# Flat pushforwards of Chern classes and the smoothing problem for cycles in
the Whitney range

Claire Voisin111The author is supported by the ERC Synergy Grant HyperK (Grant
agreement No. 854361).

###### Abstract

We investigate the smoothability of cycles in the Whitney range, that is, when
the dimension is strictly smaller than the codimension. Introducing the notion
of “flat pushforwards of Chern classes” and studying its properties, we prove
that cycles of dimension $d\leq 7$ are smoothable in a smooth variety of
dimension $>2d$, and more generally that $(d-6)!Z$, $d={\rm dim}\,Z$, is
smoothable in a variety of dimension $>2d$. We also prove that cycles of any
dimension on homogeneous varieties are smoothable in the Whitney range. We
prove more generally in all these cases that the considered cycles are flat
pushforwards of intersections of divisors on a smooth projective variety.

##  1 Introduction

Let $X$ be a smooth projective variety of dimension $n$. Following [8], we
will say that a cycle class $z\in{\rm CH}_{d}(X)$ is smoothable if it belongs
to the subgroup of $\rm{CH}_{d}(X)$ generated by the classes of
$d$-dimensional smooth subvarieties of $X$. A number of nonsmoothability
results have been proved outside of the Whitney range, that is, when $2d\geq
n$, since the question of smoothability was first raised by Borel and
Haefliger [2] for cohomology classes. Hartshorne, Rees and Thomas [8] proved
the nonsmoothability of the second Chern class $c_{2}(\mathcal{E})$ of the
tautological subbundle $\mathcal{E}$ on a Grassmannian $G(3,n),\,n\geq 6$.
Debarre [5] proved the nonssmoothability of the minimal class
$\frac{\theta^{2}}{2}$ on a very general Jacobian of curve of genus $\geq 7$,
where $\theta$ is the class of a Theta-divisor. This class of examples has
been greatly expanded in [3]. Benoist [1] exhibits examples of nonsmoothable
$d$-cycles on varieties of dimension $n$ for many possible pairs $(d,n)$
outside of the Whitney range, including the case where $2d=n$ under some
condition on the codimension $c=n-d$. In the Whitney range where $2d<n$,
Hironaka [9] proves that for any cycle $z\in{\rm CH}_{d}(X)$ of codimension
$c$, the cycle $(c-1)!z$ is smoothable, a result that has been improved later
on by Kleiman [10], who also includes the case where $2d=n$.

For the cycles themselves (as opposed to a multiple), Hironaka proved in 1968
the following result.

######  Theorem 1.1.

(Hironaka [9]) Cycles of dimension $d\leq 3$ are smoothable on smooth
varieties of dimension $n>2d$.

Our goal in this paper is to study the smoothability problem for cycles in the
Whitney range, which is mentioned in the introduction of [1] and that we state
as

######  Conjecture 1.2.

Let $X$ be a smooth projective variety of dimension $n$, defined over a field
of characteristic $0$. Then for any integer $d$ such that $2d<n$, any cycle
$z\in\rm{CH}_{d}(X)$ is smoothable.

For any smooth variety $X$, we denote by ${\rm CH}^{*}(X)_{\rm Ch}\subset{\rm
CH}^{*}(X)$ the subring generated by Chern classes of vector bundles on $X$.
The standard formula (7) combined with locally free resolutions shows that

$(c-1)!{\rm CH}^{c}(X)\subset{\rm CH}^{c}(X)_{\rm Ch}.$

However, it is well-known that ${\rm CH}^{*}(X)_{\rm Ch}$ can be a proper
subring of ${\rm CH}^{*}(X)$. We refer to [4] for an explicit example and to
Lemma 3.7 for another example. As discussed in Section 3.2, further examples
are provided by homogeneous varieties $G/H$, where $G$ is a semi-simple,
simply connected group, $H$ is a Borel subgroup and the torsion order of $G$
is not $1$ (I am grateful to Burt Totaro for explaining this to me). Cycles in
${\rm CH}_{d}(X)_{\rm Ch}$ are relevant for our subject, since, as proved by
Hironaka in [9], they are smoothable under the Whitney condition $2d<{\rm
dim}\,X$. (We will give in Section 2 an argument which involves Segre classes
and seems slightly different from Hironaka’s argument in [9].)

Let us now introduce further definitions.

######  Definition 1.3.

Let $X$ be smooth. We will denote by ${\rm CH}(X)_{\rm flpshCh}$ (for “flat
pushforward of Chern classes”), resp. ${\rm CH}(X)_{\rm smpshCh}$ (for “smooth
pushforward of Chern classes”) the subgroup generated by cycles of the form
$\pi_{*}z^{\prime}$ for a flat, resp. smooth, proper morphism
$\pi:P\rightarrow X$, with $P$ smooth, and for some cycle $z^{\prime}\in{\rm
CH}(P)_{\rm Ch}$.

Lemma 2.4 proved in Section 3.1 says that ${\rm CH}^{*}(X)_{\rm Ch}\subset{\rm
CH}^{*}(X)$ is generated by smooth pushforwards of classes of complete
intersections of divisors, so that, in Definition 1.3, we could replace “cycle
in ${\rm CH}(P)_{\rm Ch}$” by “cycle in the subring of ${\rm CH}^{*}(P)$
generated by divisors”. Further easy properties are discussed in Section 3.1.

We prove in Section 2 the following basic

######  Proposition 1.4.

(Cf. Proposition 2.9) Cycles in ${\rm CH}_{d}(X)_{\rm flpshCh}$ are smoothable
in the Whitney range $2d<{\rm dim}\,X$.

The analogous result for cycles in ${\rm CH}_{d}(X)_{\rm smpshCh}$ is a
standard Whitney type statement. Proposition 1.4 is our motivation to
introduce Definition 1.3 and study the following

######  Conjecture 1.5.

For any smooth projective variety $X$, we have

${\rm CH}(X)_{\rm flpshCh}={\rm CH}(X),$

that is, any cycle on $X$ is the push-forward, under a flat morphism
$f:Y\rightarrow X$ from a smooth projective variety $Y$, of a class
$z^{\prime}\in{\rm CH}(Y)_{\rm Ch}$.

We prove in this paper Conjecture 1.5 for cycles of dimension $\leq 7$. More
generally, we prove in Section 3.2 the following

######  Theorem 1.6.

Let $X$ be a smooth projective variety. Then for any integer $d\geq 0$,

$\displaystyle(d-6)!{\rm CH}_{d}(X)\subset{\rm CH}_{d}(X)_{\rm flpshCh},$ (1)

(where $(d-6)!:=1$ if $d\leq 6$). In particular ${\rm CH}_{d}(X)_{\rm
flpshCh}={\rm CH}_{d}(X)$ for $d\leq 7$.

One gets from Theorem 1.6 and Proposition 2.9 the following improvement and
generalization of Hironaka’s Theorem 1.1.

######  Corollary 1.7.

For any smooth variety $X$ of dimension $n$, cycles in $(d-6)!{\rm CH}_{d}(X)$
are smoothable when $2d<n$. In particular, cycles of dimension $d\leq 7$ are
smoothable if $2d<n$.

######  Remark 1.8.

In [1, Theorem 0.3], Benoist produces examples of cycles of dimension 6 and
codimension 6 that are not smoothable. Corollary 1.7 shows that, at least for
dimension 6 and codimension 6, his examples are optimal and the Whitney
condition is necessary for smoothability.

Theorem 1.6 is obtained as a consequence of the following Theorem 1.9. We will
use here and in the rest of the paper the terminology of generalized complete
intersection in $X$, for “closed algebraic subset of codimension $c$ which is
the zero-set of a section of a vector bundle of rank $c$ on $X$”.

######  Theorem 1.9.

(i) (Cf. Proposition 3.9.) Given a finite morphism $j:Y\rightarrow X$, where
$Y$ and $X$ are smooth projective and ${\rm dim}\,Y={\rm dim}\,X-1$, one has

$j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}.$

(ii) (Cf. Proposition 3.12.) Let $X$ be a smooth projective variety and let
$j:Y\hookrightarrow X$ be the inclusion of a smooth subvariety which is a
generalized complete intersection in $X$. Then $j_{*}({\rm CH}(Y)_{\rm
flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}$.

(iii) (Cf. Proposition 3.14.) Let $X$ be smooth projective and let $Y\subset
X$ be a smooth generalized complete intersection in $X$. Let
$\tau:\widetilde{X}\rightarrow X$ be the blow-up of $X$ along $Y$. Then

$\tau_{*}({\rm CH}(\widetilde{X})_{\rm flpshCh})\subset{\rm CH}(X)_{\rm
flpshCh}.$

This theorem (together with the trivial Lemmas 3.3 and 3.4) shows that the
subgroup ${\rm CH}(X)_{\rm flpshCh}\subset{\rm CH}(X)$ is stable under many
operations and this is a strong evidence for Conjecture 1.5. For example, we
have the following

######  Corollary 1.10.

Let $X$ be smooth and let $Z\subset X$ be a smooth subvariety which is a
connected component of a smooth generalized complete intersection $W$ in $X$.
Then $[Z]\in{\rm CH}(X)_{\rm flpshCh}$.

###### Proof.

Indeed, we blow-up $X$ along $W$, getting $\tau:X^{\prime}\rightarrow X$. Let
$E_{Z}\subset X^{\prime}$ be the exceptional divisor over $Z$. Theorem
1.9(iii) tells us that $\tau_{*}({\rm CH}(X^{\prime})_{\rm
flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}$. As we have
$[Z]=\pm\tau_{*}(E_{Z}^{c})$, where $c:={\rm codim}\,Z$, it follows that
$[Z]\in{\rm CH}(X)_{\rm flpshCh}$. ∎

The following corollary shows that the conjectures 1.5 and 1.2 for $d$-cycles
are implied by the following statement.

(*) Let $X$ be a smooth projective variety and $Z\subset X$ be a smooth
projective subvariety of codimension $\geq 3$. For a general hypersurface
$Y\subset X$ containing $Z$, there exist an integer $r$, a smooth projective
variety $X^{\prime}$ and a morphism $\tau:X^{\prime}\rightarrow
X_{\sharp}:=X\times\mathbb{P}^{r}$ constructed by successive blow-ups along
smooth generalized complete intersections centers, such that the proper
transform $Y^{\prime}$ of $Y=Y\times{\rm pt}\subset X_{\sharp}$ is smooth and
contains a smooth subvariety $Z^{\prime}$ birational to $Z\times{\rm pt}$ via
$\tau$.

