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Claudio Pedrini

Publications and source records attributed to Claudio Pedrini.

14 recordsLinked to original sources
arXiv2026

Kuznetsov components and transcendental motives of cubic fourfolds

Let $X \subset ¶^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$ and $Y$ are {\it Fourier -Mukai partners} and hence the categories $\sA_X$ and $\sA_Y$ are equivalent, then their transcendental motives $t(X)$ and $t(Y)$ are isomorphic. The aim of this note is to consider families of special cubic fourfolds $X$ with their FM-partners $Y$ and to give an explicit description of the isomorphism between the transcendental motives, in the case $X$ and $Y$ are rational and when they are conjecturally irrational. We also prove that ,for special cubic fourfolds $X $ in countably many Hassett divisors, with a symplectic automorphism of order 3, there exists another special cubic fourfold $Y$, an equivalence of categories $\sA^G_X \simeq \sA_{Y}$, where $\sA^G_X$ is the equivariant Kuznetsov component, and an isomorphism $t(X) \simeq t(Y)$.

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arXiv2026

Kuznetsov components ans transcendental motives of cubic fourfolds

Let $X \subset ¶^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$ and $Y$ are {\it Fourier -Mukai partners} and hence the categories $\sA_X$ and $\sA_Y$ are equivalent, then their transcendental motives $t(X)$ and $t(Y)$ are isomorphic. The aim of this note is to consider families of special cubic fourfolds $X$ with their FM-partners $Y$ and to give an explicit description of the isomorphism between the transcendental motives, in the case $X$ and $Y$ are rational and when they are conjecturally irrational. We also prove that ,for special cubic fourfolds $X $ in countably many Hassett divisors, with a symplectic automorphism of order 3, there exists another special cubic fourfold $Y$, an equivalence of categories $\sA^G_X \simeq \sA_{Y}$, where $\sA^G_X$ is the equivariant Kuznetsov component, and an isomorphism $t(X) \simeq t(Y)$.

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arXiv2026

The Chow motive of LSV hyper-Kälher manifolds

Let $X$ be a smooth cubic fourfold over $\C$ and let $π: \sJ_U \to U$, with $ U \subset (¶^5)^*$, be the Lagrangian fibration whose fibres are the smooth hyperplane sections $Y_ H = X \cap H$, with $H \in U$. There always exists a (not unique) smooth compactification $\bar \sJ \to (¶^5)^*$ which is a hyper-Kälher manifold of OG10 type. Since two different compactifications are birationally equivalent their Chow motives are isomorphic. For a general $X$ a geometrical construction of a smooth compactification $\sJ(X)$ with irreducible fibres has been described in [LSV]. In this note we prove that the Chow motive $h(\sJ(X)) $ is a direct summand of the (twisted) motive of $X^5$ and therefore is is of abelian type if $h(X)$ is of abelian type.We describe a 10 -dimensional family $\sF$ of cubics $X$ such that the compactification $\sJ(X)$ is unique, smooth, with irreducible fibres, and the Chow motive $h( \sJ(X) )$ is of abelian type.

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arXiv2025

Rational cubic fourfolds with a symplectic group of automorphisms

A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has an associated K3 surface S in the derived sense.This is equivalent to have a cohomologically associated K3 surface and therefore X is conjecturally rational. In this note we prove that cubic fourfolds with a cyclic group of symplectic automorphisms whose order is not a power of 2, are rational and belong to the Hassett divisor C_d, with d = 14, 42. We also describe rational cubic fourfolds X with a symplectic group of automorphisms G, such that (G, S_G(X), is a Lech pair where th rank of S_G equals 19 or 20.

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arXiv2025

K3 surfaces and cubic fourfolds with Abelian motive

We show that cubic fourfolds with lattice of algebraic 2-cycles of rank greater than 19 have abelian and finite dimensional (in the sense of Kimura) Chow motive. This also implies Abelianity and finite dimensionality of the motive of related hyperKahler varieties, such as the Fano variety of lines and the LLSvS 8fold. A similar remark allows us to show the Abelianity of the motive of an infinity of LSV 10folds, and of other hyperKahler 10folds associated to the twisted intermediate Jacobian fibration of cubic fourfolds with an associated K3 surface. After that, starting from certain 4-dimensional families of K3 surfaces, we construct two families of Fano varieties whose Chow motive is finite dimensional. Varieties from the first family are some quadric surface fibrations, and contain the finite dimensional transcendental motive of a K3 surface. Varieties from the second family are singular cubic fourfolds, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives.

