Infinitesimal deformation of Deligne cycle class map

TL;DR

This paper investigates the infinitesimal behavior of Deligne cycle class maps and confirms Beilinson's conjecture at the infinitesimal level, advancing understanding in algebraic geometry.

Contribution

It provides the first analysis of the infinitesimal form of Deligne cycle class maps and proves Beilinson's conjecture in this context, offering new insights.

Findings

Infinitesimal form of Deligne cycle class maps studied

Beilinson's conjecture verified at the infinitesimal level

Advances understanding of algebraic cycles and their classes

Abstract

In this note, we study the infinitesimal forms of Deligne cycle class maps. As an application, we prove that the infinitesimal form of a conjecture by Beilinson is true.

Equations

$$C{H}^p(X{)}_{\mathbb{Q}}\to C{H}^p(X_{\mathbb{C}}{)}_{\mathbb{Q}}\stackrel{r}{\to }{H}_{\mathcal{D}}^{2p}(X_{\mathbb{C}},\mathbb{Z}(p){)}_{\mathbb{Q}},$$

$$\psi :C{H}_{hom}^2(X)\to J^2(X)$$

$$C{H}^p(X)=H^p(X,K_p^M(O_X))\mathrm{modulo}\mathrm{torsion},$$

$$U\to K_p^M(O_X(U)).$$

$$H^p(X,K_p^M(O_{X[\epsilon ]}))\stackrel{\epsilon =0}{\to }{H}^p(X,K_p^M(O_X)).$$

$$\mathbb{Z}(p{)}_{\mathcal{D}}:0\to \mathbb{Z}(p)\to O_X\to \cdots \to {\Omega }_{X/\mathbb{C}}^{p-1}\to 0,$$

$$\mathbb{Z}(p{)}_{\mathcal{D}}[\epsilon ]:0\to \mathbb{Z}(p)\to O_{X[\epsilon ]}\to \cdots \to {\Omega }_{X[\epsilon ]/\mathbb{C}[\epsilon ]}^{p-1}\to 0,$$

$$\mathbb{Z}(p{)}_{\mathcal{D}}[\epsilon ]\stackrel{\epsilon =0}{\to }\mathbb{Z}(p{)}_{\mathcal{D}}.$$

$$\mathbb{Z}(p{)}_{\mathcal{D}}[\epsilon ]=\mathbb{Z}(p{)}_{\mathcal{D}}\oplus \overline{\mathbb{Z}}(p{)}_{\mathcal{D}}.$$

$$0\to O_X\to \cdots \to {\Omega }_{X/\mathbb{C}}^{p-1}\to 0,$$

$${\mathbb{H}}^{2p}(X,\mathbb{Z}(p{)}_{\mathcal{D}}[\epsilon ])\stackrel{\epsilon =0}{\to }{\mathbb{H}}^{2p}(X,\mathbb{Z}(p{)}_{\mathcal{D}}).$$

$$T_f{H}_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))={\mathbb{H}}^{2p}(X,\overline{\mathbb{Z}}(p{)}_{\mathcal{D}})={\mathbb{H}}^{2p-1}(X,\overline{\mathbb{Z}}(p{)}_{\mathcal{D}}[-1]).$$

$${\mathbb{H}}^{2p}(X,\overline{\mathbb{Z}}(p{)}_{\mathcal{D}})\cong H^{2p-1}(O_X)\oplus H^{2p-2}({\Omega }_{X/\mathbb{C}}^1)\oplus \cdots \oplus H^p({\Omega }_{X/\mathbb{C}}^{p-1}).$$

$$K_p^M(O_{X[\epsilon ]})\stackrel{\epsilon =0}{\to }{K}_p^M(O_X).$$

$$H^p(X,{\overline{K}}_p^M(O_X))\to {\mathbb{H}}^{2p}(X,\overline{\mathbb{Z}}(p{)}_{\mathcal{D}}),$$

$$\{f_1+\epsilon g_1,f_2,\cdots ,f_p\}$$

$$=\{1+\epsilon \frac{g_1}{f_1},f_2,\cdots ,f_p\}.$$

$$\frac{d_{\mathbb{C}[\epsilon ]}(1+\epsilon \frac{g_1}{f_1})}{1+\epsilon \frac{g_1}{f_1}}\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}$$

$$=\epsilon d_{\mathbb{C}}(\frac{g_1}{f_1})\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}$$

