The n-motivic t-structures for n=0,1 and 2

For a field k and an integer n∈{0,1,2}, we construct a t-structure (T⩾0Mn(k),T⩽0Mn(k)) on Voevodsky's triangulated category of motives DMeff(k), which we call the n-motivic t-structure. When n=0, this is simply the usual homotopy t-structure, but for n∈{1,2}, these are new t-structures. We will show that the category of Deligne's 1-motives can be embedded as a full subcategory in the heart of the 1-motivic t-structure. By a rather straightforward analogy, we are led to specify a class of objects in the heart of the 2-motivic t-structure which we call mixed 2-motives. We will also check that these objects form an Abelian category.