Indeed, we will prove

######  Corollary 1.11.

(See also Proposition 3.15) If the statement (*) is true for any $X$ and $Z$
of dimension $d$, Conjecture 1.5 holds for cycles of dimension $d$.

The proof of Theorem 1.6 will be obtained by establishing statement (*) when
the codimension of $Z$ in $X$ is at least $d-4$. In this case, the singular
locus of $Y$ has dimension at most $4$ and this is used in our proof.

The methods used to prove Theorems 1.6 and 1.9 do not allow us to prove the
stronger result that ${\rm CH}_{d}(X)={\rm CH}_{d}(X)_{\rm smpshCh}$ for
$d\leq 7$. In particular, the proof of the main Proposition 3.9 (Theorem
1.9(i) above) does not work if we replace the groups ${\rm CH}_{d}(Y)_{\rm
flpshCh}$ and ${\rm CH}_{d}(X)_{\rm flpshCh}$ respectively by ${\rm
CH}_{d}(Y)_{\rm smpshCh}$ and ${\rm CH}_{d}(X)_{\rm smpshCh}$. This leaves
open the following

######  Question 1.12.

Are there smooth projective varieties $X$ such that ${\rm CH}(X)\not={\rm
CH}(X)_{\rm smpshCh}$?

As follows from Theorem 1.13 below, the equality ${\rm CH}(X)={\rm CH}(X)_{\rm
smpshCh}$ is satisfied by abelian varieties but it could be that for the
example treated in Lemma 3.7, or for some homogeneous varieties, we have ${\rm
CH}_{0}(X)\not={\rm CH}(X)_{\rm smpshCh}$.

Our second main result is a proof of Conjectures 1.2 and 1.5 for homogeneous
varieties.

######  Theorem 1.13.

Let $X$ be a homogeneous projective variety. Then

(1) We have ${\rm CH}_{d}(X)={\rm CH}_{d}(X)_{\rm flpshCh}$ for any $d$. In
particular

(2) Any cycle $z\in{\rm CH}_{d}(X)$ with $2d<{\rm dim}\,X$ is smoothable.

(3) If $A$ is an abelian variety, then ${\rm CH}_{d}(A)={\rm CH}_{d}(A)_{\rm
smpshCh}$ for any $d$.

As above, item (2) follows from (1) by Proposition 2.9 since it says that
cycles in ${\rm CH}_{d}(X)_{\rm flpshCh}$ are smoothable when $2d<{\rm
dim}\,X$.

Theorem 1.13(1) is proved as part of Theorem 4.1. We also prove that, if there
exists a smooth projective $G$-equivariant completion $\overline{G}$ of $G$
which satisfies ${\rm CH}_{0}(\overline{G})={\rm CH}_{0}(\overline{G})_{\rm
smpshCh}$, then

$\displaystyle{\rm CH}_{i}(X)_{\rm smpshCh}={\rm CH}_{i}(X)$ (2)

for all $i$. We do not know if the condition above is always satisfied. The
stronger version that there always exists a smooth projective $G$-equivariant
completion $\overline{G}$ of $G$ which satisfies

$\displaystyle{\rm CH}_{0}(\overline{G})={\rm CH}_{0}(\overline{G})_{\rm Ch}$
(3)

is wrong for abelian varieties. For simply connected groups, the condition (3)
seems to be closely related to the torsion order of $G$ being $1$ (see [6] and
Section 3.2), but the precise relation is not obvious to me.

Theorem 1.13(3) will be proved in Section 4.1 and the proof of Theorem
1.13(1), which uses Theorem 1.6 for $0$-cycles, will be given in Section 4.2.
In the last section, we will give a few complements; for example we prove
Proposition 5.1 which states the stability of ${\rm CH}(\cdot)_{\rm flpshCh}$
under the inclusion map of a smooth codimension 2 subvariety. We also
establish in Theorem 5.4 a criterion for smoothability that is weaker than
being in ${\rm CH}(\cdot)_{\rm flpshCh}$ (in the Whitney range).

Thanks. I thank Olivier Benoist and Olivier Debarre for introducing me to this
subject and for interesting discussions, and Michel Brion, Laurent Manivel,
Nicolas Perrin and Burt Totaro for their help with homogeneous varieties.

##  2 Whitney type statements

We first prove the following basic Whitney type result

######  Proposition 2.1.

Let $\phi:Y\rightarrow X$ be a flat morphism between smooth varieties. Let
$n={\rm dim}\,X$. Then for a smooth subvariety $Z\subset Y$ of dimension
$d<\frac{n}{2}$ which is in general position and such that the restriction
$\phi_{\mid Z}$ is proper, $\phi_{\mid Z}:Z\rightarrow\phi(Z)$ is an
isomorphism, so the closed algebraic subset $\phi(Z)\subset X$ is smooth.
Furthermore, if $n=2d$, $\phi_{\mid Z}$ is an immersion and the image
$\phi(Z)$ has finitely many singular points.

Although this will be clear from the proof, let us first make precise what we
mean by “in general position”. This is a transversality condition with respect
to $\phi$ and its infinitesimal properties. For the proposition above, it is
enough that $Z$ is the general fiber $\mathcal{Z}_{b}$ of a family

$\displaystyle\begin{matrix}\mathcal{Z}&\stackrel{{\scriptstyle
f}}{{\rightarrow}}&Y\\\ p\downarrow\,\,\,\,\,\,&&&\\\ B&&&\end{matrix}$ (4)

where $\mathcal{Z}$ is smooth and $p$ is smooth, which is very mobile at any
point $(x,y)$ of $Z\times Z\cong\mathcal{Z}_{b}\times Z_{b}$, $x\not=y$, (that
is $f:\mathcal{Z}\rightarrow Y$ is a submersion at any point of $Z$,
$(f,f):\mathcal{Z}\times_{B}\mathcal{Z}\rightarrow Y\times Y$ is a submersion
at any point $(x,y)$ of $Z\times Z\setminus\Delta_{Z}$) and whose tangent
space is mobile at any point of $Z$, that is, the morphism

$F:\mathbb{P}(T_{\mathcal{Z}/B})\rightarrow\mathbb{P}(T_{Y})$

given by the differential of the inclusions $f_{b}:\mathcal{Z}_{b}\rightarrow
Y$, is submersive at any point of $Z$. The important fact for us is the
following

######  Remark 2.2.

Assuming $Y\subset\mathbb{P}^{N}$ is projective of dimension $m$, these
general position properties will be satisfied by a general complete
intersection of $m-d$ very ample divisors.

For the proof of Proposition 2.1, we will use the following consequence of the
“general position” assumption.

######  Lemma 2.3.

Let $Z\subset Y$ be a smooth subvariety of dimension $d$ which is in general
position in the above sense. Then,

(i) If $W\subset Y$ is a closed algebraic subvariety of codimension $>d$, $Z$
does not intersect $W$.

(ii) If $W\subset Y\times Y$ is a subvariety of codimension $>2d$, $Z\times Z$
does not intersect $W$ away from the diagonal of $Z$.

(iii) If $W\subset\mathbb{P}(T_{Y})$ is a subvariety of codimension $\geq 2d$,
$\mathbb{P}(T_{Z})$ does not intersect $W$.

###### Proof.

(i) We use the notations of (4), with $Z=\mathcal{Z}_{b}$, $b\in B$ being a
general point of $B$. As $f$ is a submersion along $\mathcal{Z}_{b}$, there
exists a Zariski neighborhood $U$ of $\mathcal{Z}_{b}$ in $\mathcal{Z}$ such
that any component of $f^{-1}(W)\cap U$ has codimension $>d$ in $U$. As ${\rm
dim}\,B={\rm dim}\,\mathcal{Z}-d$, it follows that

$p_{\mid f^{-1}(W)\cap U}:f^{-1}(W)\cap U\rightarrow B$

cannot be dominant, hence for a general $b\in B$, $\mathcal{Z}_{b}$ does not
intersect $f^{-1}(W)$, that is, $Z=f(\mathcal{Z}_{b})$ does not intersect $W$.

(ii) The argument is the same as above with $f$ replaced by
$(f,f):\mathcal{Z}\times_{B}\mathcal{Z}\rightarrow Y\times Y$. The fibers of
$(p,p):\mathcal{Z}\times_{B}\mathcal{Z}\rightarrow B$ are now of dimension
$2d$ and $(f,f)$ is a submersion away from the diagonal of $\mathcal{Z}_{b}$,
so if $W\subset Y\times Y$ has codimension $>2d$, $(f,f)^{-1}(W)$ will have
codimension $>2d$ in
$\mathcal{Z}\times_{B}\mathcal{Z}\setminus\Delta_{\mathcal{Z}}$, and will not
dominate $B$, since ${\rm dim}\,B={\rm
dim}\,(\mathcal{Z}\times_{B}\mathcal{Z})-2d$.

(iii) The argument is the same as above except that we work now with
$F:\mathbb{P}(T_{\mathcal{Z}/B})\rightarrow\mathbb{P}(T_{Y})$. We simply
observe that the fibers of the natural map
$\mathbb{P}(T_{\mathcal{Z}/B})\rightarrow B$ are of dimension $2d-1$. ∎

###### Proof of Proposition 2.1.

Let $\Delta_{Y}\subset Y\times Y$ be the diagonal and let $Y^{\prime}\subset
Y\times Y\setminus\Delta_{Y}$ be the closed algebraic subset
$Y\times_{X}Y\setminus\Delta_{Y}$. By flatness of $\phi$, we have ${\rm
codim}\,(Y^{\prime}\subset Y\times Y)=n$. As ${\rm dim}\,Z\times Z=2d<n$ and
$Z$ is in general position, $Z\times Z$ does not intersect $Y^{\prime}$ away
from the diagonal by Lemma 2.3(ii), so $\phi_{\mid Z}$ is injective. When
$n=2d$, $Z\times Z$ intersects $Y^{\prime}$ away from the diagonal in at most
finitely many points. It remains to prove the infinitesimal statement, for
which we only assume that $2d\leq n$. Let $Y_{k}\subset Y$ be the locally
closed subset of $Y$ where the rank of $\phi$ is equal to $k$. Then ${\rm
dim}\,\phi(Y_{k})\leq k$, hence by flatness, ${\rm codim}\,(Y_{k}\subset
Y)\geq n-k$, or equivalently ${\rm dim}\,Y_{k}\leq m+k$, where
$m:={\dim}\,Y-n$. In particular, for $k<d$, we get ${\rm codim}\,(Y_{k}\subset
Y)\geq n-k>n-d\geq d$, hence a $Z$ in general position does not intersect
$Y_{k}$ by Lemma 2.3(i). In general, we observe the following: Along $Y_{k}$,
we have the rank $k$ morphism

