On Finite-dimensional Motives and Murre's Conjecture

Parshin’s conjecture revisited

“…Granted the Theorem, the standard argument gives the following Corollary, see also Jannsen [10,Thm. 12.5.7].…”

Algebraic Cycles on Fano Varieties of Some Cubics

“…We refer to [21], [2], [30], [14], [17] for basics on the notion of finite-dimensional motive. An essential property of varieties with finite-dimensional motive is embodied by the nilpotence theorem:…”

“…Actually, the nilpotence property (for all powers of X) could serve as an alternative definition of finite-dimensional motive, as shown by a result of Jannsen [17,Corollary 3.9]. Conjecturally, all smooth projective varieties have finite-dimensional motive [21].…”

“…The notion of finite-dimensional motive, developed independently by Kimura and O'Sullivan [21], [2], [30], [17], [14] has given major new impetus to the field of algebraic cycles. To give but one example: thanks to this notion, we now know the Bloch conjecture is true for surfaces of geometric genus zero that are rationally dominated by a product of curves [21].…”

Numerical functors on Voevodsky’s motives

“…Some examples where it has been successfully exploited are the proof of Bloch's conjecture for surfaces dominated by a product of curves [11], the proof that rational and numerical equivalence coincide for products of elliptic curves over finite fields [9] and the proof of the Bloch-Beilinson conjecture for products of elliptic curves (over some special fields) [8].…”