Bloch's conjecture for some numerical Campedelli surfaces

Bloch's conjecture for the surface of Craighero and Gattazzo

The goal of this paper is to prove Bloch's conjecture for the numerical Godeaux surface constructed by P. Craighero and R. Gattazzo.

The Generalized Bloch Conjecture for the quotient of certain Calabi-Yau varieties

In this paper, the generalized Bloch Conjecture on zero cycles for the quotient of certain complete intersections with trivial canonical bundle is proved to hold. As an application of Bloch-Srinivas method on the decomposition of the diagonal, we compute the rational coefficient Lawson homology for 1-cycles and codimension two cycles for these quotient varieties. The (Generalized) Hodge Conjecture is proved to hold for codimension two cycles (and hence also for 2-cycles) on these quotient varieties.

Bloch-type conjectures and an example of a threefold of general type

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.

Bloch's conjecture for Enriques varieties

Enriques varieties have been defined as higher-dimensional generalizations of Enriques surfaces. Bloch's conjecture implies that Enriques varieties should have trivial Chow group of zero-cycles. We prove this is the case for all known examples of irreducible Enriques varieties of index larger than $2$. The proof is based on results concerning the Chow motive of generalized Kummer varieties.

Bloch's conjecture for Generalized Burniat Type surfaces with $p_g=0$

The aim of this article is to prove Bloch's conjecture, asserting that the group of rational equivalence classes of zero cycles of degree 0 is trivial for surfaces with geometric genus zero, for regular generalized Burniat type surfaces. The technique is the method of "enough automorphisms" introduced by Inose-Mizukami in a simplified version due to the first author.

Involutions on algebraic surfaces and the Generalised Bloch's conjecture

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.

Bloch's conjecture on certain surfaces of general type with $p_g=0$ and with an involution: the Enriques case

In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0$, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is an Enriques surface and show that the Bloch's conjecture holds for such surfaces.

Bloch's Conjecture and Chow Motives

Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences vanishes. Furthermore the cube of the ideal of homologically trivial correspondences is zero if these equivalent conditions are satisfied (e.g. if X is not of general type).

Bloch's conjecture for Inoue surfaces with p_g=0, K^2 =7

The aim of this article is to prove Bloch's conjecture (asserting that the group of rational equivalence classes of zero cycles of degree zero is trivial) for Inoue surfaces with p_g=0 and K^2 = 7. These surfaces can also be described as bidouble covers of the four nodal cubic, which allows to use the method of "enough automorphisms" introduced by Inose-Mizukami (in a simplified version).

Bloch's conjecture and valences of correspondences for K3 surfaces

Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $Γ$ acts as 0 on the Albanese kernel $T(X_Ω)$, where $Ω$ is a universal domain containing $k$. Here we prove that, for a complex K3 surface $X$, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphisms of finite order and to the existence of constant cycle curves on $X$. Finally we show that Franchetta's conjecture, as stated by K.…

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".

A note on the Bloch-Beilinson conjecture for K3 surfaces and spherical objects

For a projective K3 surface X we introduce the dense triangulated subcategory S^* of the bounded derived category D^b(Coh(X)) of coherent sheaves on X that is generated by spherical objects. For a K3 surface X over \bar Q it is shown that S^* admits a bounded t-structure if and only if the Bloch-Beilinson conjecture holds for X, i.e. CH^2(X)=Z.

The $\ell$-adic trace formula for dg-categories and Bloch's conductor conjecture

Building on recent results by A. Blanc, M. Robalo and the authors, we present an $\ell$-adic trace formula for smooth and proper dg-categories over a base $\mathbb{E}_{\infty}$-algebra $B$. We also give a variant when $B$ is only an $\mathbb{E}_2$-algebra. As an application of this trace formula, we propose a strategy of proof of Bloch's conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.

Bloch猜想在具有3阶自同构的伪射影平面上的证明

Bloch's conjecture for fake projective planes with an automorphism of order 3

本文证明了对某些一般型曲面上的对合在零维Chow群上作用为恒等，从而验证了Bloch猜想，并推广到某些伪射影平面。

Chow groups of conic bundles in $\mathbb P^5$ and the Generalised Bloch's conjecture

Consider the Fano surface of a conic bundle embedded in $\mathbb P^5$. Let $i$ denote the natural involution acting on this surface. In this note we provide an obstruction to the identity action of the involution on the group of algebraically trivial zero cycles modulo rational equivalence on the surface.