$\phi_{k}:=(\phi_{*})_{\mid Y_{k}}:T_{Y\mid
Y_{k}}\rightarrow(\phi^{*}T_{X})_{\mid Y_{k}}$

with kernel a subbundle $\mathcal{K}_{k}\subset T_{Y\mid Y_{k}}$ of corank
$k$. Let $W\subset\mathbb{P}(T_{Y})$ be the set of pairs $(y,u),\,y\in
Y,\,u\in{\rm Ker}\,\phi_{*,y}$. The stratification of $Y$ by the $Y_{k}$’s
describes $W$ as a union

$W=\sqcup_{k}\mathbb{P}(\mathcal{K}_{k}).$

As ${\rm dim}\,Y_{k}\leq k+m$ and ${\rm rk}\,\mathcal{K}_{k}=m+n-k$, we get

${\rm dim}\,\mathbb{P}(\mathcal{K}_{k})\leq 2m+n-1$

for any $k$, and thus ${\rm dim}\,W\leq 2m+n-1$. As ${\rm
dim}\,\mathbb{P}(T_{Y})=2(m+n)-1$, it follows that

${\rm codim}\,(W\subset\mathbb{P}(T_{Y}))\geq n.$

By Lemma 2.3(iii), $Z$ being of dimension $d$ and in general position with
$n\geq 2d$, $\mathbb{P}(T_{Z})$ does not intersect $W$. This means that
$\phi_{\mid Z}$ is an immersion, which concludes the proof. ∎

We will combine Proposition 2.1 with the following easy result.

######  Lemma 2.4.

Let $X$ be smooth of dimension $n$ and let $z\in{\rm CH}_{d}(X)$ be a cycle.
Assume that $z$ belongs to the subring ${\rm CH}^{*}(X)_{\rm Ch}$ of ${\rm
CH}^{*}(X)$ generated by Chern classes $c_{i}(E)$ for any coherent sheaf $E$
on $X$. Then there exist a smooth projective variety $Y$ and a smooth morphism
$f:Y\rightarrow X$ such that $z=f_{*}z^{\prime}$ in ${\rm CH}(X)$, where
$z^{\prime}\in{\rm CH}(Y)$ belongs to the subring generated by divisors on
$Y$.

###### Proof.

First of all, using finite locally free resolutions and the Whitney formula,
we know that $z$ belongs to the subring of ${\rm CH}^{*}(X)$ generated by the
Chern classes $c_{i}(E)$ for any locally free coherent sheaf $E$ on $X$.
Secondly, we can replace in this statement the Chern classes by the Segre
classes, since the total Segre and Chern classes $s(E)$ and $c(E)$ satisfy the
relation

$s(E)=c(E)^{-1},\,\,c(E)=s(E)^{-1},$

so any polynomial with integral coefficients in the Segre classes is a
polynomial with integral coefficients in the Chern classes and vice-versa.

It thus suffices to prove that any monomial $s_{i_{1}}(E_{1})\ldots
s_{i_{k}}(E_{k})\in{\rm CH}(X)$, where the $E_{i}$’s are locally free sheaves
on $X$ of rank $r_{i}$, satisfies the conclusion of Lemma 2.4. This statement
follows from the definition of Segre classes (see [7]). Indeed, let
$\pi_{i}:\mathbb{P}(E_{i})\rightarrow X$ be the projectivization of $E_{i}$
and let $H_{i}\in{\rm Pic}(\mathbb{P}(E_{i}))$ with first Chern class
$c_{1}(H_{i})\in{\rm CH}^{1}(\mathbb{P}(E_{i}))$ be the dual of its Hopf line
bundle (so that $R^{0}\pi_{i*}H_{i}=E_{i}^{*}$). Then

$\displaystyle s_{j}(E_{i})=\pi_{i*}(c_{1}(H_{i})^{j+r_{i}-1})\,\,{\rm
in}\,\,{\rm CH}(X).$ (5)

It follows from (5) and the projection formula that

$\displaystyle s_{i_{1}}(E_{1})\ldots s_{i_{k}}(E_{k})=\pi_{*}({\rm
pr}_{1}^{*}c_{1}(H_{1})^{i_{1}+r_{1}-1}\ldots{\rm
pr}_{k}^{*}c_{1}(H_{k})^{i_{k}+r_{k}-1})\,\,{\rm in}\,\,{\rm CH}(X),$ (6)

where
$\pi:\mathbb{P}(E_{1})\times_{X}\ldots\times_{X}\mathbb{P}(E_{k})\rightarrow
X$ is the fibred product of the $\pi_{i}:\mathbb{P}(E_{i})\rightarrow X$ and
${\rm pr}_{i}$ is the projection from
$\mathbb{P}(E_{1})\times_{X}\ldots\times_{X}\mathbb{P}(E_{k})$ to its $i$-th
factor. ∎

######  Corollary 2.5.

Let $X$ be smooth of dimension $n$ and let $z\in{\rm CH}_{d}(X)_{\rm Ch}$,
with $2d<n$. Then $z$ is smoothable, that is, $z$ is rationally equivalent to
a cycle $Z^{\prime}=\sum_{i}n_{i}Z^{\prime}_{i}$, where $Z^{\prime}_{i}\subset
X$ is smooth.

###### Proof.

Using Lemma 2.4, the result follows from Proposition 2.1 by Remark 2.2. ∎

######  Corollary 2.6.

(Hironaka [9], Kleiman [10]) If $X$ is smooth and $2d<{\rm dim}\,X$, any cycle
$z\in{\rm CH}_{d}(X)_{\mathbb{Q}}$ is rationally equivalent to a smooth cycle
with $\mathbb{Q}$-coefficients. More precisely $(c-1)!z$ is smoothable, where
$c:=n-d$ is the codimension of $z$.

###### Proof.

Indeed, it suffices to prove the result when $z=[Z]$ is the class of a
subvariety $Z$ of $X$ of dimension $d$. Let $\mathcal{O}_{Z}$ be the
structural sheaf of $Z$, seen as a coherent sheaf on $X$. It follows from the
Grothendieck-Riemann-Roch formula (see [7, Example 15.3.1]) that

$\displaystyle c_{n-d}(\mathcal{O}_{Z})=(-1)^{n-d-1}(n-d-1)![Z]\,\,\in\,\,{\rm
CH}_{d}(X)_{\rm Ch}\subset{\rm CH}_{d}(X),$ (7)

so Corollary 2.5 applies. ∎

######  Remark 2.7.

In [9], which does not use Segre classes but the splitting principle to reduce
Chern classes to products of divisors, the coefficient $(c-1)!$ appears
multiplied by a constant, which is possibly $1$.

######  Remark 2.8.

Kleiman in [10] argues differently by studying singularities of Schubert
varieties and proves a result which is of a different nature, as it also
includes the case where $n=2d$, which is not in the Whitney range.

Combining Lemma 2.4 and Proposition 2.1, we get the following criterion

######  Proposition 2.9.

(a) Let $\phi:Y\rightarrow X$ be a proper flat morphism between smooth
varieties. Then for any cycle $z\in{\rm CH}_{d}(Y)_{\rm Ch}$ with $2d<n={\rm
dim}\,X$, the class $z^{\prime}=\phi_{*}z$ is smoothable on $X$.

(b) The same conclusion holds if we only assume that, away from a closed
algebraic subset $Y_{0}\subset Y$ of codimension $>d$, $Y$ is smooth and
$\phi$ is flat.

###### Proof.

(a) By Lemma 2.4, the cycles $z$ as above are of the form $\pi_{*}z^{\prime}$
for a smooth proper morphism $\pi:P\rightarrow Y$ and for some cycle
$z^{\prime}\in{\rm CH}(P)$ which is a combination with integral coefficients
of intersections of divisors on $P$. By Remark 2.2, Proposition 2.9 applies to
$z^{\prime}$ and the flat morphism $\phi\circ\pi:P\rightarrow X$, proving the
statement.

(b) We do the same construction as before and observe that, since
$\pi:P\rightarrow Y$ is smooth and proper, the singular locus of $P$ and the
locus where $\phi\circ\pi$ is not flat are contained in $\pi^{-1}(P_{0})$,
which has codimension $>d$. A $d$-dimensional complete intersection of
divisors in general position in $P$ does not intersect $P_{0}$, hence is
contained in $P\setminus P_{0}$, and we can apply Proposition 2.1 to
$(\phi\circ\pi)_{\mid P\setminus P_{0}}:P\setminus P_{0}\rightarrow X$. ∎

##  3 Flat pushforwards of Chern classes

###  3.1 Comments on Definition 1.3

We start with the following remarks on Definition 1.3.

######  Remark 3.1.

By Lemma 2.4, in the definition of ${\rm CH}(X)_{\rm flpshCh}$ and ${\rm
CH}(X)_{\rm smpshCh}$, we can replace “intersections of divisors on $P$” by
“elements of ${\rm CH}(P)_{\rm Ch}$”. Indeed, if $Y$ is smooth,
$p:Y\rightarrow X$ is a proper flat morphism, and $z\in{\rm CH}_{d}(Y)_{\rm
Ch}$, there exist by Lemma 2.4 a smooth variety $P$ and a smooth morphism
$p^{\prime}:P\rightarrow Y$ such that

$z=p^{\prime}_{*}(w)\,\,{\rm in}\,\,{\rm CH}_{d}(Y),$

where $w$ belongs to the subring of ${\rm CH}(P)$ generated by divisors. The
morphism $p\circ p^{\prime}:P\rightarrow X$ is flat and projective, and
$z=(p\circ p^{\prime})_{*}w$.

######  Remark 3.2.

If $f:Y\rightarrow X$ is a flat (resp. smooth) morphism between smooth
projective varieties, one has $f_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm
CH}(X)_{\rm flpshCh}$, resp. $f_{*}({\rm CH}(Y)_{\rm smpshCh})\subset{\rm
CH}(X)_{\rm smpshCh}$.

Let us now establish a few elementary facts.

######  Lemma 3.3.

(a) One has ${\rm CH}(X)_{\rm Ch}\subset{\rm CH}(X)_{\rm smpshCh}\subset{\rm
CH}(X)_{\rm flpshCh}$.

(b) The subgroup ${\rm CH}(X)_{\rm smpshCh}$ is a subring of ${\rm CH}(X)$.

