No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On Grothendieck's generalized Hodge conjecture for a family of threefolds with trivial canonical bundle
Journal für die reine und angewandte Mathematik
... • hypersurfaces in threefolds that are products of three curves, one of which is hyperelliptic, • odd-dimensional complete intersections of 4 quadrics -generalizing the Bardelli example [3]. For simplicity we have only considered involutions since then all invariants can easily be calculated, but it will be clear that the method of construction allows for many more examples of varieties admitting all kinds of finite abelian groups of automorphisms. ...
... On the image of L the two operators L and Λ are inverses. So, since 3 ...
... A threefold of general type with finite dimensional motive. In [39] one of the authors investigated a quasi-smooth threefold X which is a complete intersection of three degree 6 hypersurfaces in the weighted projective space P = P(2 4 , 3 3 ) and showed that A 0 (X) = Q. Let us check that this example can also be treated within the present framework. ...
Let X be a complete intersection inside a variety M with finite dimensional motive and for which the Lefschetz-type conjecture B(M) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps are injective for . We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.
... With respect to the generalized Hodge conjecture, in sub-section 3.3 we show several equivalences between the classical case and the L-version (involving Lichtenbaum cohomology and integral Hodge structures) in different weights and levels using characterizations through the Hodge conjecture (étale and classical setting) and the effectiveness of motives. In the last subsection we mention the consequences of the equivalence between the classical andétale version of the generalized Hodge conjecture regarding Bardelli's example in [Bar91]. ...
... Bardelli's example. Let us recall the example presented in [Bar91] of a certain threefold X where GHC(3, 1, X) Q holds. Let σ : P 7 → P 7 be the involution defined as σ(x 0 : . . . ...
... There exists a smooth irreducible curve C, of genus 33, obtained as the intersection of two nodal surfaces, and anétale double covering C → C such that H 1 ( C, Q) − → H 3 (X, Q) − is surjective, where the first group is the anti-invariant part of the involution τ : C → C associated to the double covering and the later group is the anti-invariant part associated to the involution σ. Notice that by [Bar91,Fact 2.4.1] if we assume that X is a very general threefold, then H 3 (X, Q) + and H 3 (X, Q) − are perpendicular with respect to the cup product on H 3 (X, Q) and H 3,0 (X) ⊂ H 3 (X, C) + therefore H 3 (X, Q) − is a polarized Hodge structure perpendicular to H 3,0 (X) i.e. a polarized sub-Hodge structure of H 3 (X, Q) of level 1. The isogeny α : Prym( C → C) → J(X) − , where J(X) − is the projection of H 1,2 (X) − into J 2 (X), is the correspondence that induces the isomorphism H 1 ( C, Q) − → H 3 (X, Q) − , but in the case of integral coefficients the image of the correspondence is a subgroup of index 2. From the previous results we have the following equivalences: GHC(3, 1, X) Q holds for H 3 (X, Q) − ⇐⇒ H 2,2 (Γ × X) ∩ H 1 (Γ, Q) ⊗ H 3 (X, Q) − is alg. ...
We define the category of \'etale Chow motives as the \'etale analogue of Grothendieck motives and proved that it embeds in . This construction provides a characterization of the generalized Hodge conjecture in terms of an \'etale analogue of it. Finally we study the non-algebraic classes in the Atiyah-Hirzebruch, Benoist-Ottem and K\'ollar counter-examples in terms of Lichtenbaum classes and we give a characterization of their preimages in the Lichtenbaum cohomology groups.
... • hypersurfaces in threefolds that are products of three curves, one of which is hyperelliptic, • odd-dimensional complete intersections of 4 quadrics -generalizing the Bardelli example [3]. For simplicity we have only considered involutions since then all invariants can easily be calculated, but it will be clear that the method of construction allows for many more examples of varieties admitting all kinds of finite abelian groups of automorphisms. ...
... On the image of L the two operators L and Λ are inverses. So, since 3 ...
... A threefold of general type with finite dimensional motive. In [39] one of the authors investigated a quasi-smooth threefold X which is a complete intersection of three degree 6 hypersurfaces in the weighted projective space P = P(2 4 , 3 3 ) and showed that A 0 (X) = Q. Let us check that this example can also be treated within the present framework. ...
Let X be a complete intersection inside a variety M with finite dimensional motive and for which the Lefschetz-type conjecture B(M) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps are injective for . We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.
... This fibre is semistable of Type III and has seven components arranged as a tetrahedron with three truncated vertices. (M 6 ) By Table 3, over the threefoldX 6 contains three curves of c A 1 singularities, denoted C A 1 1 , C A 1 2 , C A 1 3 , two curves of c A 2 's, denoted C A 2 1 , C A 2 1 , and two curves of c A 3 's, denoted C A 3 1 , C A 3 2 , which form sections of the fibration. There are also two further curves of c A 1 singularities lying in the fibre over λ = ∞, given by D 1 := {y = z = 0, λ = ∞} and D 2 := {z = t = 0, λ = ∞}. ...