$$=\epsilon \frac{f_1{d}_{\mathbb{C}}{g}_1-g_1{d}_{\mathbb{C}}{f}_1}{f_1^2}\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}.$$

$$\{1+\epsilon \frac{g_1}{f_1},f_2,\cdots ,f_p\}\to \frac{f_1{d}_{\mathbb{C}}{g}_1-g_1{d}_{\mathbb{C}}{f}_1}{f_1^2}\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}.$$

$$\omega =\frac{g_1}{f_1}\frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}\in {\Omega }_{X/\mathbb{C}}^{p-1},$$

$$d_{\mathbb{C}}(\omega )$$

$$=\frac{(f_1{f}_2\cdots f_p)d_{\mathbb{C}}{g}_1-g_1{d}_{\mathbb{C}}(f_1{f}_2\cdots f_p)}{(f_1{f}_2\cdots f_p{)}^2}\wedge d_{\mathbb{C}}{f}_2\wedge \cdots \wedge d_{\mathbb{C}}{f}_p$$

$$=(\frac{d_{\mathbb{C}}{g}_1}{f_1{f}_2\cdots f_p}-\frac{g_1{f}_2\cdots f_p{d}_{\mathbb{C}}{f}_1}{(f_1{f}_2\cdots f_p{)}^2})\wedge d_{\mathbb{C}}{f}_2\wedge \cdots \wedge d_{\mathbb{C}}{f}_p.$$

$${\tilde{\wedge }}^{p}dlog(\{1+\epsilon \frac{g_1}{f_1},f_2,\cdots ,f_p\})=d_{\mathbb{C}}(\omega ),$$

$$\text{setcounter}MaxMatrixCols11\begin{array}{ccccccccc}{O}_X& \to & \cdots & \to & {\Omega }_{X/\mathbb{C}}^{p-2}& \to & {\Omega }_{X/\mathbb{C}}^{p-1}& \to & 0\\ {\alpha }_1\downarrow & & \downarrow & & {\alpha }_{p-1}\downarrow & & {\alpha }_p\downarrow & & \downarrow \\ O_X& \stackrel{-{\delta }_1}{\to }& \cdots & \stackrel{-{\delta }_{p-2}}{\to }& {\Omega }_{X/\mathbb{C}}^{p-2}& \stackrel{-{\delta }_{p-1}}{\to }& {\Omega }_{X/\mathbb{C}}^p\oplus {\Omega }_{X/\mathbb{C}}^{p-1}& \stackrel{-{\delta }_p}{\to }& {\Omega }_{X/\mathbb{C}}^{p+1}\oplus {\Omega }_{X/\mathbb{C}}^p\cdots ,\end{array}$$

$$\{1+\epsilon \frac{g_1}{f_1},f_2,\cdots ,f_p\}\to (-1{)}^{p-1}\omega ,$$

$$\frac{d_{\mathbb{C}}(1+\epsilon \frac{g_1}{f_1})}{1+\epsilon \frac{g_1}{f_1}}\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p}$$

$$=(\epsilon d_{\mathbb{C}}(\frac{g_1}{f_1})+\frac{g_1}{f_1}{d}_{\mathbb{C}}(\epsilon ))\wedge \frac{d_{\mathbb{C}}{f}_2}{f_2}\wedge \cdots \wedge \frac{d_{\mathbb{C}}{f}_p}{f_p};$$

$${\overline{K}}_p^M(O_X)\stackrel{{\wedge }^{p}dlog}{\to }{\Omega }_{X[\epsilon ]/\mathbb{C}}^p\stackrel{\rfloor \frac{\partial }{\partial \epsilon }{\mid }_{\epsilon =0}}{\to }{\Omega }_{X/\mathbb{C}}^{p-1},$$

$$0\to 0\to \cdots \to 0\to {\overline{K}}_p^M(O_X),$$

$$\text{setcounter}MaxMatrixCols11\begin{array}{ccccccccc}0& \to & 0& \to & \cdots & \to & 0& \to & {\overline{K}}_p^M(O_X)\\ 0\downarrow & & 0\downarrow & & 0\downarrow & & 0\downarrow & & \beta \downarrow (2.4)\\ 0& \to & O_X& \to & \cdots & \to & {\Omega }_{X/\mathbb{C}}^{p-2}& \to & {\Omega }_{X/\mathbb{C}}^{p-1}.\end{array}$$

Full text

1. 1 Introduction
2. 2 Main results

1. Introduction

In [1], Beilinson made the following conjecture:

