Irrationality of Motivic series of Chow varieties

In this paper we extend the Euler–Chow series for Chow varieties to Chow motives. In both series it is very natural to ask when the series is rational. We give an example where the extended series is not rational. Partially supported by program JSPS-CONCYT.     

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.     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.     

On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

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.     

Reconstructing vector bundles on curves from their direct image on symmetric powers

Let C be an irreducible smooth complex projective curve, and let E be an algebraic vector bundle of rank r on C. Associated to E, there are vector bundles ${{\mathcal F}_n(E)}$ of rank nr on S ⁿ (C), where S ⁿ (C) is the n-th symmetric power of C. We prove the following: Let E ₁ and E ₂ be two semistable vector bundles on C, with genus ${(C)\, \geq\, 2}$ . If ${{\mathcal F}_n(E_1)\,\simeq \, {\mathcal F}_n(E_2)}$ for a fixed n, then ${E_1 \,\simeq\, E_2}$ .     

Generalized Poincaré Classes and Cubic Equivalences

Let Y be a smooth projective algebraic surface over ?, and T(Y) the kernel of the Albanese map CH₀(Y)_deg0 → Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E₁ × E₂, such as an abelian or a Kummer surface, then one can easily construct an abelian variety B and a surjective 'regular' homomorphism B^?z2 → T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P⁴. This implies a sort of boundedness result for T(Y). It is natural to ask if this is the case for any smooth projective algebraic surface Y ? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 (2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k—cycles on any projective algebraic manifold X, and arrive at, from a conjectural standpoint, necessary and sufficient cohomological conditions on X for which the question can be answered affirmatively.     

Relations Among Heegner Cycles on Families of Abelian Surfaces

We compute relations of rational equivalence among special codimension 2 cycles on families of Abelian surfaces using elements of a higher Chow group. These relations are similar to those between Heegner points and special divisors obtained by Zagier, Van der Geer and others.     

Some filtrations on higherK-theory and related invariants

We lift Bloch's higher Chow construction from the level of simplicial sets to the level of simplicial spaces. We construct a simplicial space that becomes isomorphic to the Bloch/Chow complex when the functor ₀ is applied in each degree. The homotopy groups of this space are theE ₂-terms in an Atiyah-Hirzebruch spectral sequence converging to algebraicK-theory. TheseE ₂-terms map nontrivially to the expected higher Chow groups. We define and compute several intermediate invariants associated to our simplicial space.     

Tensor powers of symmetric algebras

We prove that over an integral domain a module is projective iff an appropriate tensor power of its symmetric algebra is an integral domain. Further, we show that contracting parts of primary decom­positions of the zero ideal in appropriate tensor powers of a symmetric algebra one obtains families of ideals canonically associated to a mod­ule, having the same radical as Fitting ideals. More precisely, we prove that those new ideals lie between the annihilators of exterior powers of the module and their radicals. An immediate consequence of our re­sults is a way to recover the radicals of Fitting ideals of a module from the symmetric algebra of that module (with its grading forgotten).     

q-Torsion freeness of symmetric powers

Letq be an integer ≥1, we study theq-torsion freeness of the symmetric powers of a module of projective dimension ≥2, by using the approximation complex of the module.     

Kodaira dimension of symmetric powers

We compute the plurigenera and the Kodaira dimension of the th symmetric power of a smooth projective variety . As an application we obtain genus estimates for the curves lying on .

     

Gromov-Witten invariants of jumping curves

Given a vector bundle on a smooth projective variety , we can define subschemes of the Kontsevich moduli space of genus-zero stable maps parameterizing maps such that the Grothendieck decomposition of has a specified splitting type. In this paper, using a ``compactification' of this locus, we define Gromov-Witten invariants of jumping curves associated to the bundle . We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on . Our construction is a generalization of jumping lines for vector bundles on . Since for the tautological bundle of the Grassmannians the invariants are enumerative, we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls.

     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

On the tautological ring of a Jacobian modulo rational equivalence

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of whose ramification points are all of order d, and then for hyperelliptic curves.      

On rational invariants of the group

We prove rationality of the field of invariants in several variables of a minimal irreducible representation of a simple algebraic group of type over an algebraically closed field of characteristic zero.

     

Minors of symmetric and exterior powers

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.