(c) The subgroup ${\rm CH}(X)_{\rm flpshCh}$ is a module over the ring ${\rm
CH}(X)_{\rm smpshCh}$.

(d) If we work over an algebraically closed field of characteristic $0$, (in
particular over $\mathbb{C}$), we have ${\rm CH}(X)_{\rm alg}\subset{\rm
CH}(X)_{\rm Ch}$, where ${\rm CH}(X)_{\rm alg}$ denotes the group of cycles
algebraically equivalent to $0$. The definition 1.3 thus makes sense for the
groups of cycles modulo algebraic equivalence instead of the Chow groups.

For example, in the case of $0$-cycles, the condition that $z$ belongs to
${\rm CH}_{0}(X)_{\rm flpshCh}$ or ${\rm CH}_{0}(X)_{\rm smpshCh}$ depends
only on the degree of $z$, assuming $X$ is projective and connected and the
field is algebraically closed of characteristic $0$.

###### Proof of Lemma 3.3.

(a) The second inclusion is obvious since smoothness implies flatness.

(b) and (c) Let $p_{1}:P_{1}\rightarrow X$, $p_{2}:P_{2}\rightarrow X$ be
proper morphisms with $P_{1},\,P_{2},\,X$ smooth and assume $p_{1}$ is smooth,
$p_{2}$ is flat. Then $P_{12}:=P_{1}\times_{X}P_{2}$ is smooth. If $Z_{1}$,
resp. $Z_{2}$ are intersections of divisors on $P_{1}$, resp. $P_{2}$, their
pull-backs $Z^{\prime}_{1}$, resp. $Z^{\prime}_{2}$ to $P_{12}$ via the
projections

$p^{\prime}_{1}:P_{12}\rightarrow P_{2},\,\,p^{\prime}_{2}:P_{12}\rightarrow
P_{1}$

are also intersections of divisors, and the projection formula gives

$p_{1*}Z_{1}\cdot p_{2*}Z_{2}=p_{12*}(Z^{\prime}_{1}\cdot
Z^{\prime}_{2})\,\,{\rm in}\,\,{\rm CH}(X),$

where $p_{12}:P_{1}\times_{X}P_{2}\rightarrow X$ is the natural morphism. This
proves (b) and (c) since $p_{12}$ is flat and it is smooth if $p_{2}$ is
smooth.

(d) Indeed, it suffices by (a) to prove that ${\rm CH}^{p}(X)_{\rm
alg}\subset{\rm CH}^{p}(X)_{\rm Ch}$. By formula (7), we have $(p-1)!{\rm
CH}^{p}(X)\subset{\rm CH}^{p}(X)_{\rm Ch}$ for any smooth $X$ and any closed
algebraic subset $Z\subset X$ of codimension $p$. It follows that

${\rm CH}^{p}(X)_{\rm alg}\subset{\rm CH}^{p}(X)_{\rm Ch}$

since the group ${\rm CH}^{p}(X)_{\rm alg}$ of cycles algebraically equivalent
to zero is divisible, hence contained in $(p-1)!{\rm CH}^{p}(X)$, if we are
over an algebraically closed field of characteristic $0$. ∎

Another useful lemma is the following

######  Lemma 3.4.

Let $\phi:Y_{1}\rightarrow Y_{2}$ be a morphism, with $Y_{1},\,Y_{2}$ smooth.
Then

(i) One has

$\displaystyle\phi^{*}({\rm CH}(Y_{2})_{\rm smpshCh})\subset{\rm
CH}(Y_{1})_{\rm smpshCh},$ (8)

(ii) If $\phi$ is smooth, then

$\displaystyle\phi^{*}({\rm CH}(Y_{2})_{\rm flpshCh})\subset{\rm
CH}(Y_{1})_{\rm flpshCh}.$ (9)

###### Proof.

Let $\psi:W\rightarrow Y_{2}$ be a flat (resp. smooth) projective morphism.
Then

$\psi_{1}:W_{1}:=W\times_{Y_{2}}Y_{1}\rightarrow Y_{1}$

is flat (resp. smooth). Furthermore, if either $\psi$ is smooth (Case (i)) or
$\phi$ is smooth (Case (ii)), $W_{1}$ is smooth.

Let $\phi_{1}:W_{1}\rightarrow W$ be the first projection. If $\gamma\in{\rm
CH}(W)_{\rm Ch}$, we have $\phi_{1}^{*}\gamma\in{\rm CH}(W_{1})_{\rm Ch}$, and
furthermore we have by [7, Proposition 1.7]

$\psi_{1*}(\phi_{1}^{*}\gamma)=\phi^{*}(\psi_{*}\gamma)\,\,{\rm in}\,\,{\rm
CH}(Y_{1}).$

This proves (8) and (9). ∎

######  Remark 3.5.

It is not clear if we can remove the assumption on $\phi$ in (ii). The problem
is to guarantee the smoothness of the fibred product. In Proposition 4.2, we
show how to go around this problem in some cases.

###  3.2 Some cases of Conjecture 1.5

We will give in this section the proof of Theorem 1.6, which implies in
particular Conjecture 1.5 for cycles of dimension $\leq 7$. We first note the
following

######  Corollary 3.6.

Conjecture 1.5 is satisfied by smooth varieties of dimension $\leq 10$.

###### Proof.

If $Z\subset X$ is a subvariety of codimension $p$ of a smooth variety, one
has the formula (7)

$c_{p}(\mathcal{O}_{Z})=(-1)^{p-1}(p-1)![Z]\,\,{\rm in}\,\,{\rm CH}^{p}(X),$

hence for $p\leq 2$, the class $[Z]\in{\rm CH}^{p}(X)$ belongs to ${\rm
CH}^{p}(X)_{\rm Ch}$. As cycles of dimension $d\leq 7$ belong to ${\rm
CH}_{d}(X)_{\rm flpshCh}$ by Theorem 1.6, we conclude that ${\rm CH}(X)={\rm
CH}(X)_{\rm flpshCh}$ when ${\rm dim}\,X\leq 10$ since this forces the cycles
to have either dimension $\leq 7$ or codimension $\leq 2$. ∎

Theorem 1.6 is interesting when the considered cycles do not belong to ${\rm
CH}(X)_{\rm Ch}$. Besides the case of $0$-cycles on very general abelian
varieties with high degree polarization (see [4]), some hypersurfaces in
projective space provide such examples. For example, we have

######  Lemma 3.7.

Let $X\subset\mathbb{P}^{4}$ be a very general hypersurface of degree $64$.
Then the class of a point $x\in X$ does not belong to ${\rm CH}_{0}(X)_{\rm
Ch}$. More precisely, for any vector bundle $E$ on $X$, the degree ${\rm
deg}\,c_{3}(E)$ is divisible by $2$.

###### Proof.

Let $E$ be a vector bundle of rank $r$ on $X$. By the Hirzebruch-Riemann-Roch
formula, the holomorphic Euler-Poincaré characteristic of $E$ is given by the
formula

$\displaystyle\chi(X,E)=\int_{X}{\rm Ch}(E){\rm To}(X)=\alpha c_{3}(E)+\beta
c_{2}(E)c_{1}(E)+\gamma c_{2}(E)c_{1}(X)+q(E),$ (10)

where the constants $\alpha,\,\beta,\,\gamma,\,\delta,\,\zeta$ independent of
$E$ are rational and the quantity $q(E)$ is the part of the Riemann-Roch
polynomial (in the Chern classes of $E$) which involves only $c_{1}(E)$ and
the rank of $E$, and is an integer since it is equal to

$\int_{X}(r-1){\rm To}_{3}(X)+{\rm Ch}({\rm det}\,E){\rm
To}(X)=(r-1)\chi(X,\mathcal{O}_{X})+\chi(X,{\rm det}\,E).$

The constants $\alpha$ and $\beta$ are obtained by expressing ${\rm
Ch}_{3}(E)$ as a polynomial in the Chern classes $c_{i}(E)$. One gets

$\displaystyle\alpha=\frac{1}{2},\,\beta=-\frac{1}{2}.$ (11)

Finally, the constant $\gamma$ is obtained by expressing ${\rm Ch}_{2}(E){\rm
To}_{1}(X)$ using the Chern classes of $E$. One gets

$\displaystyle\gamma=-\frac{1}{2}.$ (12)

It is proved by Kollár (see [12]) that for $X$ as above, any curve $C\subset
X$ has degree divisible by $2$. It follows that the numbers
$\int_{X}c_{2}(E)c_{1}(E)$ and $\int_{X}c_{2}(E)c_{1}(X)$ are even. We thus
deduce from (11) and (12) that $\chi(X,E)=\frac{1}{2}\int_{X}c_{3}(E)+C$,
where $C$ is an integer. Thus the degree of $c_{3}(E)$ has to be an integer. ∎

######  Remark 3.8.

We see from the proof above that the obstruction to the existence of a vector
bundle $E$ on $X$ (or more generally an element of $K_{0}(X)$) with ${\rm
deg}\,c_{3}(E)=1$ comes from the defect of the integral Hodge conjecture for
degree $4$ Hodge classes on $X$. Conversely, if the integral Hodge conjecture
for degree $4$ Hodge classes on $X$ holds true, then the generator $\alpha\in
H^{2}(X,\mathbb{Z})$ is algebraic, $\alpha=[Z]$, for some $1$-cycle $Z\in{\rm
CH}_{1}(X)$. As we have

${\rm CH}_{1}(X)={\rm CH}^{2}(X)={\rm CH}^{2}(X)_{\rm Ch},$

the cycle $Z$ belongs to ${\rm CH}^{2}(X)_{\rm Ch}$, so the cycle $H\cdot Z$
belongs to ${\rm CH}^{3}(X)_{\rm Ch}$. Hence there exists a degree $1$ element
in ${\rm CH}^{3}(X)_{\rm Ch}$ in this case.

Other examples of smooth projective varieties $X$ for which ${\rm
CH}(X)\not={\rm CH}(X)_{\rm Ch}$ are given by generalized flag manifolds for
certain affine algebraic groups with torsion index $>1$ (see [6] and [17],
[18] where this notion is discussed and computed for many groups). Merkurjev
proved in [14] that for a simply connected semisimple algebraic group $G$, and
for a closed subgroup $H$, the $K_{0}$-ring of $G/H$ is generated by classes
of homogeneous vector bundles on $G/H$ that come from representations of $H$.
If furthermore $H$ is a Borel subgroup of $G$, then homogeneous vector bundles
on $G/H$ coming from representations of $H$ are direct sums of line bundles.
In the last case, it follows that the subgroup

${\rm CH}_{0}(G/H)_{\rm Ch}\subset{\rm CH}_{0}(G/H)$

is also the subgroup generated by products of divisor classes. By definition
of the torsion index of $G$, the index of the later subgroup is a multiple of
the torsion index of $G$. The computations in [17], [18] thus give plenty of
examples where ${\rm CH}_{0}(G/H)_{\rm Ch}\subset{\rm CH}_{0}(G/H)$ is a
proper subgroup.

We now turn to the proof of Theorem 1.6. We will prove it in several steps,
each one adding some arguments to the previous one. We first prove the
following weaker statement, namely the inclusion

$\displaystyle(d-1)!{\rm CH}_{d}(X)\subset{\rm CH}_{d}(X)_{\rm flpshCh}.$ (13)

###### Proof of (13).

Let $X$ be smooth projective of dimension $n$ and let $Z\subset X$ be a
subvariety of dimension $d$. We choose a desingularization
$\widetilde{Z}\rightarrow Z$ of $Z$ and an embedding $\widetilde{Z}\subset
X\times\mathbb{P}^{m}$ over $X$ for some $m$. As the projection
$p_{X}:X\times\mathbb{P}^{m}\rightarrow X$ to $X$ is flat, it suffices to
prove that

$\displaystyle(d-1)!\widetilde{Z}\in{\rm CH}(X\times\mathbb{P}^{m})_{\rm
flpshCh},$ (14)

as it implies by Remark 3.2 that $(d-1)!{Z}\in{\rm CH}(X)_{\rm flpshCh}$,
which is the contents of (13). In other words, letting

$Z^{\prime}=\widetilde{Z},\,X^{\prime}=X\times\mathbb{P}^{m},$

we reduced to the case of the class of a smooth subvariety $Z^{\prime}\subset
X^{\prime}$, which we treat now. If ${\rm dim}\,X^{\prime}\leq 2d$, this is
finished by formula (7) since then ${\rm codim}\,(Z^{\prime}\subset
X^{\prime})\leq d$. If not, let $Y$ be a smooth general complete intersection
of sufficiently ample hypersurfaces in $X^{\prime}$ containing $Z^{\prime}$,
with $Y$ of dimension $2d$. Such $Y$ exists since $Z^{\prime}$ is smooth of
dimension $d$ and ${\rm dim}\,X^{\prime}\geq 2d$. As ${\rm
codim}\,(Z^{\prime}\subset Y)=d$, we have $(d-1)![Z^{\prime}]_{Y}\in{\rm
CH}_{d}(Y)_{\rm Ch}$ by formula (7), where $[Z^{\prime}]_{Y}$ is the class of
$Z^{\prime}$ in $Y$. Let $j:Y\hookrightarrow X^{\prime}$ be the inclusion of
$Y$. By Corollary 3.10 stated below applied to the sequence of inclusions

$Y_{1}\cap Y_{2}\cap\ldots\cap Y_{r+1}\hookrightarrow Y_{1}\cap
Y_{2}\cap\ldots\cap Y_{r}$

for $r\leq N-1$, we have

$j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X^{\prime})_{\rm flpshCh},$

hence $(d-1)![Z^{\prime}]\in{\rm CH}_{d}(X^{\prime})_{\rm flpshCh}$ since
$[Z]=j_{*}[Z^{\prime}]_{Y}$. ∎

Corollary 3.10 is obtained as an immediate consequence of the following

######  Proposition 3.9.

Let $X$ be a smooth variety and $j:Y\rightarrow X$ be a finite morphism, where
$Y$ is smooth projective with ${\rm dim}\,Y={\rm dim}\,X-1$. Then

$j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}.$

######  Corollary 3.10.

If $X$ is smooth projective of dimension $n$ and $j:Y\rightarrow X$ is the
inclusion of a smooth subvariety of codimension $c$, such that there exist
inclusions

$Y=Y_{c}\subset Y_{c-1}\subset\ldots\subset Y_{0}=X,$

where $Y_{r}$ is smooth of dimension $n-r$ for any $0\leq r\leq n-c$, one has

$j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}.$

###### Proof of Proposition 3.9.

Let $T$ be the smooth projective variety obtained by blowing-up the graph
$\Gamma_{j}$ of $j$ in $Y\times X$. Let

$\tau:T\rightarrow Y\times X$

be the blow-up map and let ${\rm pr}_{Y},\,{\rm pr}_{X}$ be the two
projections from $Y\times X$ to $Y$ and $X$. We denote

$p:={\rm pr}_{Y}\circ\tau:T\rightarrow Y,\,\,q:={\rm
pr}_{X}\circ\tau:T\rightarrow X$

the two natural morphisms.

######  Lemma 3.11.

The morphism $p$ is smooth and the morphism $q$ is flat.

###### Proof.

Indeed, the fiber of $p$ over $y\in Y$ is the blow-up of $X$ along $y$, which
is smooth. The fiber of $q$ over $x\in X$ is isomorphic to $Y$ when $x\not\in
j(Y)$, hence it has dimension $n-1$, $n={\rm dim}\,X$. We claim that all the
fibers of $q$ have dimension $\leq n-1$. To see this, we observe that

$q^{-1}(x)=\tau^{-1}(Y\times\\{x\\})$

is the set-theoretic union of several components, some being contained in the
exceptional divisor $E$ over $\Gamma_{j}$ and mapping via $\tau$ to
$(Y\times\\{x\\})\cap\Gamma_{j}$, the other being birational to $Y$. The
component which is birational to $Y$ has dimension $\leq n-1$. The other
components are also of dimension $\leq n-1$, since the morphism $\tau_{\mid
E}:E\rightarrow\Gamma_{j}\cong Y$ has fibers of dimension $n-1$, and

$(Y\times\\{x\\})\cap\Gamma_{j}\cong j^{-1}(x)\subset Y\cong\Gamma_{j}$

has dimension $0$ because $j$ is finite. This proves the claim. The fibers are
thus equidimensional, hence $q$ is flat since both $T$ and $X$ are smooth. ∎

For any class $w\in{\rm CH}_{d}(Y)_{\rm flpshCh}$, there exist by definition a
(not necessarily connected but equidimensional) smooth projective variety $W$,
a flat morphism $\phi:W\rightarrow Y$, and divisors
$D_{1},\ldots,\,D_{N-d}\in{\rm CH}^{1}(W)$, $N:={\rm dim}\,W$, such that

$\displaystyle w=\phi_{*}(D_{1}\ldots D_{N-d})\,\,{\rm in}\,\,{\rm
CH}_{d}(Y).$ (15)

We now observe that

$\displaystyle j_{*}w=\Gamma_{j*}(w)={\rm pr}_{X*}({\rm
pr}_{Y}^{*}z^{\prime}\cdot\Gamma_{j})$ (16)

in ${\rm CH}_{d}(X)$. Furthermore, we have as usual

$\displaystyle\Gamma_{j}=\pm\tau_{*}E^{n}\,\,{\rm in}\,\,{\rm CH}^{n}(Y\times
X),\,n={\rm dim}\,X.$ (17)

Let now

$W_{T}:=W_{\times Y}T$

with first projection $p_{W}$ to $W$, second projection $p_{T}$ to $T$ and
morphism

$\psi:=q\circ p_{T}:W_{T}\rightarrow X.$

Combining (15), (16), and (17), and applying the projection formula, we get

$\displaystyle j_{*}w=\pm\psi_{*}(p_{W}^{*}(D_{1}\ldots D_{N-d})\cdot
p_{T}^{*}E^{n})\,\,{\rm in}\,\,{\rm CH}_{d}(X),$ (18)

which proves that $j_{*}w$ belongs to ${\rm CH}_{d}(X)_{\rm flpshCh}$, since
$W_{T}$ is smooth by Lemma 3.11, and the morphism $\psi$ is flat, being the
composition of the two flat morphisms $q$ and $p_{T}$. The proof of
Proposition 3.9 is finished. ∎

For the proof of Theorem 1.6, we prove now the following stability results
extending Proposition 3.9.

######  Proposition 3.12.

Let $X$ be smooth projective and let $j:Y\hookrightarrow X$ be the inclusion
of a smooth projective subvariety which is the zero-set of a transverse
section $\sigma$ of a vector bundle $E$ on $X$. Then

$\displaystyle j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm
flpshCh}.$ (19)

###### Proof.

We prove the result by induction on the rank of $E$, the case of rank $1$
being a particular case of Proposition 3.9. Let $E$ be a rank $r$ vector
bundle on $X$ and let $\pi:\mathbb{P}(E^{*})={\rm Proj}\,({\rm
Sym}^{*}E)\rightarrow X$ be the projectivization of $E^{*}$. Let
$\pi^{*}E\rightarrow\mathcal{H}$ be the quotient line bundle on
$\mathbb{P}(E^{*})$. The section $\pi^{*}\sigma$ projects to section
$\sigma^{\prime}$ of $\mathcal{H}$ and we have

######  Lemma 3.13.

(i) The $0$-locus of $\sigma^{\prime}$ is a smooth hypersurface $X^{\prime}$
of $\mathbb{P}(E^{*})$.

(ii) Furthermore, the induced section $\sigma^{\prime\prime}$ of $F:={\rm
Ker}\,(\pi^{*}E\rightarrow\mathcal{H})$ on $X^{\prime}$ is transverse with
$0$-locus $\pi^{-1}(Y)=\mathbb{P}(E^{*}_{\mid Y})\subset
X^{\prime}\subset\mathbb{P}(E^{*})$.

###### Proof.

(i) The vanishing locus of $\sigma^{\prime}$ is a $\mathbb{P}^{r-2}$-bundle
over the open subset $X\setminus Y$ of $X$ where $\sigma\not=0$, hence it is
smooth over $X\setminus Y$. It obviously contains $\pi^{-1}(Y)$ and it remains
to show that it is smooth there, which is easy.

