An algebraic $SL_2$-vector bundle over $R_2$ as a variety

Murre’s Conjecture for a Rational Homogeneous Bundle over a Variety

In this paper, we investigate Murre’s conjectures on the structure of rational Chow groups for a rational homogeneous bundle Z → S over a smooth variety. Absolute Chow-Künneth projectors are exhibited for Z whenever S has a Chow–Künneth decomposition.

Vector bundles over algebraic curves

Universal Vector Bundle over the Reals

Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR × Pic d(XR) if and only if χ(L) is odd for L ∈ Picd(XR). There is a universal quaternionic algebraic line bundle over X × Pic(X) if and only if the degree d is odd. Tak...

The Index of an Algebraic Variety

Let K be the field of fractions of a Henselian discrete valuation ring OK . Let XK/K be a smooth proper geometrically connected scheme admitting a regular model X/OK . We show that the index δ(XK/K) of XK/K can be explicitly computed using data pertaining only to the special fiber Xk/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to hori...

Algebraic Cycles on an Abelian Variety

It is shown that to every Q-linear cycle α modulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle α modulo rational equivalence on A lying above α, characterised by a condition on the spaces of cycles generated by α on products of A with itself. The assignment α 7→ α respects the algebraic operations and pullback and push forward along homomorphism...