Algebraic Cycles Research Articles

ALGEBRAIC CYCLES AND TATE CLASSES ON HILBERT MODULAR VARIETIES

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL 2(𝔸E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M ét (π∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M ét (π∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M ét (π∞) is spanned by algebraic cycles.

Algebraic cycles and local quantum cohomology

We review the Hodge theory of some classic examples from mirror symmetry, with an emphasis on what is intrinsic to the A-model, and on interesting open questions and problems. In particular, we illustrate the construction of a quantum Z-local system on the cohomology of the total space of the canonical bundle of P^2, and suggest how this should be related to the higher algebraic cycles studied in our 2011 CNTP article Algebraic K-theory of toric hypersurfaces.

Weak Lefschetz for Chow groups: Infinitesimal lifting

Let X be a smooth projective variety over an algebraically closed field k of characteristic zero and Y ⊂ X a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map CH(X)Q → CH(Y )Q is an isomorphism for all p < dim(Y )/2. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous K-cohomology group on the formal completing of X along Y . This splits the conjecture into two smaller conjectures: one consisting of an algebraization problem and the other dealing with infinitesimal liftings of algebraic cycles. We give a complete proof of the infinitesimal part of the conjecture.

A category of kernels for equivariant factorizations and its implications for Hodge theory

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space. Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths’ classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category. Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.

Voevodsky’s motives and Weil reciprocity

We describe Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil reciprocity, Voevodsky's category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

The Tautological Ring of the Moduli Space M2,nrt

The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mumford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincare duality. The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra. In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result. In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well.

CHOW STABILITY OF CANONICAL GENUS 4 CURVES

In this paper, we give sufficient conditions on a canonical genus 4 curve for it to be Chow (semi)stable. A Deligne-Mumford stable curve is a complete connected curve C having ample dualising sheaf !C and admitting only nodes as singularities. An n- canonical curve C ⊂ P N is a Deligne-Mumford stable curve of arithmetic genus g embedded by the complete linear system |! ⊗n C | where N = (2n−1)(g−1)−1 if n ≥ 2, and N = g − 1 if n = 1. Let Chowg,n be the closure of the locus of the Chow forms of n-canonical curves of arithmetic genus g in the Chow variety of algebraic cycles of dimension 1 and degree 2g−2 in P N. The natural action of SLN+1 on P N induces an action on Chowg,n. Denote the corresponding GIT (Geometric Invariant Theory) quotient space by Chowg,n//SLN+1. To understand this quotient space as a parameter space of curves with some geometric properties, we need to find Chow stability conditions. Mumford showed that, for n ≥ 5 and g ≥ 2, the Chow stable curves are pre- cisely Deligne-Mumford stable curves and there is no strictly Chow semistable curve (cf. (14)). This implies that the quotient space is precisely the moduli space of Deligne-Mumford stable curves M4. The cases when n = 3 and g ≥ 3 were concerned by Schubert in (16). He proved that a 3-canonical curve of genus g ≥ 3 is Chow stable if and only if it is pseudo-stable and also showed that there is no strictly Chow semistable curve, and thus the quotient space is the moduli space of pseudo-stable curves M ps . A pseudo-stable curve is a complete connected curve C satisfying the following properties.

Corrigendum to “Motives and representability of algebraic cycles on threefolds over a field”

We correct a mistake in Lemma 3.1(5) in J. Algebraic Geom. 21 (2012), 347–374.

Algebraic cycles satisfying the Maurer-Cartan equation and the unipotent fundamental group of curves

AbstractWe address the question of lifting the étale unipotent fundamental group of curves to the level of algebraic cycles and show that a sequence of algebraic cycles whose sum satisfies the Maurer-Cartan equation would do the job. For any elliptic curve with the origin removed and the curve $\double-struck(G)_m\$, we construct such a sequence of algebraic cycles whose image under the cycle map gives rise to the étale unipotent fundamental group of the curve.