(ii) The vanishing locus of $\sigma^{\prime\prime}$ on $X^{\prime}$ equals
scheme-theoretically the vanishing locus of the section $\pi^{*}\sigma$ of
$\pi^{*}E$ on $\mathbb{P}(E^{*})$, hence equals $\pi^{-1}(Y)$. It is thus
smooth of codimension $r-1$ in $X^{\prime}$. ∎

Denoting by $\pi_{Y}:\mathbb{P}(E^{*}_{\mid Y})\rightarrow Y$ the restriction
of $\pi$ over $Y$, we know by Lemma 3.4 that $\pi^{*}_{Y}:{\rm
CH}(Y)\rightarrow{\rm CH}(\mathbb{P}(E^{*}_{\mid Y}))$ maps ${\rm CH}(Y)_{\rm
flpshCh}$ to ${\rm CH}(\mathbb{P}(E^{*}_{\mid Y}))_{\rm flpshCh}$. Denoting by

$j^{\prime}:\mathbb{P}(E^{*}_{\mid Y})\hookrightarrow
X^{\prime},\,\,j^{\prime\prime}:X^{\prime}\hookrightarrow\mathbb{P}(E^{*})$

the inclusion maps, we get, first by the induction hypothesis on the rank and
lemma 3.13, and secondly by Proposition 3.9, that

$j^{\prime}_{*}({\rm CH}(\mathbb{P}(E^{*}_{\mid Y}))_{\rm flpshCh})\subset{\rm
CH}(X^{\prime})_{\rm flpshCh},\,\,j^{\prime\prime}_{*}({\rm
CH}(X^{\prime})_{\rm flpshCh})\subset{\rm CH}(\mathbb{P}(E^{*}))_{\rm
flpshCh}.$

We conclude that the map $\gamma:=j^{\prime\prime}_{*}\circ
j^{\prime}_{*}\circ\pi_{Y}^{*}:{\rm CH}(Y)\rightarrow{\rm
CH}(\mathbb{P}(E^{*}))$ has the property that

$\gamma({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(\mathbb{P}(E^{*}))_{\rm
flpshCh}.$

Recalling from [7, Proposition 1.7] that $\gamma=\pi^{*}\circ j_{*}$, we thus
proved that

$\pi^{*}\circ j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm
CH}(\mathbb{P}(E^{*}))_{\rm flpshCh}.$

Let $h=c_{1}(\mathcal{H})\in{\rm CH}^{1}(\mathbb{P}(E^{*}))$. By Lemma 3.3, we
have

$h^{r-1}{\rm CH}(\mathbb{P}(E^{*}))_{\rm flpshCh}\subset{\rm
CH}(\mathbb{P}(E^{*}))_{\rm flpshCh}$

and by Remark 3.2, $\pi_{*}({\rm CH}(\mathbb{P}(E^{*}))_{\rm
flpshCh})\subset{\rm CH}(X){\rm flpshCh}$. As $\pi_{*}(h^{r-1}\pi^{*}z)=z$ for
any $z\in{\rm CH}(X)$, we conclude that for any $z\in{\rm CH}(Y)_{\rm
flpshCh}$,

$j_{*}z=\pi_{*}(h^{r-1}\pi^{*}(j_{*}z))\in{\rm CH}(X)_{\rm flpshCh}.$

∎

Another consequence of proposition 3.9 is the following

######  Proposition 3.14.

Let $X$ be smooth projective and let $Y\subset X$ be a smooth projective
subvariety which is the $0$-set of a transverse section $\sigma$ of a vector
bundle $E$ on $X$. Let $\tau:\widetilde{X}\rightarrow X$ be the blow-up of $X$
along $Y$. Then

$\displaystyle\tau_{*}({\rm CH}(\widetilde{X})_{\rm flpshCh})\subset{\rm
CH}(X)_{\rm flpshCh}.$ (20)

###### Proof.

Let $\pi:\mathbb{P}(E)={\rm Proj}({\rm Sym}^{*}E^{*})\rightarrow X$ be the
projectivization of $E$. Then the section $\sigma$ gives a rational section
$X\dashrightarrow\mathbb{P}(E)$ of $\pi$, whose image is isomorphic to
$\widetilde{X}$. Furthermore, as shows a local computation,
$\widetilde{X}\subset\mathbb{P}(E)$ is the zero-set of a transverse section
$\overline{\sigma}$ of the quotient vector bundle $F:=\pi^{*}E/\mathcal{S}$ on
$\mathbb{P}(E)$, namely, $\overline{\sigma}$ is the projection of
$\pi^{*}\sigma$ in $\pi^{*}E/\mathcal{S}$, where $\mathcal{S}\subset\pi^{*}E$
is the tautological subbundle. We have

$\displaystyle\tau_{*}=\pi_{*}\circ j_{*}:{\rm
CH}(\widetilde{X})\rightarrow{\rm CH}(X),$ (21)

where $j:\widetilde{X}\rightarrow\mathbb{P}(E)$ is the inclusion map. By
Proposition 3.12, we have

$j_{*}({\rm CH}(\widetilde{X})_{\rm flpshCh})\subset{\rm
CH}(\mathbb{P}(E))_{\rm flpshCh}$

and $\pi_{*}({\rm CH}(\mathbb{P}(E))_{\rm flpshCh})\subset{\rm CH}(X)_{\rm
flpshCh}$ by Remark 3.2. Hence (21) implies (20). ∎

We now use these results to prove Corollary 1.11. More precisely, let us prove
the following statement.

######  Proposition 3.15.

Let $Y$ be smooth projective of dimension $2d-k$, and $r$ a large integer.
Denote $Y_{\sharp}:=:=Y\times\mathbb{P}^{r}$. Let $Z\subset Y$ be closed
smooth of dimension $d$. Assume that for a general hypersurface
$Y^{\prime}\subset Y$ containing $Z$, there exist a smooth projective variety
$\widetilde{Y}$ and a morphism $\tau:\widetilde{Y}\rightarrow Y_{\sharp}$
obtained by successive blow-ups $Y_{i+1}\rightarrow Y_{i}$,
$Y_{0}:=Y_{\sharp}$ along smooth closed algebraic subsets $W_{i}\subset Y_{i}$
which are generalized complete intersections on $Y_{i}$, such that

(1) the proper transform $\widetilde{Y^{\prime}}$ of $Y^{\prime}\times{\rm
pt}$ in $\widetilde{Y}$ is smooth and a generalized complete intersection, and

(2) the proper transform $Z^{\prime}$ of $Z$ is smooth, birational to $Z$.

Then

$\displaystyle(d-k-2)![Z]\in{\rm CH}_{d}(Y)_{\rm flpshCh}.$ (22)

If the statement above is satisfied for any $Z,\,Y$ as above with ${\rm
codim}\,(Z\subset Y)\geq 3$, then

$\displaystyle[Z]\in{\rm CH}_{d}(Y)_{\rm flpshCh}.$ (23)

###### Proof.

Let $\tilde{j}:\widetilde{Y^{\prime}}\hookrightarrow\widetilde{Y}$ be the
inclusion. As $\widetilde{Y^{\prime}}$ is a generalized complete intersection
in $\widetilde{Y}$ by (1), Proposition 3.12 gives

$\displaystyle\tilde{j}_{*}({\rm CH}(\widetilde{Y^{\prime}})_{\rm
flpshCh})\subset{\rm CH}_{d}(\widetilde{Y})_{\rm flpshCh}.$ (24)

By iterated applications of Proposition 3.14,

$\displaystyle\tau_{*}{\rm CH}_{d}(\widetilde{Y})_{\rm flpshCh}\subset{\rm
CH}_{d}(Y_{\sharp})_{\rm flpshCh}.$ (25)

Finally, by (7),

$\displaystyle(d-k-2)![Z^{\prime}]\in{\rm CH}_{d}(\widetilde{Y^{\prime}})_{\rm
flpshCh}$ (26)

since ${\rm codim}\,(Z^{\prime}\subset\widetilde{Y^{\prime}})=d-k-1$. As we
have

$\displaystyle[Z\times{\rm pt}]=(\tau\circ\tilde{j})_{*}[Z^{\prime}]\,\,{\rm
in}\,\,{\rm CH}_{d}(Y),$ (27)

(24), (25) and (26) imply that $(d-k-2)![Z\times{\rm pt}]\in{\rm
CH}_{d}(Y_{\sharp})_{\rm flpshCh}$, which implies (22) by applying the first
projection $Y_{\sharp}\rightarrow Y$.

If for any pair $(Z,Y)$ as above with ${\rm dim}\,Z=d$, ${\rm dim}\,Y\geq
d+3$, and any general hypersurface $Y^{\prime}\subset Y$ containing $Z$, there
exists a $\widetilde{Y}$ satisfying the stated properties, the proof above
provides inductively varieties $\widetilde{Y^{\prime}_{k}}$ of dimension
$2d-k$, $k\leq d-2$ containing a smooth subvariety $Z_{k}$ of dimension $d$
birational to $Z$, and morphisms

$j_{k}:\widetilde{Y^{\prime}_{k}}\rightarrow\widetilde{Y^{\prime}_{k-1}}$

such that

$\displaystyle j_{k*}({\rm CH}(\widetilde{Y^{\prime}_{k}})_{\rm
flpshCh})\subset{\rm CH}(\widetilde{Y^{\prime}_{k-1}})_{\rm flpshCh}$ (28)

and $[Z_{k-1}]=j_{k*}[Z_{k}]$. Furthermore, $\widetilde{Y^{\prime}_{0}}=Y$ and
$Z_{0}=Z$. For $k=d-2$, one has ${\rm dim}\,\widetilde{Y^{\prime}_{d-2}}=d+2$,
so $[Z_{d-2}]\in{\rm CH}(\widetilde{Y^{\prime}_{d-2}})_{\rm flpshCh}$, hence
$[Z]\in{\rm CH}_{d}(Y)_{\rm flpshCh}$ by (28). ∎

Let us show how Proposition 3.15 for $k=0$ immediately gives the inclusion

$\displaystyle(d-2)!{\rm CH}_{d}(X)_{\rm flpshCh}\subset{\rm CH}_{d}(X)_{\rm
flpshCh}$ (29)

for any smooth projective variety $X$.

###### Proof of (29).

As before, if $Z\subset X$ is any subvariety, we replace it by a
desingularization $\widetilde{Z}\subset X\times\mathbb{P}^{N}$, hence we can
assume without loss of generality that $Z$ is smooth. $Z$ is then contained in
a smooth subvariety $Y\subset X$ of dimension $2d$, which is a complete
intersection of smooth hypersurfaces. By iterated applications of Proposition
3.9, it suffices to prove that the class $z$ of $Z$ in $Y$ has the property
that

$\displaystyle(d-2)!z\in{\rm CH}_{d}(Y)_{\rm flpshCh}.$ (30)

######  Lemma 3.16.

A general sufficiently ample hypersurface $Y^{\prime}\subset Y$ containing $Z$
has isolated ordinary quadratic singularities.

