Pure motives, mixed motives and extensions of motives associated to singular surfaces

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky’s category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky’s formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert–Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar.