Wenchuan Hu
In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.
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Let $\otimes$ be the map which classifies the tensor product of two line bundles, an extension of this map to the space of all codimension 1 algebraic cycles is constructed. It is proved that this extension cannot exist in codimension greater than 1.
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $\leq 6$. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.
We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(¶^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d.
In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a simi…
We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.
In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat v…
In this text we are going to discuss the relation between monodromy and algebraic cycles.
In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the hom…
Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are r…
In this paper we show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg-MacLane spaces ${\cal K}_{2q} \equiv K({\Bbb Z},2) \times K({\Bbb Z}, 4) \times ... \times K({\Bbb Z}, 2q)$ have models which are limits of complex projective varieties. Precisely, we have ${\cal K}_{2q} = \lim_{d\to\infty}{\cal C}_d^q({\Bbb P}^n)$ where ${\cal C}_d^q({\Bbb P}^n)$ denotes the Chow variety of effective cycles of codimension $q$ and degree $d$ on ${\Bbb P}^n$. It is natural to ask which elements in the homology of ${\cal K}_{2q}…
We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian.
For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.
We construct algebraic virtual cycles that give us the cohomological field theories of Fan-Jarvis-Ruan invariants by integral transformations.
We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[(-1)*C] in JC modulo algebraic equivalence is torsion.
It is shown that to every Q-linear cycle \barαmodulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle αmodulo rational equivalence on A lying above \barα. The assignment \barα-> αrespects the algebraic operations and pullback and push forward along homomorphisms of abelian varieties.
Following an idea of Totaro, we prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of $\ell$-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.
Periods of join algebraic cycles
本文证明在特定维数和次数下，光滑超曲面模空间包含无穷多个与线性周期同余维数的 Hodge 轨迹，并构造了由 $0$ 维周期连接而成的 Hodge 周期，建立了连接周期类公式及 Artin Gorenstein 代数张量积刻画。
We characterize universally generalizing morphisms which satisfy descent of algebraic cycles integrally as those universally generalizing morphisms which are surjective with generically reduced fibres. In doing so, we introduce a naive pull-back of cycles for arbitrary morphisms between noetherian schemes, which generalizes the classical pull-back for flat morphisms, and then prove basic properties of this naive pull-back.
We study a formal deformation problem for rational algebraic cycle classes motivated by Grothendieck's variational Hodge conjecture. We argue that there is a close connection between the existence of a Chow-Künneth decomposition and the existence of expected deformations of cycles. This observation applies in particular to abelian schemes.
In this note we are going to consider a smooth projective surface equipped with an involution and study the action of the involution at the level of Chow group of zero cycles.
For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersur…
The motivic fundamental group of a punctured elliptic curve and algebraic cycles
本文在椭圆动机DG范畴中研究穿孔椭圆曲线的动机基本群，利用Schur复形给出其消解，并构造出类似于Bloch-Totaro闭链的代数闭链。
Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn-Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.
Using fundamental results of Deligne, we prove a nilpotence theorem for algebraic cycles and use this to prove a torsion nilpotence result for correspondences on surfaces.
We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.
This is a survey on unramified cohomology with a view towards its applications to rationality problems.
We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a "patch" -- a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.
We give a theorem of Leray-Hirsch type for Chow groups and use it to study the Hogde and Grothendieck's standard conjectures for algebraic fiber bundles of Leray-Hirsch type. Morevoer, the Hodge conjecture for product varieties will also be considered.
We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.
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 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.
This note is about an old conjecture of Voisin, which concerns zero--cycles on the self-product of surfaces of geometric genus one. We prove this conjecture for surfaces with $p_g=1$ and $q=2$.
We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.
In this short note, we simply collect some known results about representing algebraic cycles by various kind of "nice" (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. We also add a few elementary and easy observations on these representation problems that we were not able to locate in the literature.
We present a collection of algebraic equivalences between tautological cycles on the Jacobian $J$ of a curve, i.e., cycles in the subring of the Chow ring of $J$ generated by the classes of certain standard subvarieties of $J$. These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly.
We 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.
The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.
We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If $K/k$ is an algebraic extension, then the kernel of $A_1(X_k)\to A_1(X_K)$ is at most ${\mathbb Z}/2{\mathbb Z}$. If $k$ is finite, then the image equals the subgroup of Galois invariant cycles. This paper replaces Sections~2--3 of 2211.15915.v.1 and Sections~2--3 of 2211.15911.v.1. The other Sections are retained in the revis…
We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.
This note is about cycle-theoretic properties of the Fano variety of lines on a smooth cubic fivefold. The arguments are based on the fact that this Fano variety has finite-dimensional motive. We also present some results concerning Chow groups of Fano varieties of lines on certain cubics in other dimensions.
We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.
We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
In this paper, we shall give a candidate for the t-structure on the triangulated category of mixed motives due to Voevodsky.
In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations. We also provide the ph…
We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2…
This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.
This paper is for the proceedings of the Chen-Chow Conference held in Tianjin, China in October 2000. The goal of the paper is to produce and survey evidence for a connection between Chen's work on iterated integrals on the one hand, and algebraic cycles and motives on the other. The paper is expository, and begins with an introduction to Chen's work. Topics covered include loop space de Rham theorems; iterated integrals of currents; the works of Carlson-Clemmens-Morgan and Bruno Harris on periods of mixed Hodge structures on homotopy groups and algebraic cycles; the cobar construction, algebr…
In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…
We focus on Voisin's conjecture on 0-cycles on the self-product of surfaces of geometric genus one, which arises in the context of the Bloch-Beilinson filtration conjecture. We verify this conjecture for the family of Todorov surfaces of type $(2,12)$, giving an explicit description of this family as quotient surfaces of the complete intersection of four quadrics in $\mathbb{P}^{6}$. We give some motivic applications.
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.
In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.
We show that Gushel-Mukai fivefolds admit 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 these varieties injects into cohomology.
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.
We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).
In this text we prove that if a smooth cubic in $\PR^5$ has its Fano variety of lines birational to the Hilbert scheme of two points on a K3 surface, then there exists a smooth projective curve or a smooth projective surface embedded in the Fano variety, such that the kernel of the push-forward (at the level of zero cycles ) induced by the closed embedding is torsion.
We present a restricted version of some affine Jacobi's residue formula (on an affine algebraic variety) with applications to higher dimensional (and affine) analogues of Wood's (or Reiss's) relations about the interpolation of pieces of analytic manifolds. We also compare algebraic and analytic constructions of residue currents along analytic cycles and sketch some of their main properties. This work extend previous work of the authors in the affine space.
Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push forward along those morphisms X^l -> X^m for which each component X^l -> X is a projection. It is shown that these Q-vector subspaces are all finite-dimensional, provided that the Q-linear Chow motive of X is a direct summand of that of an abelian variety.
Motivated by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture about 0-cycles on self-products of surfaces of geometric genus one. We verify Voisin's conjecture for the family of Todorov surfaces with $K^2=2$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. As a by-product, we prove that certain Todorov surfaces have finite-dimensional motive.
Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of zeta functions associated with groups of cycles. We also consider an analogue in Arakelov geometry.
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