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arXiv2024

K3 surfaces associated to a cubic fourfold

Let $X\subset ¶^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different ways. If there is an equivalence of categories $\sA_X \simeq D^b(S,α)$ where $\sA_X$ is the Kuznetsov component of $D^b(X)$ and $α$ is a Brauer class, or if there is an isomorphism between the transcendental motive $t(X)$ and the (twisted ) transcendental motive of a K3 surface$S$. In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.

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arXiv2020

A family of special cubic fourfolds with motive of abelian type

In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperkähler varieties, such as the Fano variety of lines and the LLSvS 8fold.

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arXiv2019

The transcendental motive of a cubic fourfold

In this note we introduce the transcendental part $t(X)$ of the motive of a cubic fourfold $X$ and prove that it is isomorphic to the (twisted) transcendental part $h_2^{tr}(F(X))$ in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines $F(X)$. Then we prove that $t(X)$ is isomorphic to the Prym motive associated to the surface $S_l \subset F(X)$ of lines meeting a general line $l$. If $X$ is a special cubic fourfold in the sense of Hodge theory, and $F(X)\cong S^{[2]}$, with $S$ a $K3$, then we show that $t(X) \cong t_2(S)(1)$, where $t_2(S)$ is the transcendental motive. Therefore the motive $h(X)$ is finite dimensional if and only if $S$ has a finite dimensional motive. If $X$ is very general then $t(X)$ cannot be isomorphic to the (twisted) transcendental motive of a surface. We relate the existence of an isomorphism $t(X) \cong t_2(S)(1)$ to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a fibration over $\mathbf{P}^2$, whose fibers are either quadrics or del Pezzo surfaces of degree 6, and prove the isomorphism $t_2(S)(1) \cong t(X)$, with $S$ a K3 surface.

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arXiv2017

On the rationality and the finite dimensionality of a cubic fourfold

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. If $X$ is special, in the sense of B.Hasset, and $F(X) \simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we show that the Chow motive $h(X)$ contains as a direct summand a "transcendental motive" $t(X)$ such that $t(X)\simeq t_2(S)(1)$. The motive of $X$ is finite dimensional if and only if $S$ has a finite dimensional motive, in which case $t(X)$ is indecomposable. Similarly, if $X$ is very general and the motive $h(X)$ is finite dimensional, then $t(X)$ is indecomposable

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arXiv2015

Bloch's conjecture and valences of correspondences for K3 surfaces

Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $Γ$ acts as 0 on the Albanese kernel $T(X_Ω)$, where $Ω$ is a universal domain containing $k$. Here we prove that, for a complex K3 surface $X$, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphisms of finite order and to the existence of constant cycle curves on $X$. Finally we show that Franchetta's conjecture, as stated by K.O'Grady, holds true for the family of polarized K3 surfacees of genus $g$, if $ 3 \le g \le 6$

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arXiv2011

On the finite dimensionality of a K3 surface

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by S.I Kimura, has several geometric consequences. For a complex surface of general type with p_g = 0 it is equivalent to Bloch's conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we prove some results on Kimura's conjecture for complex K3 surfaces. If X has a large Picard number, i.e 19 or 20, then the motive of X is finite dimensional. If X has a non-symplectic group acting trivially on algebraic cycles then the motive of X is finite dimensional. If X has a symplectic involution i, i.e a Nikulin involution, then the finite dimensionality of h(X) implies h(X) is isomorphic to h(Y), where Y is a desingularization of the quotient surface X= X/< i >. We give several examples of K3 surfaces with a Nikulin involution such that the above isomorphism holds, so giving some evidence to Kimura's conjecture in this case.

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arXiv2011

Derived categories of coherent sheaves and motives of K3 surfaces

Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational coefficients. In this paper we prove the conjecture in the case X is a K3 surface admitting an elliptic fibration (a case that always occurs if the Picard rank of X is at least 5) with finite-dimensional Chow motive. We also relate this result with a conjecture by Huybrechts showing that, for a K3 surface with a symplectic involution, the finite-dimensionality of its motive implies that the involution acts as the identity on the Chow group of 0-cycles. We give examples of pairs of K3 surfaces with the same finite-dimensional motive but not derived equivalent.

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arXiv2003

Finite dimensional motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself implies finite dimensionality of M(X). We also show (Theorem 27) that, for a surface X with trivial geometrical genus, the motive M(X) is finite dimensional if and only if the Chow group of 0-cycles of X is finite dimensional in the sense of Mumford, i.e. iff the Bloch Conjecture holds for X.

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