtained which satisfy Murre's conjectures and the motivic Hard Lefschetz theorem is verified.

Abstract. We show that ample line bundles L on a g-dimensional simple abelian variety A, satisfying h 0 (A, L) > 2 g · g!, give projective normal embeddings, for all g ≥ 1.