Absolute Chow-Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties
TL;DR: In this paper, the existence of a Chow-Kuenneth decomposition for a rational homogeneous bundle over a smooth variety is investigated and the same conclusion holds for a class of log homogeneous varieties, studied by Brion.
Abstract: In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\\to S$ over a smooth variety, defined over complex numbers. Chow-K\\\"unneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.
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TL;DR: In this article, the authors studied the Chow groups of X in terms of the Chow group of B and of the fibres of f and showed that when X and B are smooth projective and when f is a flat quadric fibration, the Chow motive is built from the motives of varieties of dimension less than the dimension of B.
Abstract: Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is "built" from the motives of varieties of dimension less than the dimension of B.
84 citations
01 Apr 2017
TL;DR: In this paper, a relation between the Chow motives of a smooth cubic hypersurface and the Fano variety of lines on it was established, which implies that if X has finite-dimensional motive (in the sense of Kimura), then F also has finitedimensional motive.
Abstract: Let X be a smooth cubic hypersurface, and let F be the Fano variety of lines on X. We establish a relation between the Chow motives of X and F. This relation implies in particular that if X has finite-dimensional motive (in the sense of Kimura), then F also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension 3 and 5, and of certain cubics in other dimensions.
27 citations
TL;DR: In this article, a relation between the Chow motives of a smooth cubic hypersurface and the Fano variety of lines on it was established, and it was shown that if a smooth cube has finite-dimensional motives, then a Fano line on it also has finite dimensions.
Abstract: Let $X$ be a smooth cubic hypersurface, and let $F$ be the Fano variety of lines on $X$. We establish a relation between the Chow motives of $X$ and $F$. This relation implies in particular that if $X$ has finite-dimensional motive (in the sense of Kimura), then $F$ also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension $3$ and $5$, and of certain cubics in other dimensions.
23 citations
TL;DR: In this paper, the Chow group of 0-cycles on complex varieties of geometric genus one is verified, and some new examples of surfaces for which Voisin's conjecture is verified.
Abstract: Inspired by the Bloch–Beilinson conjectures, Voisin has formulated a conjecture concerning the Chow group of 0-cycles on complex varieties of geometric genus one. This note presents some new examples of surfaces for which Voisin’s conjecture is verified.
22 citations
TL;DR: In this article, some conjectures about Chow groups of varieties of geometric genus one are studied and some examples are given of Calabi-Yau threefolds where these conjectures can be verified using the theory of finite-dimensional motives.
Abstract: We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi–Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
19 citations
References
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Book•
01 Jan 1991
TL;DR: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions are discussed in this paper.
Abstract: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
2,919 citations
01 Jan 1983
TL;DR: In this paper, a Chow ring for the moduli space M g of curves of genus g and its compactification M g is defined, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes.
Abstract: The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification M g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here the basic classes are the Chern classes of the tautological or universal bundle that lives over the Grassmannian, and the most basic cycles are the loci of linear spaces satisfying various Schubert conditions: the so-called Schubert cycles. However, since Harris and I have shown that for g large, M g is not unir.ational [H-M] it is not possible to expect that M g has a decomposition into elementary cells or that the Chow ring of M g is as simple as that of the Grassmannian. But in the other direction, J. Harer [Ha] and P. Miller [Mi] have strong results indicating that at least the low dimensional homology groups of M g behave nicely. Moreover, it appers that many geometrically natural cycles are all expressible in terms of a small number of basic classes.
867 citations
Book•
01 Jan 2006
TL;DR: Infinitesimal deformations as discussed by the authors have been studied in formal deformation theory, and examples of deformation functors can be found in Hilbert and Quot schemes, e.g.
Abstract: Infinitesimal deformations.- Formal deformation theory.- Examples of deformation functors.- Hilbert and Quot schemes.
533 citations
01 Jan 1994
TL;DR: In this article, the authors consider the properties of vector bundles and show that vector bundles can be classified into three classes: vector bundles, line bundles, and vector bundles with line bundles.
Abstract: 1. Basic properties of nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 1.A. Nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 1.B. Nef vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 11
431 citations
282 citations