###### Proof.

By Bertini, the singularities of $Y^{\prime}$ are on $Z$, and they correspond
to the zeroes of the differential

$\displaystyle d\sigma\in H^{0}(Z,N_{Z}^{*}(Y^{\prime}))$ (31)

along $Z$ of the defining equation $\sigma$ of $Y^{\prime}$. The section (31)
being generic, its zero locus is transverse, hence the singular locus of
$Y^{\prime}$ is zero-dimensional. In fact, the transversality of the section
$d\sigma$ also implies that the singularities are ordinary quadratic (see
Lemma 3.17 below). ∎

By Lemma 3.16, the hypersurface $Y^{\prime}$ is desingularized by a single
blow-up along the finite set $W\subset Y^{\prime}$ of its singular points. We
choose now a general $0$-dimensional complete intersection $W^{\prime}\subset
Y$ containing $W$. We have $W^{\prime}=W\cup W^{\prime\prime}$, where the set
$W^{\prime\prime}$ is disjoint from $Y^{\prime}$. It follows that the blow-up
$\widetilde{Y}$ of $Y$ along $W^{\prime}$ contains the blow-up
$\widetilde{Y^{\prime}}$ of $Y^{\prime}$ along $W$ as a smooth hypersurface.
It is clear that the proper transform $Z^{\prime}$ of $Z$ under this blow-up
is smooth birational to $Z$, so the assumptions of Proposition 3.15 are
satisfied (with $k=0,\,r=0$) and (30) follows. ∎

We now prove the improved inclusion

$\displaystyle(d-5)!{\rm CH}_{d}(X)\subset{\rm CH}_{d}(X)_{\rm flpshCh}.$ (32)

###### Proof of (32).

The proof mimics the argument above but the desingularization step is more
involved. Let $Z\subset X$ be a subvariety of dimension $d$. By the previous
proof, we can assume that $Z$ is smooth and that there exists a smooth
projective variety $X^{\prime}$ of dimension $2d-1$, mapping to $X$ via a
morphism $j^{\prime}:X^{\prime}\rightarrow X$ such that $j^{\prime}_{*}({\rm
CH}(X^{\prime})_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}$ and containing
a smooth subvariety $Z^{\prime}\subset X^{\prime}$ which is birational to $Z$
via $j$. We are now going to reduce step by step the dimension of $X^{\prime}$
by taking hypersurfaces and desingularizing them, until we have constructed a
variety $X^{\prime\prime}$ of dimension $2d-4$ containing a smooth subvariety
$Z^{\prime\prime}$ of dimension $d$, together with a morphism
$j^{\prime\prime}:X^{\prime\prime}\rightarrow X$ such that
$j^{\prime\prime}({\rm CH}(X^{\prime\prime})_{\rm flpshCh})\subset{\rm
CH}(X)_{\rm flpshCh}$ and the restriction of $j^{\prime\prime}$ to
$Z^{\prime\prime}$ is a birational morphism $Z^{\prime\prime}\rightarrow Z$.
We explain only the last step, the previous ones being similar and in fact
easier. We thus assume that $X$ is smooth of dimension $2d-3$ and $Z\subset X$
is smooth of dimension $d$. Let $Y\subset X$ be a general hypersurface
containing $Z$.

######  Lemma 3.17.

$Y$ has ordinary quadratic singularities along a $3$-dimensional smooth
subvariety $T\subset Z$ and is smooth away from $Z$.

###### Proof.

This follows from Bertini away from $Z$. Along $Z$, the differential $df$ of
the local equations defining $Y$ provides a section

$df\in H^{0}(Z,N^{*}_{Z/X}(Y)).$

Taking $Y$ sufficiently ample, the morphism $df$ is arbitrary and the vector
bundle $N^{*}_{Z/X}(Y)$ is generated by sections, hence the vanishing locus of
$df$ is a smooth subvariety $T\subset Z$ of codimension $d-3$, hence dimension
$3$. This implies that the singularities of $Y$ are ordinary quadratic along
$T$. This is proved by taking local holomorphic coordinates
$z_{1},\ldots,\,z_{2d-3}$ on $X$, such that $Z$ is defined by equations
$z_{j}=0$ for $j\geq d+1$. In these coordinates, the local holomorphic
equation $f$ defining $Y$ takes the form

$\displaystyle f=\sum_{j=d+1}^{2d-3}z_{j}g_{j}.$ (33)

The locus $T$ is the locus of points $z\in Z$ where the functions $g_{j}$ all
vanish. The locus $T$ is smooth of codimension $d-3$ in $Z$ if and only if the
restriction to $Z$ of the differentials $dg_{j}$ are independent, or
equivalently the differentials $dg_{j}$ are independent modulo the $dz_{j}$,
$j\geq d+1$. This is clearly equivalent by (33) to the fact that $f$ has
ordinary quadratic singularities along $T$. ∎

The variety $Y$ can thus be desingularized by blowing-up the subvariety $T$ in
$X$. Unfortunately, $T$ is not a generalized complete intersection in $X$,
which we need in order to apply Proposition 3.15. We correct this as follows.
First of all, we replace $X$ by $X_{\sharp}:=X\times\mathbb{P}^{r}$ for some
$r\geq 3$, and $Y$ by $Y\times{\rm pt}$. $Y\subset X_{\sharp}$ is a
generalized complete intersection, and we are going to desingularize $Y$ by
successive blow-ups $\tau_{i}:X_{i}\rightarrow X_{i-1},\,i\leq
4,\,X_{0}=X_{\sharp}$ along smooth generalized complete intersections, keeping
the property that the proper transforms $Y_{i}\subset X_{i}$ is a smooth
generalized complete intersection. In other words, we are going to construct a
resolution of singularities of $Y\times{\rm pt}\subset X_{\sharp}$ satisfying
the conditions of Proposition 3.15, and (32) will follow from Proposition
3.15.

Let $m$ be the dimension of $X_{\sharp}$ (which we can assume arbitrarily
large). By choosing $m-3$ general hypersurfaces in $X_{\sharp}$ containing
$T$, we construct a complete intersection subvariety

$T_{1}=T\cup T^{\prime},$

and the two components $T,\,T^{\prime}$ are smooth and meet transversally
along a smooth surface $S\subset Y=Y\times{\rm pt}\subset X_{\sharp}$. This is
where we use the fact that ${\rm dim}\,T\leq 3$, as it was observed by
Hironaka in [9] that the residual subvariety $T^{\prime}$ obtained by general
liaison starting from a smooth subvariety of dimension $\leq 3$ is smooth. The
fact that the two components intersect transversally along a smooth surface
$S$ is proved as Lemma 3.17. Furthermore, $m$ being large, we can assume that
$T^{\prime}$ intersects $Y$ only along $S=T\cap T^{\prime}$. We now choose
$m-2$ general hypersurfaces in $X_{\sharp}$ containing $S$. Their intersection
is a reducible surface

$S_{1}=S\cup S^{\prime},$

where the residual component $S^{\prime}$ is a smooth surface, the
intersection $S\cap S^{\prime}$ is as before a smooth curve $C\subset Y$, and
$S^{\prime}$ intersects $Y$ only along $C$. We next choose $m-1$ general
hypersurfaces in $X_{\sharp}$ containing $C$. Their intersection is a
reducible curve

$C_{1}=C\cup C^{\prime},$

where the second component $C^{\prime}$ is a smooth curve, the intersection
$C\cap C^{\prime}$ is a reduced $0$-dimensional subscheme $W\subset Y$, and
$C^{\prime}$ intersects $Y$ only along $W$. We finally choose $m$ general
hypersurfaces in $X_{\sharp}$ containing $W$. Their intersection is a reduced
$0$-dimensional subscheme $W_{1}\subset X$, which intersects $Y$ along $W$.

We now blow-up $X_{\sharp}$ along $W_{1}$, getting $X_{1}$, then blow-up the
proper transform $\widetilde{C}_{1}$ of $C_{1}$ in $X_{1}$ getting $X_{2}$,
then blow-up the proper transform $\widetilde{S}_{1}$ of $S_{1}$ in $X_{2}$,
getting $X_{3}$, and finally blow-up the proper transform $\widetilde{T}_{1}$
of $T_{1}$ in $X_{3}$. This gives us a variety $X_{4}$ birational to
$X_{\sharp}$. We will denote by $\tau_{i}:X_{i}\rightarrow X_{i-1}$ (with
$X_{0}:=X_{\sharp}$) the blow-up maps and by $\tau:X_{4}\rightarrow
X_{\sharp}$ their composition.

We observe that, after blowing-up $W_{1}$, the proper transforms of the curves
$C$ and $C^{\prime}$ are disjoint, so $\widetilde{C}_{1}\subset X_{1}$ is in
fact smooth. Similarly, after blowing-up $\widetilde{C}_{1}$, the proper
transforms of the surfaces $S$ and $S^{\prime}$ are disjoint, so
$\widetilde{S}_{1}\subset X_{2}$ is in fact smooth. Finally, after blowing-up
$\widetilde{S}_{1}$, the proper transforms of $T$ and $T^{\prime}$ are
disjoint, hence $\widetilde{T}_{1}\subset X_{3}$ is smooth. It follows that
$X_{4}$ is smooth. As argued above, in order to complete the proof of (32), it
now suffices to prove

######  Lemma 3.18.

(i) For any $1\leq i\leq 4$, the blow-up maps $\tau_{i}:X_{i}\rightarrow
X_{i-1}$ is the blow-up along a smooth center which is a generalized complete
intersection in $X_{i-1}$.

(ii) The proper transform $Y_{4}\subset X_{4}$ of $Y$ is smooth and is a
generalized complete intersection in $X_{4}$.

Lemma 3.18 follows from the following general

######  Lemma 3.19.

Let $N$ be a smooth variety and $M\subset N$ be a subvariety which has
ordinary quadratic singularities along its singular locus $M_{1}\subset M$ (in
particular $M_{1}$ is a smooth subvariety of $N$). Let $T\subset N$ be a
smooth subvariety, which is a disjoint union

$T=T_{1}\sqcup T_{2},$

where $T_{1}\subset M_{1}$ and $T_{2}$ is disjoint from $M$. Let
$\widetilde{N}_{T}$ be the blow-up of $N$ along $T$ and
$\widetilde{M}\subset\widetilde{N}_{T}$ be the proper transform of $M$. Then

(i) The singularities of $\widetilde{M}$ are ordinary quadratic along the
proper transform of $M_{1}$.

(ii) If $M$ is a generalized complete intersection in $N$, $\widetilde{M}$ is
a generalized complete intersection in $\widetilde{N}_{T}$.

