Let $X$ be a product of smooth projective curves over a finite unramified extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has good reduction and that $X$ has a $k$-rational point. We propose the following conjecture. The kernel of the Albanese map $CH_0(X)^0\rightarrow\text{Alb}_X(k)$ is $p$-divisible. When $p$ is an odd prime, we prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidence for a local-to-global conjecture for zero-cycles of Colliot-Thélène and Sansuc (\cite{Colliot-Thelene/Sansuc1981}), and Kato and Saito (\cite{Kato/Saito1986}).
If $X$ is an abelian variety over a field and $L$ is an invertible sheaf, we know that the degree of the 0-cycle $L^g$ is divisible by $g!$. As a 0-cycle, it is not, even over a field of cohomological dimension 1. But we show that over a finite field there is perhaps some hope.
The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kah…
In this paper, we establish the following criterion for divisibility in the local ring of those quasianalytic function germs at zero which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two such function germs is that of their Taylor series at zero in the formal power series ring.
We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these res…
Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The sequence {D_{nR}) for n \ge 1 is called the ``elliptic divisibility sequence of R.'' Let P,Q in E(k(T)) be independent points. We conjecture that deg (gcd(D_{nP},D_{mQ})) is bounded for m,n \ge 1, and that gcd(D_{nP},D_{nQ}) = gcd(D_{P},D_{Q}) for infinitely many n \ge 1. We prove these conjectures in the case that j(E) is in k. More generally, we prove analo…
Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is surjective. We relate this restriction map to the $p$-adic cycle class map.
We study the Chow group of zero-cycles on singular varieties using the cdh topology. We define the cdh versions of the zero-cycles and albanese maps. We prove results comparing these groups for a singular variety with the similar groups on the resolution of singularities. We use these to prove some results about the known Chow group of zero-cycles on surfaces and threefolds and some cases of arbitrary dimension.
In this paper, we study $\mathbb{A}^1$-equivalence classes of zero cycles on open complex algebraic surfaces. We prove the logarithmic version of Mumford's theorem on zero cycles and prove that log Bloch's conjecture holds for quasiprojective surfaces with log Kodaira dimension $-\infty$.
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.
We study local-global principles for zero-cycles on K3 surfaces defined over number fields. We follow an idea of Liang to use the trivial fibration over the projective line.
Given a smooth variety $X$ and an effective Cartier divisor $D \subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\rm CH}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|…
We construct a cycle class map from the higher Chow groups of 0-cycles to the relative $K$-theory of a modulus pair. We show that this induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and relative $K$-theory of truncated polynomial rings over a regular semi-local ring, essentially of finite type over a characteristic zero field.
We prove the existence of a canonical zero-cycle on a Calabi-Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, n=dim X, is proportional to the class of a point on a rational curve in X.
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.
We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups over local and finite fields.
We study the deformations of the Chow group of zero-cycles of the special fibre of a smooth scheme over a henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization theorem for thickened zero-cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.
We study the restriction map to the closed fiber of a regular projective scheme over an excellent henselian discrete valuation ring, for a cohomological version of the Chow group of relative zero-cycles. Our main result extends the work of Saito--Sato to general perfect residue fields.
We use the elements in $K$-cohomology groups which are constructed by Flach and Mildenhall to obtain a finiteness result for the torsion part of the Chow group of a self-product of a modular curve.
We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.
We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Thélène for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our approach also allows to reprove the ramified global class field theory of Kato and Saito. Finally, we apply the Kato conjectures to study the $p$-adic cycle class map over henselian discrete valuation rings of mixed characteristic and to deduce finiteness theorems for arithmetic schemes in low degree.
Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi-Yau varieties. We reformulate Voisin's conjecture in the setting of hyperkähler varieties, and we prove this reformulated conjecture for one family of hyperkähler fourfolds.
Garbagnati has constructed certain surfaces of general type that are bidouble planes as well as double covers of K3 surfaces. In this note, we study the Chow groups (and Chow motive) of these surfaces.
An old conjecture of Voisin describes how $0$-cycles of a surface $S$ should behave when pulled-back to the self-product $S^m$ for $m>p_g(S)$. We exhibit some surfaces with large $p_g$ that verify Voisin's conjecture.
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin's conjecture.
For a generalized Kummer variety X of dimension 2n, we will construct for each 0 < i < n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch-Beilinson filtration for generalized Kummer varieties.
Let $G$ be a smooth connected linear algebraic group and $X$ be a $G$-torsor. Totaro asked: if $X$ admits a zero-cycle of degree $d \geq 1$, then does $X$ have a closed étale point of degree dividing $d$? While the literature contains affirmative answers in some special cases, we give an example to show that the answer is negative in general.
In this paper we extend Gazaki's results on the Chow groups of abelian varieties to the higher Chow groups. We introduce a Gazaki type filtration on the higher Chow group of zero-cycles on an abelian variety, whose graded quotients are connected to the Somekawa type $K$-group. Via the étale cycle map, we will compare this filtration with a filtration on the étale cohomology induced by the Hochschild-Serre spectral sequence. As an application over local fields, we obtain an estimate of the kernel of the reciprocity map.
We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Bloch's conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.