Conjecture 1.1** (Conjecture 2.4.2.1 [1]).**

Let X be a smooth projective variety defined over a number field k, then for each positive integer $p$, the rational Chow group $CH^{p}(X)_{\mathbb{Q}}$ injects into Deligne cohomology of $X_{\mathbb{C}}$, where $X_{\mathbb{C}}:=X\times_{k}\mathbb{C}$. Concretely, if a class in $CH^{p}(X)_{\mathbb{Q}}$ vanishes in Deligne cohomology $H_{\mathcal{D}}^{2p}(X_{\mathbb{C}},\mathbb{Z}(p))_{\mathbb{Q}}$ under the composition

[TABLE]

where the right arrow $r$ is the cycle class map for Deligne cohomology, then it is 0.

This conjecture is very difficult to approach, and up to now there is not a single example with dimension $X\geq 2$ with large Chow ring $CH(X\times_{k}\mathbb{C})$ for which this conjecture has been verified.111See the first paragraph of page 1 [4]. Esnault and Harris [4] suggest a modest conjecture (see Theorem 0.1 [4]) which follows from Conjecture 1.1 and has been proved in a particular case (see Theorem 0.2 [4]) by using $l$-adic cohomology.

The main result of this note is to study the infinitesimal form of the Deligne cycles class map, see Lemma 2.9 and Theorem 2.11. As an application, we prove that the infinitesimal form of the Conjecture 1.1 is true, see Theorem 2.13.

In a companion paper [12], as a further application, we show that the infinitesimal form of the following conjecture (due to Griffiths-Harris) is true,

Conjecture 1.2** ([9]).**

Let $X\subset\mathbb{P}_{\mathbb{C}}^{4}$ be a general hypersurface of degree $d\geq 6$, we use

[TABLE]

to denote the Abel-Jacobi map from algebraic 1-cycles on $X$ homologically equivalent to zero to the intermediate Jacobian $J^{2}(X)$, $\psi$ is zero.

2. Main results

Let $X$ be a smooth projective variety defined over $\mathbb{C}$ and $\mathbb{C}[\varepsilon]/(\varepsilon^{2})$ be the ring of dual numbers, we use $X[\varepsilon]:=X\times\mathrm{Spec}(\mathbb{C}[\varepsilon]/(\varepsilon^{2}))$ to denote the first order infinitesimal deformation of $X$. The classical definition of Chow groups can not recognize nilpotents, so to overcome this deficiency, for each positive integer $p$, one uses the following Soulé’s variant of Bloch-Quillen identification to study the infinitesimal deformations of Chow groups,

[TABLE]

where $K^{M}_{p}(O_{X})$ is the Milnor K-theory sheaf associated to the presheaf

[TABLE]

Using the identification (2.1), one considers $H^{p}(X,K^{M}_{p}(O_{X[\varepsilon]}))$ as the first order infinitesimal deformation of $CH^{p}(X)$ and defines,

Definition 2.1**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the formal tangent space to $CH^{p}(X)$, denoted $T_{f}CH^{p}(X)$, is defined to be the kernel of the natural map

[TABLE]

It is known that the formal tangent space $T_{f}CH^{p}(X)$ can be identified with $H^{p}(X,\Omega_{X/\mathbb{Q}}^{p-1})$, where $\Omega_{X/\mathbb{Q}}^{p-1}$ is the absolute differential.

Deligne cohomology $H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$ is defined to be the hypercohomology of the Deligne complex $\mathbb{Z}(p)_{\mathcal{D}}$ (in analytic topology) :

[TABLE]

where $\mathbb{Z}(p)=(2\pi i)^{p}\mathbb{Z}$ is in degree 0. The infinitesimal deformation of this complex, denoted $\mathbb{Z}(p)_{\mathcal{D}}[\varepsilon]$, has the form,

[TABLE]

where $\mathbb{Z}(p)$ is still equal to $(2\pi i)^{p}\mathbb{Z}$.

