Griffiths Group Research Articles

Infinite torsion in Griffiths groups

We show that there are smooth complex projective varieties with infinite 2 -torsion in their third Griffiths groups. It follows that the torsion subgroup of Griffiths groups is in general not finitely generated, thereby solving a problem of Schoen from 1992.

Experiments with Ceresa classes of cyclic Fermat quotients

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the L L -function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.

Unramified cohomology, integral coniveau filtration and Griffiths groups

Unramified cohomology, integral coniveau filtration and Griffiths groups

Limits and singularities of normal functions

We find a new method to detect the linearly independence of \({\mathbb {R}}\)-regulator indecomposable \(K_1\)-cycles which is based on the singularities and limits of admissible normal functions. We also construct a collection of higher Chow cycles on certain surfaces in \({\mathbb {P}}^3\) of degree \(d \geqslant 4\) which degenerate to an arrangement of d planes in general position. By applying our method, we show that these higher Chow cycles are enough to show the surjectivity of the real regulator map when \(d = 4\). Hence our construction gives a new explicit proof of the Hodge-\({\mathcal {D}}\)-Conjecture for a certain type of K3 surfaces. As an application, we also construct a general semistable degeneration family of degree \(d + 1\) threefolds in \({\mathbb {P}}^3 \,{\times }\, {\mathbb {P}}^1 \,{\times }\, {\mathbb {P}}^1\) such that a codimension 1 stratum is a surface of the above type. The real regulator indecomposability of our higher Chow cycles implies that the Griffiths group of the general fiber of these threefolds is non-trivial.

Motives and the Pfaffian–Grassmannian equivalence

We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds, and verifications of Kimura's finite-dimensionality conjecture, Voevodsky's smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.

Albanese kernels and Griffiths groups

We describe the Griffiths group of the product of a curve $C$ and a surface $S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$. When $C$ is a hyperplane section of $S$ varying in a Lefschetz pencil, we prove the nonvanishing in $\text{Griff}(C\times S)$ of a modification of the graph of the embedding $C\hookrightarrow S$ for infinitely many members of the pencil, provided the ground field $k$ is of characteristic $0$, the geometric genus of $S$ is $>0$, and $k$ is large or $S$ is "of motivated abelian type".

Generalised Heegner cycles and the complex Abel–Jacobi map

Generalised Heegner cycles were introduced in Bertolini et al. (Duke Math J 162(6), 1033–1148, 2013) as a variant of Heegner cycles on Kuga–Sato varieties. The first main result of this article is a formula for the image of these cycles under the complex Abel–Jacobi map in terms of explicit line integrals of modular forms on the complex upper half-plane. The second main theorem uses this formula to show that the Chow group and the Griffiths group of the product of a Kuga–Sato variety with an elliptic curve with complex multiplication are not finitely generated. More precisely, it is shown that the subgroup generated by the image of generalised Heegner cycles has infinite rank in the group of null-homologous cycles modulo both rational and algebraic equivalence.

2-Cycles sur les hypersurfaces cubiques de dimension 5

We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel–Jacobi map induces an isomorphism between the Chow group of algebraically trivial 2-cycles and the intermediate Jacobian.

The derived functors of unramified cohomology

We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.

Lefschetz theorems for torsion algebraic cycles in codimension 2

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 dim⁡Y≥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.

Torsion 1-cycles and the coniveau spectral sequence

We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree 2{\dim}(X)-3 . We also describe in terms of \mathcal H -cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.

2-Cycles on higher Fano hypersurfaces

Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth.

P-adic L-functions and the coniveau filtration on Chow groups

Abstract We give evidence for the refined version of the Beilinson–Bloch conjecture involving coniveau filtrations, by studying several infinite families of CM motives (indexed by the integers r ≥ 1 {r\geq 1} ) that are irreducible of Hodge type ( 2 ⁢ r + 1 , 0 ) + ( 0 , 2 ⁢ r + 1 ) {(2r+1,0)+(0,2r+1)} and whose L-functions vanish at the center. In each case, we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration. Consequently, we obtain new explicit examples of cycles on varieties over number fields that are nontorsion in the Griffiths group. The proof of the main result relies crucially on p-adic Hodge theoretic methods, and in particular on the relation between images of homologically trivial cycles under the p-adic Abel–Jacobi map and the values of p-adic L-functions.

On the Space of Conics on Complete Intersections

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold, the fivefold quartic double solid, and the fivefold intersection of a quadric and a cubic. We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a smooth general complete intersection FCY manifolds, is also infinitely generated. In the case of the fivefold quartic double solid, we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for the fivefold intersection of a quadric and a cubic there is a noncommutative CY 3-fold, such that the Griffiths group of the intersection surjects on to the Griffiths group of noncommutative 3-fold. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back to honest algebraic varieties such as products of elliptic curves and K3-surfaces.

Algebraic cobordism theory attached to algebraic equivalence

AbstractBased on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

Classes de cycles motiviques étales

Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (\'etale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Th\'el\`ene and Voisin. If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. If k is the separable closure of a finite field and X comes from a variety over a finite field as described above, then D = 0 as soon as H^3(X,Q_l) is entirely of coniveau > 0, but an example of Schoen shows that this condition is not necessary.

Algebraic cycles on the generic abelian fourfold with polarization of type (1,2,2,2)

In this paper we construct a non-torsion element in the higher Griffiths group Griff 3,2(A4) for the generic abelian fourfold A4 with polarization of type (1,2,2,2). The key idea is to use the fact that A4 can be realized as a generalized Prym variety which naturally contains some curves, i.e. dimension 1 cycles.

Birational invariants defined by Lawson homology

New birational invariants for a projective manifold are defined by using Lawson homology. These invariants can be highly nontrivial even for projective threefolds. Our techniques involve the weak factorization theorem of Wlodarczyk and tools developed by Friedlander, Lawson, Lima-Filho and others. A blowup formula for Lawson homology is given in a separate section. As an application, we show that for each n\geq 5, there is a smooth rational variety X of dimension n such that the Griffiths groups Griff}_p(X) are infinitely generated even modulo torsion for all p with 2\leq p\leq n-3.

Lawson homology, morphic cohomology and Chow motives

AbstractIn this paper, Lawson homology and morphic cohomology are defined for Chow motives. As consequences, we rederive the projective bundle formula proved by Friedlander and Gabber, the blowup formula for Lawson homology by the first author, and a formula for certain homogeneous projective varieties, including those admitting cellular decompositions. We also define rational coefficient Lawson homology and morphic cohomology for Chow motives of finite quotient varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. We also discuss generic finite maps, in particular, we give examples of self‐product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim