arXiv subjects
Publications and source records attributed to Robert Laterveer.
Let $Y$ be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on $Y\times Y$ that is Abel-Jacobi trivial but non-torsion in the Chow group $A^4(Y\times Y)$. As a consequence, $Y$ does not admit a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-K\"unneth decomposition modulo algebraic equivalence.
The Beauville-Voisin conjecture is about the Chow ring of hyper-K\"ahler varieties. We consider a variant version taking place in the ring of algebraic cycles modulo algebraic equivalence. We prove this variant version is true in codimension at least 3 for Fano varieties of lines on cubic fourfolds, and for double EPW sextics.
A strong version of the Beauville-Voisin conjecture asserts that for hyper-K\"ahler varieties, the subring of the Chow ring generated by divisors, Chern classes and Lagrangian constant cycle subvarieties should inject into cohomology. We verify this in codimension larger than two for Hilbert squares of K3 surfaces, for Fano varieties of lines in cubic fourfolds, and for double EPW sextics.
We state several questions, and prove some partial results, about the Chow ring $A^\ast(X)$ of complete intersections in projective space. For one thing, we prove that if $X$ is a general Calabi-Yau hypersurface, the intersection product $A^2(X)\cdot A^i(X)$ is one-dimensional, for any $i>0$. We also show that quintic threefolds have a multiplicative Chow-K\"unneth (MCK) decomposition. We wonder whether all Calabi-Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.
Let $X$ be a double EPW sextic, and $\iota$ its anti-symplectic involution. We relate the $\iota$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we also relate the anti-invariant part of the Chow motive of $X$ with the motive of a Gushel-Mukai fourfold. As an application, we obtain a similar result for certain Fano varieties of lines in cubics with infinite-order birational automorphisms.
Cayley and Oguiso have constructed certain quartic K3 surfaces $S$, with automorphisms $g$ of infinite order. We show that when $g$ is symplectic (resp. anti-symplectic), it acts as the identity (resp. minus the identity) on the degree zero part of the Chow group of zero-cycles of $S$.
Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite dimensional motive, in the sense of Kimura. We also establish the Kuga-Satake Hodge conjecture for S, as well as Voisin'conjecture concerning zero-cycles. As a consequence, we obtain Kimura finite dimensionality, the Kuga-Sataka Hodge conjecture, and Voisin's conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism.
The Chow rings of hyper-Kähler varieties are conjectured to have a particularly rich structure. In this paper, we formulate a conjecture that combines the Beauville-Voisin conjecture regarding the subring generated by divisors and the Franchetta conjecture regarding generically defined cycles. As motivation, we show that this Beauville-Voisin-Franchetta conjecture for a hyper-Kähler variety $X$ follows from a combination of Grothendieck's standard conjectures for a very general deformation of $X$, Murre's conjecture (D) for $X$ and the Franchetta conjecture for $X^3$. As evidence, beyond the case of Fano varieties of lines on smooth cubic fourfolds, we show that this conjecture holds for codimension-2 and codimension-8 cycles on Lehn-Lehn-Sorger-van Straten eightfolds. Moreover, we establish that the subring of the Chow ring generated by primitive divisors injects into cohomology.
We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology.
Gushel-Mukai sixfolds are an important class of so-called Fano-K3 varieties. In this paper we show that they admit a multiplicative Chow-Künneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double EPW sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of Gushel-Mukai sixfolds.
We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatighenti, Manivel and Tanturri, have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. It follows that the Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side result, we obtain some criteria for the Franchetta property of blown-up projective varieties.
We show that prime Fano threefolds $Y$ of genus 10 have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
We show that the hyper-K\"ahler varieties of OG10-type constructed by Laza-Sacc\`a-Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this conjecture. The main technical assumption of this general result is that the Lagrangian fibration satisfies the hypotheses of Ng\^o's support theorem. Verifying that the LSV tenfolds do satisfy those hypotheses is of independent interest. Another point of independent interest of the paper is the definition and the study of the Lefschetz standard conjecture in the relative setting, and its relation to the classical absolute case.
We exhibit several families of Fano threefolds with a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argument, we observe that double covers of projective spaces admit a multiplicative Chow-Künneth decomposition.
We exhibit a 2-dimensional family of non-hyperelliptic curves of genus 5, called Humbert curves, for which the tautological ring injects into cohomology. In particular, Humbert curves have a multiplicative Chow-Künneth decomposition (in the sense of Shen-Vial), and their Ceresa cycle is torsion.
O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fourfolds of discriminant 26, we prove O'Grady's generalized Franchetta conjecture for g=14, providing the first evidence beyond Mukai models.
Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
We show that prime Fano threefolds $Y$ of genus 8 have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.