Definition 2.2**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the tangent complex to the Deligne complex $\mathbb{Z}(p)_{\mathcal{D}}$, denoted $\overline{\mathbb{Z}}(p)_{\mathcal{D}}$, is defined to be the kernel of the natural map

[TABLE]

Since the map $X\to X[\varepsilon]$ has a retraction $X[\varepsilon]\to X$, $\Omega_{X[\varepsilon]/\mathbb{C}[\varepsilon]}^{i}=\Omega_{X/\mathbb{C}}^{i}\oplus\varepsilon\Omega_{X/\mathbb{C}}^{i}$, where $i=0,\cdots,p-1$. The tangent complex to the Deligne complex is a direct summand of the thickened Deligne complex

[TABLE]

One easily sees that

Lemma 2.3**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the tangent complex (to the Deligne complex $\mathbb{Z}(p)_{\mathcal{D}}$) $\overline{\mathbb{Z}}(p)_{\mathcal{D}}$ is of the form

[TABLE]

where $O_{X}$ is in degree 1 and $\Omega_{X/\mathbb{C}}^{p-1}$ is in degree $p$.

We consider the hypercohomology $\mathbb{H}^{2p}(X,\mathbb{Z}(p)_{\mathcal{D}}[\varepsilon])$ of the complex $\mathbb{Z}(p)_{\mathcal{D}}[\varepsilon]$ as the infinitesimal deformation of the Deligne cohomology $H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$ and define,

Definition 2.4**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the formal tangent space to the Deligne cohomology $H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$, denoted $T_{f}H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$, is defined to be the kernel of the natural map

[TABLE]

By the definition, the formal tangent space to the Deligne cohomology is the hypercohomology of the tangent complex $\overline{\mathbb{Z}}(p)_{\mathcal{D}}$:

[TABLE]

The following isomorphism is a standard fact in complex geometry,

Lemma 2.5** (cf. Proposition on page 17 of [6]).**

With the notations above, one has the isomorphism

[TABLE]

Let $\overline{K}^{M}_{p}(O_{X})$ denote the kernel of the natural map

[TABLE]

Since the map $X\to X[\varepsilon]$ has a retraction $X[\varepsilon]\to X$, $K^{M}_{p}(O_{X[\varepsilon]})=K^{M}_{p}(O_{X})\oplus\overline{K}^{M}_{p}(O_{X})$. By Definition 2.1, the tangent space $T_{f}CH^{p}(X)$ is $H^{p}(X,\overline{K}^{M}_{p}(O_{X}))$. Next, we would like to construct a map between tangent spaces

[TABLE]

which is the infinitesimal form of the Deligne cycle class map $CH^{p}(X)\to H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$.

An element of $\overline{K}^{M}_{p}(O_{X})$ is of the form $\prod_{i}\{f^{i}_{1}+\varepsilon g^{i}_{1},\cdots,f^{i}_{p}+\varepsilon g^{i}_{p}\}\in K^{M}_{p}(O_{X[\varepsilon]})$ such that $\prod_{i}\{f^{i}_{1},\cdots,f^{i}_{p}\}=1\in K^{M}_{p}(O_{X})$. We are reduced to looking at $\{f_{1}+\varepsilon g_{1},\cdots,f_{p}+\varepsilon g_{p}\}\in K^{M}_{p}(O_{X[\varepsilon]})$ such that $\{f_{1},\cdots,f_{p}\}=1\in K^{M}_{p}(O_{X})$. For simplicity, we assume that $g_{2}=\cdots=g_{p}=0$ and have

[TABLE]

Applying $\wedge^{p}dlog$ to $\{1+\varepsilon\dfrac{g_{1}}{f_{1}},f_{2},\cdots,f_{p}\}$, where $d=d_{\mathbb{C}[\varepsilon]}$, one obtains

[TABLE]

This gives a map from $\overline{K}^{M}_{p}(O_{X})$ to $\Omega_{X/\mathbb{C}}^{p}$,

Definition 2.6**.**

One defines a map $\tilde{\wedge}^{p}dlog:\overline{K}^{M}_{p}(O_{X})\to\Omega_{X/\mathbb{C}}^{p}$ by

[TABLE]

On the other hand, let

[TABLE]

one checks that

[TABLE]

Comparing $d_{\mathbb{C}}(\omega)$ with (2.2), one sees that

Lemma 2.7**.**

With the notations above, one has

[TABLE]

where $\omega=\dfrac{g_{1}}{f_{1}}\dfrac{d_{\mathbb{C}}f_{2}}{f_{2}}\wedge\cdots\wedge\dfrac{d_{\mathbb{C}}f_{p}}{f_{p}}$.