Around rationality of cycles

Abstract We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

Variations on a theme of rationality of cycles

Abstract We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

P-adic deformation of algebraic cycle classes

We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle on the closed fibre lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.

On algebraic cycles on a fibre product of families of K3-surfaces

We prove the Hodge conjecture and the standard conjecture of Lefschetz type for fibre squares of smooth projective non-isotrivial families of -surfaces over a smooth projective curve under the assumption that the rank of the lattice of transcendental cycles on a generic geometric fibre of the family is an odd prime. We prove the Hodge conjecture for a fibre product of two non-isotrivial families of -surfaces (possibly with degenerations) under the condition that, for every point of the curve, at least one family has non-singular fibre over this point, and the rank of the lattice of transcendental cycles on a generic geometric fibre of one family is odd and not equal to the corresponding rank for the other.

L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, II

Abstract Ceci est le second volet d'une série de deux articles visant à construire et étudier des groupes de Galois motiviques dans le cadre des motifs triangulés. Ces groupes de Galois motiviques ont été construits dans le premier volet et leurs algèbres de fonctions régulières ont été décrites explicitement en termes de formes différentielles ou de cycles algébriques. Dans le présent article, nous avons regroupé des compléments importants au premier volet. Dans une première partie, nous décrivons le lien entre le groupe de Galois motivique et le groupe de Galois usuel d'un sous-corps de 𝔻. Dans une deuxième partie, nous développons les bases d'une théorie de la ramification pour les groupes de Galois motiviques en construisant des groupes de décomposition et d'inertie motiviques associés à une place géométrique. On introduit également la notion de groupe de Galois motivique relatif pour une extension K / k $K/k$ qui mesure la différence entre les groupes de Galois motiviques de k et K. Ce dernier s'avère plus accessible que son analogue absolu. En effet, on montrera que c'est un quotient du complété pro-algébrique du pro-groupe fondamental topologique de la pro-variété hom k ( K , ℂ ) $\hom _k(K,\mathbb {C})$ , du moins lorsque l'extension K / k $K/k$ est de type fini.

Around rationality of integral cycles

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for Z/2Z-coefficients. Those results have already been proved by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠2.

The argument cycle and the coamoeba

We investigate the coamoeba of a complex algebraic variety V ⊂ (ℂ*) n through the study of initial forms of the defining ideal. By use of a universal Gröbner basis, we prove that the closure of the coamoeba is included in the union of coamoebas corresponding to all initial ideals. We also study complete intersections V of dimension n/2 more closely to get a lower bound for the multiplicity in V of a given point θ on the n:th torus. For this purpose, we associate a certain algebraic cycle, the argument cycle, to V and θ , and study its homology. In particular, we give a method to approximate the coamoeba when n = 2.

Algebraic cycles and fibrations

Let f:X \rightarrow B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f . One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is «built» from the motives of varieties of dimension less than the dimension of B .

Limit Cycle Identification in Nonlinear Polynomial Systems

We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.

On the [formula omitted]-adic section conjecture

We prove that the analog of the Grothendieck anabelian section conjecture for curves holds true over a p-adic local field, if the following two assertions hold true. First, every section of the arithmetic fundamental group of a hyperbolic curve over a p-adic local field has an algebraic cycle class. Second, given a section of the cuspidally abelian absolute Galois group of the function field of such a curve, the corresponding section of the maximal Z/pZ-elementary abelian quotient, and a neighborhood of the latter, then the index of the corresponding curve is prime-to-p.

Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch–Beilinson–Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology, that is, the cohomology, of smooth projective varieties. In this paper, given a smooth projective complex variety X, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of X has an incidence on the Chow groups of X. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low-dimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a 4-fold of general type with a trivial Chow group of zero-cycles and we prove Murre's conjectures for 3-folds dominated by a product of curves, for 3-folds rationally dominated by the product of three curves, for rationally connected 4-folds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura–O'Sullivan's notion of finite-dimensionality. Assuming the standard conjectures on algebraic cycles, the former is known to imply the latter. We show that the missing ingredient for finite-dimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups.