Cycles on the generic abelian threefold

H Clemens and C Schoen gave examples of three-folds where the group of codimension two cycles modulo algebraic equivalence has infinite rank. This paper provides yet another example of the same phenomenon.     

Divided powers in Chow rings and integral Fourier transforms

We prove that for any monoid scheme M over a field with proper multiplication maps M×M→M, we have a natural PD-structure on the ideal CH>0(M)⊂CH_∗(M) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to N²-torsion, where N=1+⌊log₂(3g)⌋. As a consequence we obtain, over , a PD-structure (for the intersection product) on N²⋅a, where a⊂CH(J) is the augmentation ideal. We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.     

Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

Planar polynomial vector fields having first integrals and algebraic limit cycles

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.     

Remarks on Hard Lefschetz conjectures on Chow groups Dedicated in celebration of SCIENCE CHINA on the occasion of its 60th birthday

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we show that they are equivalent to the well-known conjectures of Beauville and Murre.     

Some computations of algebraic cycle homology

We compute the algebraic cycle homology for codimension 1 cycles on a variety over a perfect field; our computation agrees with Nart's computation of Bloch's higher Chow groups for codimension 1 cycles. We interpret algebraic cycle homology in terms of sheaves for Voevodsky's h-topology and use this to adapt a recent result of Suslin-Voevodsky: we establish for a complex variety that algebraic cycle homology with Z/n coefficients is naturally isomorphic to singular homology with Z/n coefficients.Partially supported by the NSF and NSA Grant #MDA904-90-H-4006.     

Lower Bounds on Genera of Subvarieties of Generic Hypersurfaces

Abstract

A second-order invariant of C. Voisin gives a powerful method for bounding from below the geometric genus of a k-dimensional subvariety of a degree dhypersurface in complex projective n-space. This work uses the Voisin method to establish a general bound, which lies behind recent results of G. Pacienza and Z. Ran.     

Algebraic cycles and very special cubic fourfolds

Informed by the Bloch–Beilinson conjectures, Voisin has made a conjecture about 0-cycles on self-products of Calabi–Yau varieties. In this note, we consider variant versions of Voisin’s conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

On the generic kernel filtration for modules of constant Jordan type

Let ${E \cong (\mathbb{Z}/p)^2}$ be an elementary abelian p-group of rank two k an algebraically closed field of characteristic p, and let J =?J(kE). We investigate finitely generated kE-modules M of constant Jordan type and their generic kernels ${\mathfrak{K}(M)}$ . In particular, we answer a question posed by Carlson, Friedlander, and Suslin regarding whether or not the submodules ${J^{-i} \mathfrak{K}(M)}$ have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or ${J^2 \mathfrak{K}(M) = 0}$ . We also show that this question has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that ${J^{-1} \mathfrak{K}(M)}$ does not have constant Jordan type.     

Algebraic cycles and topology of real Enriques surfaces

For a real Enriques surface Y we prove that every homology class in H₁(Y (R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface Y.     

The Tate conjecture for powers of ordinary cubic fourfolds over finite fields

Recently, Levin proved the Tate conjecture for ordinary cubic fourfolds over finite fields. In this paper, we prove the Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is based on properties of so-called polynomials of K3-type introduced by the author about 12 years ago.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Lefschetz Motives and the Tate Conjecture

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.     

Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic algebras

We compute degrees of algebraic cycles on certain Severi-Brauer varieties and apply it to show that:

Cycles and Spectra

Some recent work on spaces of algebraic cycles is surveyed. The main focus is on spaces of real and quaternionic cycles and their relation to equivariant Eilenberg- MacLane spaces.Dedicated to IMPA on the occasion of its 50^th anniversary     

Nilpotency of Connes'' periodicity operator and the idempotent conjectures

For certain classes of discrete groups we verify the idempotent conjectures for various group algebras by the method of cyclic cohomology. In particular, the Banach ¹ (G) of a torsion free word hyperbolic groupG of Gromov contains no nontrivial idempotents. Moreover, the range of any tracial state onK ₀(¹(G)) is .Sponsored in part by a grant from the National Science Foundation.     

On mixed Hessians and the Lefschetz properties

We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of a paper by Maeno and Watanabe for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property. We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefschetz Property; we construct minimal examples of such algebras and we give bounds, depending on the degree, for their existence. Artinian Gorenstein algebras presented by quadrics were conjectured to satisfy WLP in two papers by Migliore and Nagel, and in a previous paper we constructed the first counter-examples.     

Cyclic cohomology and the family Lefschetz theorem

Any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra. We prove that such cocycle furnishes additive maps from the vertically equivariant K-theory to the scalars. This enables to associate to any closed current on the base of the fibration, a Lefschetz formula for fiber-preserving isometries. Using geometric operators on the base, we deduce the integrality of some characteristic numbers. Received: 28 June 2001 / Published online: 1 February 2002