For any separated algebraic space $X/S$ we construct a separated algebraic space $Γ^d(X/S)$ -- the space of divided powers -- which parameterizes zero cycles of degree $d$ on $X$. The space of divided powers for an affine scheme is given by the spectrum of the algebra of divided powers. In characteristic zero or when $X/S$ is flat, the constructed space coincides with the symmetric product $Sym^d(X/S)$. We also prove several fundamental results on the kernels of multiplicative polynomial laws necessary for the construction of $Γ^d(X/S)$.
The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First we prove a Beauville-Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilb…
The paper has two parts. First we prove that the specialization maps on R-equivalence and on the Chow group of zero cycles are isomorphisms for families over a local, Henselian, Dedekind ring when the special fiber is smooth and separably rationally connected. This is pretty much the best one can hope for. The second part contains examples of unirational quartic hypersurfaces whith infinitely many R-equivalence classes. There are also examples of quartics with points in a large odd degree field extension but no points in any smaller odd degree extension.
Using the Gille-Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer's theorem), for twisted forms of maximal orthogonal Grassmannians (theorem of Bayer-Fluckiger and Lenstra), for E6- (Rost's theorem), and E7-varieties.
In this paper, we apply incidence divisors constructed through Archimedean height paring to prove that Griffiths' conjecture on incidence equivalence is correct for a smooth projective variety with first non-vanishing cohomology. (Incidence divisor is insignificant. Theorem 5.1 is significant, but the proof of it presented here is incorrect.)
An old conjecture of Voisin describes how $0$-cycles on a surface $S$ should behave when pulled-back to the self-product $S^m$ for $m>p_g(S)$. We show that Voisin's conjecture is true for a $3$-dimensional family of surfaces of general type with $p_g=q=2$ and $K^2=7$ constructed by Cancian and Frapporti, and revisited by Pignatelli-Polizzi.
We show that given a smooth projective variety X over C with dim(X) > 2, an ample line bundle O(1) on X and an integer n > 1, any n distinct points on a generic hypersurface of degree d in X are linearly independent in CH_0(X) if d >> 0. This generalizes a result of C. Voisin.
Under some hypotheses on the singular type of the one-parameter family of elliptic curves in a primitively polarized $K3$ surface $S$ determined by its polarization (which is expected to be true for a very general polarized $K3$ surface), we give a more geometric proof of the fact that the second Chern class of $S$ is equal to $24 \cdot o_S$ in the Chow group of $0$-cycles where $o_S$ is the Beauville-Voisin canonical $0$-cycle.
For $Y \subset X$ a locally complete intersection of codimension p, Spencer Bloch [2] constructed the semi-regularity map $π: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(Ω_{X/k}^{p-1})$. As an analogue, we construct a map $\tildeπ: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(Ω_{X/\mathbb{Q}}^{p-1})$, without assuming local complete intersections. While the semi-regularity map $π$ is expected to be injective, we show $\tildeπ$ is a zero map. We use this zero map to interpret how to eliminate obstructions to deforming cycles, an idea by Mark Green and Phillip Griffiths in [9].
Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all $\mathbb{Q}$-homology planes and for open smooth surfaces which are not of log general type.
We study a conjecture, due to Voisin, on 0-cycles on varieties with $p_g=1$. Using Kimura's finite dimensional motives and recent results of Vial's on the refined (Chow-)Künneth decomposition, we provide a general criterion for Calabi-Yau manifolds of dimension at most $5$ to verify Voisin's conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to $5$.
This paper is a work in progress on Bloch's conjecture asserting the vanishing of the Pontryagin product of a $ p $ codimensional cycle on an abelian variety by $ p+1 $ zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss, in particular, the case of $ 3 $ dimensional cycles.
We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sa…
We prove an extension of the Kato-Saito class field theory for smooth projective schemes over a finite field to schemes with singularities. As an application, we obtain Bloch's formula for the Chow groups of 0-cycles on such schemes. We identify the Chow group of 0-cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasi-projective schemes over algebraically closed fields. This completes the missing $p$-torsion part in the torsion theorem of Spiess and Szamuely.
In this paper, we study the Brauer-Manin pairing of smooth proper varieties over local fields, and determine the $p$-adic part of the kernel of one side. We also compute the $A_0$ of a potentially rational surface which splits over a wildly ramified extension.
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.
We show the surjectivity of a specialisation map on higher $(0,1)$-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture stated in an article of Kerz, Esnault and Wittenberg saying that base change holds for such schemes in general for motivic cohomology in degrees $(i,d)$ for fixed $d$ being the relative dimension over the base. Furthermore, the specialisation map we study is related to a finiteness conjecture for the $n$-torsion of $CH_0(X)$, where $X$ is a variety over a $p$-adic field.
The Beauville-Voisin conjecture predicts the existence of a filtration on projective hyper-Kähler manifolds opposite to the conjecture Bloch-Beilinson filtration, called the Beauivlle-Voisin filtration. Voisin has introduced a filtration on zero cycles of an arbitrary projective hyper-Kähler manifold. On moduli space of stable objects of a projective K3 surface, there are other candidates constructed by Shen-Yin-Zhao, Barros-Flapan-Marian-Silversmith and more recently by Vial from different point of views. According to the work of Vial, all of them are proved to be equivalent except Voisin's f…
Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O'Grady's filtration on the $\mathrm{CH}_0$-group of the $K3$ surface. This solves a conjecture of O'Grady and improves on previous results of Huybrechts, O'Grady, and Voisin. Then we propose a cand…
