GALOIS ACTIONS ON HOMOTOPY GROUPS OF ALGEBRAIC VARIETIES

We study the Galois actions on the `-adic schematic and Artin-Mazur homotopy groups of algebraic varieties.For proper varieties of good reduction over a local field K , we show that the `-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever `is not equal to the residue characteristic p of K .For quasiprojective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence.When `D p , a slightly weaker result is proved by comparing the crystalline and p -adic schematic homotopy types.Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups ét n .X x K / ˝y Z Q `.