Algebraic Cycles Research Articles

A note on Tate’s conjectures for abelian varieties

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

Determining monodromy groups of abelian varieties

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford–Tate group, $$\ell $$ -adic monodromy groups, and the Sato–Tate group. Assuming the Mumford–Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.

On flat pullbacks for Chow groups

It is a fundamental property of the Chow groups of algebraic schemes that they are contra-functorial with respect to flat morphisms between schemes. While the pullback homomorphism is easy to define at the level of algebraic cycles, the crucial step is to show that the pullback of cycles preserves rational equivalence, so that it descends to the Chow groups. The purpose of this note is to give a natural sheaf theoretic proof of the preservation of rational equivalence under flat pullback on cycles.

On the degree of algebraic cycles on hypersurfaces

Abstract Let X ⊂ ℙ 4 {X\subset\mathbb{P}^{4}} be a very general hypersurface of degree d ≥ 6 {d\geq 6} . Griffiths and Harris conjectured in 1985 that the degree of every curve C ⊂ X {C\subset X} is divisible by d. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d. Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being d = 5005 {d=5005} . The set of these degrees d has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over ℚ {\mathbb{Q}} that satisfy the conjecture.

Torsion codimension cycles on supersingular abelian varieties

AbstractWe prove that torsion codimension $2$ algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension $2$ on supersingular abelian varieties over the algebraic closure of finite fields.

Height pairing on higher cycles and mixed Hodge structures

For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized complex and intersect properly. In a special case, this mixed Hodge structure is an oriented biextension, and its height agrees with the higher archimedean height pairing introduced in a previous paper by the first two authors. We also compute a non-trivial example of this height given by Bloch–Wigner dilogarithm function. Finally, we study the variation of mixed Hodge structures of Hodge–Tate type, and show that the height extends continuously to degenerate situations.

Regularization of closed positive currents and extension theory

In this note we extend to any bidimension (p,p) the Demailly theorem of regularization of closed positive (1,1)-currents on a compact Kähler manifold X of dimension n. When the manifold X is projective, we get explicitly a closed regularization with bounded negative part, constructed by using the space Cp(X) of effective algebraic cycles of X of dimension p. This space can be injected in the space of divisors of Cn−p−1(X) and we arrive at an intrinsic construction of the Skoda potential associated with a closed positive current of X. On another hand, in the case of a divisor D of X, we give an explicit bound for the degree of an irreducible component of the singular locus Dsing, involving the geometry of X. Lastly when X is embedded in the projective space PN, we prove the existence of a closed current extending in the generalized sense a given closed current of X, by using here a space of cycles. As an application, we obtain a characterization of the cohomology classes contained in some algebraic hypersurface of X.

Chern character and obstructions to deforming cycles

Green-Griffiths observed that we could eliminate obstructions to deforming divisors. Motivated by recent work of Bloch-Esnault-Kerz on deformation of algebraic cycle classes, we use Chern character to generalize Green-Griffiths' observation and to show how to eliminate obstructions to deforming cycles of codimension p.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

Spherical varieties and norm relations in Iwasawa theory

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving “vertical” norm-compatibility relations for such classes (where the level varies at a fixed prime p), treating the case of Betti and étale cohomology at once, and revealing an unexpected relation to the theory of spherical varieties. This machinery can be used to construct many new examples of norm-compatible families, potentially giving rise to new constructions of both Euler systems and p-adic L-functions: examples include families of algebraic cycles on Shimura varieties for U(n)×U(n+1) and U(2n) over the p-adic anticyclotomic tower.

Algebraic cycles and Fano threefolds of genus 8

We show that prime Fano threefolds Y of genus 8 have a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, a certain tautological subring of the Chow ring of powers of Y injects into cohomology.

Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point

Abstract The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6k + 1, k ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.

Deligne-Beilinson cohomology of the universal K3 surface

Abstract O’Grady’s generalised Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarised K3 surfaces. In [4], this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4th DB-cohomology group of universal oriented polarised K3 surfaces with at worst an $A_1$ -singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O’Grady’s original conjecture in DB cohomology.

Differential systems with algebraic and non-algebraic limit cycles explicitly given

Differential systems with algebraic and non-algebraic limit cycles explicitly given

Explicit Non Algebraic Limit Cycle for a Discontinuous Piecewise Differential Systems Separated by One Straight Line and Formed by Linear Center and Linear System Without Equilibria

Abstract In this paper, we deal with the discontinuous piecewise differential linear systems formed by two differential systems separated by a straight line when one of these two differential systems is a linear without equilibria and the other is a linear center. We are going to show that the maximum number of crossing limit cycles is one, and if exists, it is non algebraic and analytically given.

EXISTENCE AND COMPUTATION OF INVARIANT ALGEBRAIC CURVES FOR PLANAR QUADRATIC DIFFERENTIAL SYSTEMS

<abstract> Some necessary conditions are given for the existence of invariant algebraic curves for planar quadratic differential systems in a special canonical form. An efficient algorithm is then designed for computations of invariant algebraic curves. From the algorithm, a quadratic differential system is found with two Hopf bifurcations as the parameter varies, each leading to an invariant algebraic limit cycle of degree 5. A family of degree 6 invariant algebraic limit cycles is also produced. To further demonstrate the capability of the algorithm, we provide a quadratic system with a family of degree 7 invariant algebraic curves enclosing one or two centers, and a system possessing a degree 16 irreducible invariant algebraic curve with a singular point of multiplicity 8 on the curve. </abstract>

Chow rings of stacks of prestable curves I

Abstract We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21]. In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

Remarks on generating series for special cycles on orthogonal Shimura varieties

In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F:\Q|=d, of signature ((m,2)^{d_+},(m+2,0)^{d-d_+}), 1\le d_+<d. For each n, 1\le n\le m, there are special cycles Z(T) in S, of codimension nd_+, indexed by totally positive semi-definite matrices with coefficients in the ring of integers O_F. The generating series for the classes of these cycles in the cohomology group H^{2nd_+}(S) are Hilbert-Siegel modular forms of parallel weight m/2+1. One can form analogous generating series for the classes of the special cycles in the Chow group CH^{nd_+}(S). For d_+=1 and n=1, the modularity of these series was proved by Yuan-Zhang-Zhang. In this note we prove the following: Assume the Bloch-Beilinson conjecture on the injectivity of Abel-Jacobi maps. Then the Chow group valued generating series for special cycles of codimension nd_+ on S is modular for all n with 1\le n\le m.

Non-tautological Hurwitz cycles

We show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on {overline{{mathcal {M}}}}_{g,N}, assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds

AbstractThis article is about Lehn–Lehn–Sorger–van Straten eightfolds$Z$and their anti-symplectic involution$\iota$. When$Z$is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of$\iota$on the Chow group of$0$-cycles of$Z$. The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.