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Register a new userMotives and representability of algebraic cycles on threefolds over a field
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We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2. This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.
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Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdans motivic integration and Ayoubs equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Blochs conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.
In this short note, we simply collect some known results about representing algebraic cycles by various kind of nice (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. We also add a few elementary and easy observations on these representation problems that we were not able to locate in the literature.
We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor from the bounded derived category of a smooth projective variety over a field to the category of L-modules, to the case where L is a field extension of the base field k of the variety X, with L of transcendence degree less than or equal to one or L purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor between the bounded derived categories of coherent sheaves of X and Y, with X and Y smooth projective and Y of dimension less than or equal to one or Y a rational surface. We show that for any such F there exists a generic kernel A in the derived category of the product, such that F is isomorphic to the Fourier-Mukai transform with kernel A after composing both with the pullback to the generic point of Y.
Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism between the Chow group $CH_r(X)_0$ of degree $0$ dimension $r$ algebraic cycles modulo rational equivalence on $X$, and the group of sections of the sheaf of $mathbb A^1$-path connected components of the loop space of $BC_r^{infty }(X)$ at $Spec(k)$. Equivalently, $CH_r(X)_0$ is isomorphic to the group of sections of the $S^1wedge mathbb A^1$-fundamental group $Pi _1^{S^1wedge mathbb A^1}(BC_r^{infty }(X))$ at $Spec(k)$.
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