Robert Laterveer
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View article: Questions on the Chow ring of complete intersections
Questions on the Chow ring of complete intersections Open
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 $…
View article: The Beauville–Voisin conjecture for double EPW sextics
The Beauville–Voisin conjecture for double EPW sextics Open
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View article: Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations
Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations Open
We show that the hyper‐Kähler varieties of OG10‐type constructed by Laza–Saccà–Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this c…
View article: Some More Fano Threefolds with a Multiplicative Chow–Künneth Decomposition
Some More Fano Threefolds with a Multiplicative Chow–Künneth Decomposition Open
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 cohomo…
View article: A 9-dimensional family of K3 surfaces with finite dimensional motive
A 9-dimensional family of K3 surfaces with finite dimensional motive Open
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 Ho…
View article: Some motivic properties of Gushel–Mukai sixfolds
Some motivic properties of Gushel–Mukai sixfolds Open
Gushel–Mukai (GM) 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. A…
View article: The Beauville-Voisin conjecture for double EPW sextics
The Beauville-Voisin conjecture for double EPW sextics Open
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.
View article: Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations
Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations Open
We show that the hyper-Kähler varieties of OG10-type constructed by Laza-Saccà-Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this c…
View article: Some more Fano threefolds with a multiplicative Chow-Künneth decomposition
Some more Fano threefolds with a multiplicative Chow-Künneth decomposition Open
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 cohomo…
View article: On the Chow Ring of Fano Fourfolds of K3 type
On the Chow Ring of Fano Fourfolds of K3 type Open
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 the…
View article: Special Cubic Four-Folds, K3 Surfaces, and the Franchetta Property
Special Cubic Four-Folds, K3 Surfaces, and the Franchetta Property Open
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 f…
View article: On the Chow ring of some Lagrangian fibrations
On the Chow ring of some Lagrangian fibrations Open
Let $X$ be a hyperk\\"ahler variety admitting a Lagrangian fibration.\nBeauville's "splitting property" conjecture predicts that fibres of the\nLagrangian fibration should have a particular behaviour in the Chow ring of\n$X$. We study this…
View article: Some Motivic Properties of Gushel-Mukai Sixfolds
Some Motivic Properties of Gushel-Mukai Sixfolds Open
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…
View article: Algebraic cycles and Fano threefolds of genus 8
Algebraic cycles and Fano threefolds of genus 8 Open
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.
View article: Some new Fano varieties with a multiplicative Chow-Künneth decomposition
Some new Fano varieties with a multiplicative Chow-Künneth decomposition Open
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 …
View article: On the Motive of Codimension 2 Linear Sections of Gr(3, 6)
On the Motive of Codimension 2 Linear Sections of Gr(3, 6) Open
11 pages, to appear in Tokyo J. Math., comments welcome. arXiv admin note: text overlap with arXiv:2009.11061, arXiv:2105.03171
View article: Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds
Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds Open
This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$ . When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ …
View article: Algebraic cycles and Lehn-Lehn-Sorger-van Straten eightfolds
Algebraic cycles and Lehn-Lehn-Sorger-van Straten eightfolds Open
This article is about Lehn-Lehn-Sorger-van Straten eightfolds $Z$, and their anti-symplectic involution $ι$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $ι$ on the C…
View article: Algebraic cycles and intersections of three quadrics
Algebraic cycles and intersections of three quadrics Open
Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y d…
View article: Algebraic cycles and intersections of three quadrics
Algebraic cycles and intersections of three quadrics Open
Let $Y$ be a smooth complete intersection of three quadrics, and assume the dimension of $Y$ is even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers o…
View article: On the Chow ring of Fano varieties on the Fatighenti–Mongardi list
On the Chow ring of Fano varieties on the Fatighenti–Mongardi list Open
Conjecturally, Fano varieties of K3 type admit a multiplicative\nChow-K\\"unneth decomposition, in the sense of Shen-Vial. We prove this for many\nof the families of Fano varieties of K3 type constructed by\nFatighenti-Mongardi. This has i…
View article: Zero-cycles on Garbagnati surfaces
Zero-cycles on Garbagnati surfaces Open
Garbagnati has constructed certain surfaces of general type that are bidouble\nplanes as well as double covers of K3 surfaces. In this note, we study the Chow\ngroups (and Chow motive) of these surfaces.\n
View article: The generalized Franchetta conjecture for some hyper-Kähler varieties, II
The generalized Franchetta conjecture for some hyper-Kähler varieties, II Open
We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn–Lehn–Sorger–van Straten (LLSS). As a corollary, we establish the Beauville–Voisin conjecture for very general LLS…
View article: Motives and the Pfaffian–Grassmannian equivalence
Motives and the Pfaffian–Grassmannian equivalence Open
We consider the Pfaffian-Grassmannian equivalence from the motivic point of\nview. The main result is that under certain numerical conditions, both sides of\nthe equivalence are related on the level of Chow motives. The consequences\ninclu…
View article: On the Chow ring of some Lagrangian fibrations
On the Chow ring of some Lagrangian fibrations Open
Let $X$ be a hyperkähler variety admitting a Lagrangian fibration. Beauville's "splitting property" conjecture predicts that fibres of the Lagrangian fibration should have a particular behaviour in the Chow ring of $X$. We study this conje…
View article: On the motive of codimension 2 linear sections of $\hbox{Gr}(3,6)$
On the motive of codimension 2 linear sections of $\hbox{Gr}(3,6)$ Open
We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\hbox{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also …
View article: On the Chow ring of some special Calabi–Yau varieties
On the Chow ring of some special Calabi–Yau varieties Open
We consider Calabi–Yau n -folds X arising from certain hyperplane arrangements. Using Fu–Vial’s theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of X generated by divisors,…