K3曲面上的（反）自等价性的布洛赫猜想

Bloch's conjecture for (anti-)autoequivalences on K3 surfaces

本文通过引入扭曲K3曲面的反射自等价性概念并利用Cartan-Dieudonné型分解，证明了皮卡数至少为3的K3曲面上所有（反）辛自等价性的布洛赫猜想，并将其推广到超凯勒簇的双有理自同构及反辛对合情形。

关于Goncharov猜想与Bloch-Kriz混合塔克动机的co-Lie代数关系的论点

On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

研究通过混合塔克动机的co-Lie代数定义从Goncharov群B_n(F)到动机多项式对数的线性映射，支持Beilinson和Soulé关于字段K-群消失的猜想。

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for abelian varieties

We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $γ$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to th…

Bloch's conjecture for Catanese and Barlow surfaces

Catanese surfaces are regular surfaces of general type with $p_g=0$. They specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result for the Barlow surfaces. We will also give a conditional application (more precisely, assuming the variational Hodge conjecture) of the same method to the Chow motive of low degree $K3$ surfaces.

扭曲阿贝尔曲面间等价的Bloch猜想及其应用

Bloch's conjecture for equivalences between twisted abelian surfaces and applications

本文通过扭曲阿贝尔曲面构造自等价在Albanese核上的自然作用，证明所有（反）辛自等价的Bloch猜想，并应用于扭曲模Kum_n型簇的辛双有理自同构，同时引入Shen-Yin-Zhao型滤子并与Voisin滤子比较。

On a multiplicative version of Bloch's conjecture

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Kunneth decomposition for these varieties.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II

We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.

Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups

We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a…

$\mathbb{A}^1$-equivalence of zero cycles on surfaces II

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.

强BSD对于abelian曲面和Bloch-Beilinson猜想

Strong BSD for abelian surfaces and the Bloch-Beilinson conjecture

本文在BSD成立的前提下，证明了特定abelian曲面上的Bloch-Beilinson猜想。

On the splitting of the Bloch-Beilinson filtration

For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Nam…

有效零闭链与Bloch-Beilinson滤过

Effective zero-cycles and the Bloch-Beilinson filtration

本文将Voisin关于两点有理等价的猜想推广到高次有效零闭链，在阿贝尔簇与hyper-Kähler流形上给出证据，并对K3曲面上半稳定层模空间给出显式判据，同时提出关于对角线所属对应子代数的第二个猜想。

Equivariant Bloch-Kato conjecture and non-abelian Iwasawa Main Conjecture

This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture.

A refined Beilinson-Bloch conjecture for motives of modular forms

We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.

Bloch猜想在$p_g=q=0, K^2=9$曲面上的成立

Bloch's conjecture on surfaces with $p_g=q=0, K^2=9$

本文证明了Bloch猜想对满足$p_g=q=0$且$K^2=9$的所有光滑复射影曲面成立，给出了该猜想在特定数值条件下的完整证明。

混合特征下光滑与半稳定概形的Bloch-Ogus理论

Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

本文研究了混合特征下光滑与半稳定概形上对偶同调理论的Bloch-Ogus理论与Gersten猜想，证明了光滑情形下的猜想及半稳定情形的特例，并由此导出优秀离散赋值环上局部环的Galois符号映射的满射性。

On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…

Bloch's conjecture revisited

Let $X$ be a non-singular projective complex surface. We can show that Bloch's conjecture (i.e., that if $p_g=0$ then the Albanese kernel vanishes) is equivalent to the following statement: If $p_g(X)=0$ then for any given Zariski open $U\subset X$ and $ω\in H^2(U,{\bf C})$ there is a smaller Zariski open $V\subset U$ such that $$ω\mid_V =ω'+ζ$$ where $ω'\in F^2H^2(V,{\bf C})$ and $ζ$ is integral.

