Zero-cycles and the Cayley–Oguiso automorphism Bini, Gilberto; Laterveer, Robert
Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche,
06/2024, Letnik:
70, Številka:
2
Journal Article
Recenzirano
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
.
This paper presents two approaches to reducing problems on 2-cycles on a smooth cubic hypersurface X over an algebraically closed field of characteristic ≠2, to problems on 1-cycles on its variety of ...lines F(X). The first one relies on osculating lines of X and Tsen-Lang theorem. It allows to prove that CH2(X) is generated, via the action of the universal P1-bundle over F(X), by CH1(F(X)). When the characteristic of the base field is 0, we use that result to prove that if dim(X)≥7, then CH2(X) is generated by classes of planes contained in X and if dim(X)≥9, then CH2(X)≃Z. Similar results, with slightly weaker bounds, had already been obtained by Pan (Math Ann 1–28, 2016). The second approach consists of an extension to subvarieties of X of higher dimension of an inversion formula developped by Shen (J Algebraic Geom 23:539–569, 2014, Rationality, universal generation and the integral Hodge conjecture, arXiv:1602.07331) in the case of 1-cycles of X. This inversion formula allows to lift torsion cycles in CH2(X) to torsion cycles in CH1(F(X)). For complex cubic 5-folds, it allows to prove that the birational invariant provided by the group CH3(X)tors,AJ of homologically trivial, torsion codimension 3 cycles annihilated by the Abel–Jacobi morphism is controlled by the group CH1(F(X))tors,AJ which is a birational invariant of F(X), possibly always trivial for Fano varieties.
On the Chow ring of certain Fano fourfolds Laterveer, Robert
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica,
12/2020, Letnik:
19, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Abstract We prove that certain Fano fourfolds of K3 type constructed by Fatighenti–Mongardi have a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of these ...fourfolds.
The notion of modulus is a striking feature of Rosenlicht–Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch–Esnault, ...Park, Rülling, Krishna–Levine. Recently, Kerz–Saito introduced a notion of Chow group of 0-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of 0-cycles with modulus and show their torsion and divisibility properties.
Modulus is being brought to sheaf theory by Kahn–Saito–Yamazaki in their attempt to construct a generalization of Voevodsky–Suslin–Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.
Let Y be a smooth projective variety over C, and X be a smooth hypersurface in Y. We prove that the natural restriction map on Chow groups of codimension two cycles is an isomorphism when restricted ...to the torsion subgroups provided dimY≥5. We prove an analogous statement for a very general hypersurface X⊂P4 of degree ≥5. In the more general setting of a very general hypersurface X of sufficiently high degree in a fixed smooth projective four-fold Y, under some additional hypothesis, we prove that the restriction map is an isomorphism on ℓ-primary torsion for almost all primes ℓ. As a consequence, we obtain a weak Lefschetz theorem for torsion in the Griffiths groups of codimension 2 cycles, and prove the injectivity of the Abel–Jacobi map when restricted to torsion in this Griffiths group, thereby providing a partial answer to a question of Nori.
Let $X$ be a hyperk\"ahler 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 conjectural behaviour for two very classical examples of Lagrangian fibrations.
An old conjecture of Voisin describes how zero-cycles on a variety
X
should behave when pulled-back to the self-product
X
m
for
m
larger than the geometric genus of
X
. Using complete intersections ...of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin’s conjecture.
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