The following commutative diagram, which gives a quasi-isomorphism from the upper complex to the bottom one, is the tangent to the commutative diagram in Section 2.7 [5] (page 56),

[TABLE]

where $\alpha_{i}(x)=(-1)^{i-1}(x)$ for $1\leq i\leq p-1$ and $\alpha_{p}(x)=(-1)^{p-1}(d_{\mathbb{C}}x,x)$; $\delta_{i}(x)=dx$ for $1\leq i\leq p-2$, $\delta_{p-1}(x)=(0,dx)$ and $\delta_{p}(x,y)=(-dx,-x+dy)$. For $(d_{\mathbb{C}}\omega,\omega)\in\Omega_{X/\mathbb{C}}^{p}\oplus\Omega_{X/\mathbb{C}}^{p-1}$, there exists the unique $(-1)^{p-1}\omega\in\Omega_{X/\mathbb{C}}^{p-1}$ such that $\alpha_{p}((-1)^{p-1}\omega)=(d_{\mathbb{C}}\omega,\omega)$.

Definition 2.8**.**

One defines a map $\beta:\overline{K}^{M}_{p}(O_{X})\longrightarrow\Omega_{X/\mathbb{C}}^{p-1}$ by

[TABLE]

Where $\omega=\dfrac{g_{1}}{f_{1}}\dfrac{d_{\mathbb{C}}f_{2}}{f_{2}}\wedge\cdots\wedge\dfrac{d_{\mathbb{C}}f_{p}}{f_{p}}$.

We can see the map $\beta$ (2.4) in an alternative way 222We thank Spencer Bloch and Jerome Hoffman for comments. Firstly, applying $\wedge^{p}dlog$ to $\{1+\varepsilon\dfrac{g_{1}}{f_{1}},f_{2},\cdots,f_{p}\}$, where $d=d_{\mathbb{C}}$, one obtains

[TABLE]

Secondly, applying the truncation map $\rfloor\dfrac{\partial}{\partial\varepsilon}\mid_{\varepsilon=0}:\Omega_{X[\varepsilon]/\mathbb{C}}^{p}\to\Omega_{X/\mathbb{C}}^{p-1}$ to $(\varepsilon d_{\mathbb{C}}(\dfrac{g_{1}}{f_{1}})+\dfrac{g_{1}}{f_{1}}d_{\mathbb{C}}(\varepsilon))\wedge\dfrac{d_{\mathbb{C}}f_{2}}{f_{2}}\wedge\cdots\wedge\dfrac{d_{\mathbb{C}}f_{p}}{f_{p}}$, one obtains $(-1)^{p-1}\dfrac{g_{1}}{f_{1}}\dfrac{d_{\mathbb{C}}f_{2}}{f_{2}}\wedge\cdots\wedge\dfrac{d_{\mathbb{C}}f_{p}}{f_{p}}$. This shows that the composition

[TABLE]

agrees with the map $\beta$ (2.4).

The map $\beta$ (2.4) induces a map from the complex

[TABLE]

where $\overline{K}^{M}_{p}(O_{X})$ is in degree p, to the complex $\overline{\mathbb{Z}}(p)_{\mathcal{D}}$,

[TABLE]

This induces a map between (hyper)cohomology groups

[TABLE]

which is the infinitesimal form of the Deligne cycle class map

[TABLE]

By Lemma 2.4, $\mathbb{H}^{2p}(X,\overline{\mathbb{Z}}(p)_{\mathcal{D}})\cong H^{2p-1}(O_{X})\oplus\cdots\oplus H^{p}(\Omega_{X/\mathbb{C}}^{p-1})$ and by the diagram (2.5), we note the image of the map (2.6) lies in $H^{p}(\Omega_{X/\mathbb{C}}^{p-1})$. So the map (2.6) is indeed the composition:

[TABLE]

In summary,

Lemma 2.9**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the infinitesimal form of the Deligne cycle class map

[TABLE]

is given by (2.6) $\lambda:H^{p}(X,\overline{K}^{M}_{p}(O_{X}))\to\mathbb{H}^{2p}(X,\overline{\mathbb{Z}}(p)_{\mathcal{D}})$, which is the composition:

[TABLE]

Remark 2.10**.**

When we consider the infinitesimal deformation of the Deligne complex $\mathbb{Z}(p)_{\mathcal{D}}$, $\mathbb{Z}(p)$ is fixed so that it does not appear in the tangent complex $\overline{\mathbb{Z}}(p)_{\mathcal{D}}$. This explains why the construction of $\lambda$ (2.6) is simpler than the known construction [3, 5, 10] of the Deligne cycle class map.

In the following, we consider $H^{p}(X,\overline{K}^{M}_{p}(O_{X}))\to H^{p}(\Omega_{X/\mathbb{C}}^{p-1})$ as the infinitesimal form of the Deligne cycle class map $r:CH^{p}(X)\to H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))$, and describe it explicitly.