... The resulting threefold is non-normal along its central fibre. After performing a change of coordinates z → λz, which has the effect of normalizing the pull-back of Z and contracting the pull-back of S, we find that the resulting threefold is smooth away from the nine curves C A j i (note that the three C A 1 i 's are no longer disjoint, and all intersect at the point λ = s = x = y = 0). Blow-up the six curves C A 2 i twice each, then simultaneously resolve the three curves C A 1 i by the method of [15,Sect. ...
... This family has an irreducible fibre over λ = 0 and the threefold total space is smooth away from the six curves C A m i , which form sections of the fibration. However, there is some isolated singularity behaviour where the curves C A m i meet the fibre λ = 0: the curves C A 1 1 and C A 2 1 collide in a c A 4 singularity, the curves C A 2 2 and C A 3 1 collide in a c A 6 singularity, and the singularity along the curve C A 4 1 jumps to a cD 5 . After blowing up these three singularities, we are left with three isolated nodes (c A 1 's) in the threefold total space, corresponding to the three points where the isolated singularity behaviour occurs. ...
We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case and compute their Hodge numbers.
... While for the generalized Hodge conjecture, besides the aforementioned cases, very few are known. One class of known cases concerns algebraic varieties with an automorphism group, see for example [8] and [65]. As far as we know, besides these and some results about abelian varieties (cf. ...
... • [65] deals with some complete intersection surfaces with an automorphism group. See also [8] for a similar result about Calabi-Yau 3-folds. ...
... We note that pull-backs are functorial: (αα ′ ) * (a) = α ′ * α * (a). 8. Here we use the terminology 'partition' in an unusual way. 3 , then the representativeα : 1 → 1, 2 → 2, 3 → 1, 4 → 3, 5 → 2, and thus the corresponding morphism for any Y is α : Y 3 → Y 5 (y 1 , y 2 , y 3 ) → (y 1 , y 2 , y 1 , y 3 , y 2 ). ...
This Thesis consists of three chapters. In Chapter 1, admitting the Lefschetz standard conjecture, we prove the generalized Hodge conjecture for the kernel of the cup product with a big cohomology class, which is a sub-Hodge structure of coniveau 1. In Chapter 2, we establish a decomposition of the small diagonal of X × X × X for a Calabi-Yau complete intersection X in a projective space. As a corollary, we prove that the intersection product, modulo rational equivalence, of algebraic cycles with positive and complementary dimensions, is as degenerated as possible. In Chapter 3, we prove that a polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold acts as identity on its Chow group of 0-cycles, as is predicted by the generalized Bloch conjecture.
... While for the generalized Hodge conjecture, besides the aforementioned cases, very few are known. One class of known cases concerns about algebraic varieties with an automorphism group, see for example [Bar91] and [Voi92]. As far as we know, besides these and some results about abelian varieties (cf. ...
... • [Voi92] deals with some complete intersection surfaces with an automorphism group. See also [Bar91] for a similar result about Calabi-Yau 3-folds. ...
We study the generalized Hodge conjecture for certain sub-Hodge structure
defined as the kernel of the cup product map with a big cohomology class, which
is of Hodge coniveau at least 1. As predicted by the generalized Hodge
conjecture, we prove that the kernel is supported on a divisor, assuming the
Lefschetz standard conjecture.
We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case and compute their Hodge numbers.
Maschke's Calabi-Yau threefold is the double cover of projective three space
branched along Maschke's octic surface. This surface is defined by the lowest
degree invariant of a certain finite group acting on a four dimensional vector
space. Using this group, we show that the middle Betti cohomology group of the
threefold decomposes into the direct sum of 150 two-dimensional Hodge
substructures. We exhibit one dimensional families of rational curves on the
threefold and verify that the associated Abel-Jacobi map is non-trivial. By
counting the number of points over finite fields, we determine the rank of the
N\'eron-Severi group of Maschke's surface and the Galois representation on the
transcendental lattice of some of its quotients. We also formulate precise
conjectures on the modularity of the Galois representations associated to
Maschke's threefold and to a genus 33 curve which parametrizes rational curves
in the threefold.
In this paper for every p > 0 the universal family of the hypersurfaces of degree 2p and dimension 3 invariant under a certain action of the group of p-th roots of unity is considered. The maximal abelian rational sub-Hodge structure of weight 3 is investigated on the general element and the general Grothendieck-Hodge conjecture is checked for the general point of some special sub-families.
We verify the generalized Hodge conjecture GHC(X,5,2) for the quadratic complex of lines in projective four–space.
This paper deals with some families of projective hypersurfaces of degree 6 and dimension 3, left invariant under certain
actions of the group of the cubic roots of unity.
The general Grothendieck-Hodge conjecture, for the families above and for their intersections, is verified. In particular
certain families of elliptic curves on the general element of these families are closely investigated and their degeneration
at the Fermat point is studied. These results are used to verify the general Infinitesimal Hodge Conjecture extending to the
general type case a similar conjecture due to A. Albano and S. Katz.
It is proved that every morphism of a curve with an involution into an Abelian variety, anticommuting with the involution, factors through the associated Prym variety. This result is used to show that Jacobians of curves of genus five arise as Prym varieties associated to a certain class of curves.