Bloch's formula for 0-cycles with modulus and higher dimensional Class Field Theory

We prove Bloch's formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves. This was originally solved by Kerz and Saito in characteristic $\neq 2$.

Rationality problems and conjectures of Milnor and Bloch-Kato

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified étale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detec…

Beilinson's Hodge conjecture for smooth varieties

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to ΓH^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.

Some Calabi-Yau fourfolds verifying Voisin's conjecture

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning the behaviour of zero-cycles on self-products of Calabi-Yau varieties. This note contains some examples of Calabi-Yau fourfolds verifying Voisin's conjecture.

关于函数域上的贝利逊-布洛赫猜想

On the Beilinson-Bloch conjecture over function fields

本文研究了函数域上的贝利逊-布洛赫猜想，通过扩展贾恩森的结果并证明塔特猜想，推导出该猜想的特殊情形，并回答了Moonen关于Chow群的问题。

Goncharov's relations in Bloch's higher Chow group CH^3(F,5)

Using Totaro-Bloch-Kriz's linear fractional cycles Gangl and Muller-Stach recently prove the 5-term relations for the dilogarithm in Bloch's higher Chow group CH^2(F,3) and the Kummer-Spence relations in some group G(F) over an arbitrary field F where G(F) is isomorphic to CH^3(F,5) up to torsions under the Beilinson-Soule vanishing conjecture that CH^2(F,n)=0 for n>3. In this paper we show that Goncharov's 22-term relations for the trilogarithm also hold in G(F).

Finite dimensional motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself implies finite dimensionality of M(X). We also show (Theorem 27) that, for a surface X with trivial geometrical genus, the motive M(X) is finite …

Some results on a conjecture of Voisin for surfaces of geometric genus one

Inspired by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture concerning the behaviour of 0-cycles on self-products of varieties of geometric genus one. This note presents some new examples of surfaces for which Voisin's conjecture is verified.

On the Chow groups of Plücker hypersurfaces in Grassmannians

Motivated by the generalized Bloch conjecture, we formulate a conjecture about the Chow groups of Plücker hypersurfaces in Grassmannians. We prove weak versions of this conjecture.

非全纯变种上的一些Beilinson-Bloch猜想

The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields

本文证明了非全纯变种上Beilinson-Bloch猜想，并推导出中间Jacobians上的Birch-Swinnerton-Dyer猜想，同时推广了有限域上的Tate猜想。

On Voisin's conjecture for zero-cycles on hyperkaehler varieties

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.

On algebraic Chern classes of flat vector bundles

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex projective varities but in some cases they give a more precise information. We also study Higgs version of Bloch's conjecture and analogous problems in the positive characteristic case.

Hecke特征标与Picard曲面的Eisenstein上同调的Bloch-Beilinson猜想

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

本文研究Hecke特征标与Picard曲面的Eisenstein上同调之间的Bloch-Beilinson猜想，通过构造Hodge结构延伸并计算双延伸高度来验证其非平凡性。

A case of the deformational Hodge conjecture via a pro Hochschild-Kostant-Rosenberg theorem

Following ideas of Bloch, Esnault, and Kerz, the deformational part of Grothendieck's variational Hodge conjecture is established for proper, smooth schemes over $K[[t]]$, where $K$ is an algebraic extension of the rational numbers. The main tool is a pro Hochschild-Kostant-Rosenberg theorem for Hochschild homology.

The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjecture

We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating serie…

Zagier's conjecture on $L(E,2)$

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups $K_2(E)$ and $K_1(E)$ for an elliptic curve $E$ over an arbitrary field $k$. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on $L(E,2)$ for modular elliptic curves over $\Bbb Q$.

Algebraic cycles and very special cubic fourfolds

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.

q-Terms, singularities and the extended Bloch group

Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \cite{Ko}. We introduce the notion of a $q$-hypergeometric term (in short, $q$-term). The latter is a product of ratios of $q$-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods…

A remark on the Chow ring of Sicilian surfaces

We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for Sicilian surfaces.

Algebraic cycles on Todorov surfaces of type $(2,12)$

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.

An infinitesimal approach to the study of Cycles on Abelian Varieties

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.