It is well known that $\overline{K}^{M}_{p}(O_{X})\xrightarrow{\cong}\Omega_{X/\mathbb{Q}}^{p-1}$ and the isomorphism is given by

[TABLE]

Moreover, one has the following commutative diagram,

[TABLE]

This shows the following diagram is commutative,

[TABLE]

Passing to cohomology groups, one has the following commutative diagram,

[TABLE]

To summarize,

Theorem 2.11**.**

Let $X$ be a smooth projective variety defined over $\mathbb{C}$, for each positive integer $p$, the infinitesimal form of the Deligne cycle class map

[TABLE]

is given by

[TABLE]

where $\delta r$ is induced by the natural map $\Omega_{X/\mathbb{Q}}^{p-1}\to\Omega_{X/\mathbb{C}}^{p-1}$.

Let $X$ be a smooth projective variety over $k$, where $k$ is a field of characteristic [math]. For each positive integer $p$, one still has the Soulé’s variant of Bloch-Quillen identification

[TABLE]

where $K^{M}_{p}(O_{X})$ is the Milnor K-theory sheaf associated to the presheaf

[TABLE]

The formal tangent space to $CH^{p}(X)$ is identified with $H^{P}(\Omega_{X/\mathbb{Q}}^{p-1})$.

Corollary 2.12**.**

Let $X$ be a smooth projective variety defined over $k$, where $k$ is a field of characteristic [math], for each positive integer $p$, the infinitesimal form of the composition

[TABLE]

has the form

[TABLE]

which is

[TABLE]

If $k$ is a number field, $\Omega_{X/\mathbb{Q}}=\Omega_{X/k}$ and $H^{p}(\Omega_{X/\mathbb{Q}}^{p-1})=H^{p}(\Omega_{X/k}^{p-1})$. By base change, $H^{p}(\Omega_{X_{\mathbb{C}}/\mathbb{C}}^{p-1})\cong H^{p}(\Omega_{X/k}^{p-1})\otimes_{k}\mathbb{C}$. The map (2.9) can be rewritten as

[TABLE]

which is obviously injective. In summary,

Theorem 2.13**.**

The infinitesimal form of Conjecture 1.1 is true. To be precise, let $X$ be a smooth projective variety defined over a number field $k$, for each positive integer $p$, the infinitesimal form of the composition

[TABLE]

in Conjecture 1.1 is of the form,

[TABLE]

which is injective.

Conjecture 1.1 is part of the Bloch-Beilinson conjecture. Let $X$ be a smooth projective variety over $\mathbb{C}$, for each positive integer $p$, Bloch-Beilinson conjecture predicts that there is a filtration which has the form

[TABLE]

The first two steps are known and $F^{2}CH^{p}(X)_{\mathbb{Q}}$ is the kernel of the Deligne cycle class map

[TABLE]

An alternative way to state Conjecture 1.1 is,

Conjecture 2.14** ( cf. Implication 1.2 [7] page 478).**

If $X$ is a smooth projective variety over a number field $k$, then

[TABLE]

where $F^{2}CH^{p}(X)_{\mathbb{Q}}$ is the filtration induced from $F^{2}CH^{p}(X_{\mathbb{C}})_{\mathbb{Q}}$ under the natural map $CH^{p}(X)\to CH^{p}(X_{\mathbb{C}})$.

Theorem 2.13 suggests that this Conjecture(and Conjecture 1.1) looks reasonable at the infinitesimal level.

Remark 2.15**.**

The assumption“$k$ is a number field” in Theorem 2.13 is crucial, it guarantees $\Omega_{X/\mathbb{Q}}=\Omega_{X/k}$. This suggests that the assumption“$k$ is a number field” in Conjecture 1.1 can not be loosened. In fact, for the ground field of transcendental degree one, Green-Griffiths-Paranjape [8] has found counterexamples, extending earlier examples by Bloch, Nori and Schoen.

To understand algebraic cycles, the transcendental degree of the ground field does matter.

Acknowledgements. This note is inspired by Hélène Esnault’s talk on [4] at Tsinghua University (December 2017). The author sincerely thanks Spencer Bloch, Hélène Esnault, Jerome Hoffman and Jan Stienstra for discussions. Jerome Hoffman has read a preliminary version, his suggestions improve this note a lot. This work is partially supported by the Fundamental Research Funds for the Central Universities.