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.     

Indecomposability of certain Lefschetz fibrations

We prove that Lefschetz fibrations admitting a section of square cannot be decomposed as fiber sums. In particular, Lefschetz fibrations on symplectic 4-manifolds found by Donaldson are indecomposable. This observation also shows that symplectic Lefschetz fibrations are not necessarily fiber sums of holomorphic ones.

     

On the image of the -adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Let be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic . It is shown that the Tate conjecture implies the surjectivity of the -adic Abel-Jacobi map, , for all and almost all . For a special class of threefolds the surjectivity of is proved without assuming any conjectures.

     

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.     

Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub)

We clarify the expected properties of the slice filtration on triangulated motives from the point of view of the generalized Hodge conjecture. In the appendix, J. Ayoub proves unconditionally that the slice filtration does not respect geometric motives. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Motivic invariants of symmetric powers of curves

We study the structure of various invariants of the symmetric powers of a smooth projective curve in terms of that of the Jacobian of the curve. We generalise the results of Macdonald and Collino to various invariants including the Weil-cohomology theory, the higher Chow groups, the additive higher Chow groups and the rational K-groups.     

On the topology of a generic fibre of a polynomial function

In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.     

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 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.     

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 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.     

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.