Robert Laterveer

A remark on the motive of the Fano variety of lines of a cubic

Let X be a smooth cubic hypersurface, and let F be the Fano variety of lines on X. We establish a relation between the Chow motives of X and F. This relation implies in particular that if X has finite-dimensional motive (in the sense of Kimura), then F also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension 3 and 5, and of certain cubics in other dimensions.RésuméSoit X une hypersurface cubique lisse, et soit F la variété de Fano paramétrant les droites contenues dans X. On établit une relation entre les motifs de Chow de X et de F. Cette relation implique le fait que F a motif de dimension finie (au sens de Kimura) à condition que X a motif de dimension finie. En particulier, si X est une cubique lisse de dimension 3 ou 5, alors F a motif de dimension finie.

SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES

Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn-Sebastian, we give some new examples of varieties where Voevodskys conjecture is verified.

Some desultory remarks concerning algebraic cycles and Calabi–Yau threefolds

We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi–Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.

Hard Lefschetz for Chow groups of generalized Kummer varieties

The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen–Vial and Fu–Tian–Vial on multiplicative Chow–Künneth decompositions.

On a multiplicative version of Bloch’s conjecture

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial’s construction of a refined Chow–Künneth decomposition for these varieties.

Yet another version of Mumford’s theorem

The aim of this note is to provide a variant statement of Mumford’s theorem. This variant states that for a general variety, all Chow groups are “as large as possible”, in the sense that they cannot be supported on a divisor.

Algebraic cycles and Todorov surfaces

Motivated by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture about 0-cycles on self-products of surfaces of geometric genus one. We verify Voisins conjecture for the family of Todorov surfaces with

Some elementary examples of quartics with finite-dimensional motive

K^2=2

Correspondences and singular varieties

and fundamental group

A family of K3 surfaces having finite-dimensional motive

\mathbb{Z}/2\mathbb{Z}