We apply indeed Lemma 3.19 to the proper transforms of the loci
$C_{1},\,S_{1},\,T_{1}$ under the successive blow-ups. This guarantees us that
at each step, these proper transforms are zero-sets of sections of adequate
vector bundles on the intermediate blow-up varieties $X_{1},\ldots,\,X_{4}$,
which proves Lemma 3.18(i). We also apply Lemma 3.19 to $Y$ itself and its
successive proper transforms, using Lemma 3.17, which proves Lemma 3.18(ii).

###### Proof of Lemma 3.19.

(i) Let $t:={\rm dim}\,T$, $m_{1}:={\rm dim}\,M_{1}$, $m:={\rm dim}\,M$ and
$n:={\rm dim}\,N$. By assumption, we can construct local coordinates
$z_{1},\ldots,\,z_{n}$ on $N$ such that $M_{1}$ is defined by $z_{i}=0,\,i\geq
m_{1}+1$, $T$ is defined by $z_{i}=0,\,i\geq t+1$, and $M$ is defined by

$\displaystyle z_{i}=0,\,i\geq m+2,\,f(z):=\sum_{i=m_{1}+1}^{m+1}z_{i}^{2}=0.$
(34)

The blow-up $\widetilde{N}$ of $N$ along $T$ is defined inside

$\mathbb{P}^{n-t-1}\times N$

by the equations

$Y_{i}z_{j}=Y_{j}z_{i}$

for $i,\,j\geq t+1$, the $Y_{i}$’s being homogeneous coordinates on
$\mathbb{P}^{n-t-1}$. On the open set where $Y_{k}\not=0$, we have local
coordinates

$y_{j},\,{\rm for}\,\,j\geq t+1\,j\not=k,\,z_{k}$ $z_{l},\,\,{\rm
for}\,\,1\leq l\leq t$

on $\widetilde{N}$ and the blow-up map is given by

$z_{j}=z_{k}y_{j},$

the exceptional divisor being defined by $z_{k}=0$. We examine the various
cases $k\leq m_{1},\,m_{1}+1\leq k\leq m+1,\,k>m+1$.

If $k\leq m_{1}$, the local equations for $\widetilde{M}$ are

$\displaystyle y_{i}=0,\,i\geq m+2,\,\sum_{i=m_{1}+1}^{m+1}y_{i}^{2}=0,$ (35)

and $\widetilde{M_{1}}$ is defined by $y_{i}=0$ for any $i\geq m_{1}+1$. It
follows that $\widetilde{M}$ has ordinary quadratic singularities along
$\widetilde{M_{1}}$.

If $k>m+1$, the local equations for $\widetilde{M}$ are

$\displaystyle z_{k}=0,\,y_{i}=0,\,{\rm for}\,i\geq m+2,\,i\not=k,\,\,{\rm
and}\,\,\sum_{i=m_{1}+1}^{m+1}y_{i}^{2}=0.$ (36)

Furthermore $\widetilde{M_{1}}$ is defined by $y_{i}=0$ for any $i\geq
m_{1}+1$, $i\not=k$, and $z_{k}=0$. It follows again that $\widetilde{M}$ has
ordinary quadratic singularities along $\widetilde{M_{1}}$.

If $m_{1}+1\leq k\leq m+1$, the local equations for $\widetilde{M}$ are

$\displaystyle y_{i}=0,\,i\geq
m+2,\,1+\sum_{i=m_{1}+1,i\not=k}^{m+1}y_{i}^{2}=0,$ (37)

and $\widetilde{M_{1}}$ is defined by $y_{i}=0$ for any $i\geq m_{1}+1$. It
follows that $\widetilde{M}$ is smooth in this open set.

(ii) Let $E$ be the exceptional divisor of the blow-up map
$\tau:\widetilde{N}\rightarrow N$, and $E_{1}$ be its component over $T_{1}$.
We construct $\widetilde{\mathcal{F}}$ by modifying $\tau^{*}\mathcal{F}(-E)$
along $E_{1}$. As $T_{1}\subset M$, the section $s$ vanishes along $T_{1}$ and
has a differential

$ds:N_{T_{1}/N}\rightarrow\mathcal{E}_{\mid T_{1}}.$

This differential has corank $1$, as follows from the fact that $M$ is
singular with hypersurface singularities along $T_{1}$. We thus have a
quotient line bundle $\mathcal{L}$ of $\mathcal{E}_{\mid T_{1}}$, and denoting
$\tau_{E_{1}}:E_{1}\rightarrow T_{1}$ the restriction of $\tau$ to $E_{1}$, we
get a quotient map constructed as the composition

$q:\tau^{*}\mathcal{E}(-E_{1})\rightarrow\tau_{E_{1}}^{*}\mathcal{E}_{\mid
T_{1}}(-E_{1})\rightarrow\tau_{E_{1}}^{*}\mathcal{L}(-E_{1}).$

We set

$\widetilde{\mathcal{F}}:={\rm Ker}\,q.$

This is a vector bundle of rank $n-m$ on $\widetilde{N}$. Furthermore, we
observe that by construction, the section $\tau^{*}\sigma$ of
$\tau^{*}\mathcal{F}$ provides in fact a section $\tilde{s}$ of
$\widetilde{\mathcal{F}}\subset\tau^{*}\mathcal{F}$. It remains to check that
the vanishing locus of $\tilde{s}$ is exactly $\widetilde{M}$. This easily
follows from the local equations exhibited in the previous proof. ∎

∎

###### Proof of Theorem 1.6.

We have finally to explain how to improve the inclusion (32) to

$\displaystyle(d-6)!{\rm CH}_{d}(X)\subset{\rm CH}_{d}(X)_{\rm flpshCh}.$ (38)

We can assume that $d\geq 5$. Let $Z\subset X$ be a smooth subvariety of
dimension $d$. From the previous proof, we know that there exist a smooth
projective variety $Y$ of dimension $2d-4$ and a morphism

$j:Y\rightarrow X$

such that $j_{*}({\rm CH}(Y)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}$,
and a smooth subvariety $Z^{\prime}\subset Z$ such that $j(Z^{\prime})=Z$ and
$j_{\mid Z^{\prime}}$ induces a birational morphism between $Z^{\prime}$ and
$Z$. We now choose a general hypersurface $Y^{\prime}\subset Y$ containing
$Z^{\prime}$. By Lemma 3.17, $Y^{\prime}$ has ordinary quadratic singularities
along a closed algebraic subset $Q\subset Z^{\prime}$ which is smooth of
dimension $4$, namely $Q$ is the zero-set of the section of
$N_{Z/Y}^{*}(Y^{\prime})$ defined as the differential of the local equation
defining $Y^{\prime}$, which is well-defined along $Z^{\prime}$. Assume that,
after replacing $Y$ by $Y_{\sharp}=Y\times\mathbb{P}^{r}$, we can construct a
$4$-dimensional complete intersection $Q_{1}\subset Y$ which contains $Q$ as
an irreducible component, is the union $Q_{1}=Q\cup Q^{\prime}$, where
$Q^{\prime}$ is smooth, the union has ordinary quadratic singularities along
$T:=Q^{\prime}\cap Q$ and $Q^{\prime}$ meets $Y^{\prime}$ exactly along $T$.
Then the whole previous proof works, introducing complete intersections
$T_{1}=T\cup T^{\prime}$, $S_{1}=S\cap S^{\prime}$, where $S:=T\cap
T^{\prime}$ et caetera. In conclusion, we just have to explain how to reduce
to this case, and we will do it by a blow-up of $Y_{\sharp}$ at finitely many
points. The new problem that we meet here is that liaison of varieties of
dimension $\geq 4$ introduces points where the above description of $Q_{1}$
fails. Indeed, the general 4-dimensional complete intersection $Q_{1}\subset
Y_{\sharp}$ containing $Q$ as an irreducible component is the zero-set of a
section

$\sigma\in H^{0}(Y_{\sharp},\mathcal{E})$

where $\mathcal{E}$ is a vector bundle of rank $m-4$, $m:={\rm
dim}\,Y_{\sharp}$, on $Y_{\sharp}$, whose differential along $Q$ provides a
morphism

$d\sigma:N_{Q/Y_{\sharp}}\rightarrow\mathcal{E}_{\mid Q}.$

As ${\rm dim}\,Q=4$, there are finitely points $y_{i}\in Q$, where the
morphism $d\sigma$ has corank $2$, and where the closed algebraic subset
$Q_{1}=Q\cup Q^{\prime}$ does not have ordinary quadratic singularities along
a smooth hypersurface $Q\cap Q^{\prime}$ in $Q$.

We fix this point as follows.

######  Lemma 3.20.

Let $\widetilde{Y}$ be the blow-up of the finitely many points $y_{i}$ in
$Y_{\sharp}$ and at a number of other points which are not on
$Y^{\prime}=Y^{\prime}\times{\rm pt}$. Then

(i) The proper transform $\widetilde{Q}_{1}$ of $Q_{1}$ in $\widetilde{Y}$ has
ordinary quadratic singularities along a $3$-dimensional smooth subvariety
$T$.

(ii) Furthermore, $\widetilde{Q}_{1}$ is the zero-set of a section of a rank
$d-4$ vector bundle on $\widetilde{Y}$.

###### Proof.

(i) At the points $y_{i}$, the embedding dimension of $Q_{1}$ is 6, so we can
choose local coordinates $z_{1},\ldots,\,z_{2d-4}$ on $Y_{\sharp}$ centered at
$y_{i}$, such that $Q_{1}$ is contained in $M:=\\{z_{j}=0,\,j\geq 7\\}$, and
$Q$ is defined by the equations $z_{j},\,j\geq 5$. The two equations for
$Q_{1}$ in $M$ have vanishing differential at $y_{i}$, and they vanish along
$Q$. In an adequate choice of coordinates on $M$, they take the simplest form,
namely

$\displaystyle z_{5}z_{1}+z_{6}z_{2}=0,\,z_{5}z_{3}+z_{6}z_{4}=0.$ (39)

Along the exceptional divisor of $\widetilde{Y}$ over $y_{i}$, we have
homogeneous coordinates $Z_{i}$ and two situations:

\- In the Zariski open set where $Z_{1}\not=0$, we have coordinates
$z_{1},\,z^{\prime}_{2},\ldots,\,z^{\prime}_{6}$ on the proper transform
$\widetilde{M}$, the morphism $\tau$ being given by
$\tau^{*}z_{i}=z_{1}z^{\prime}_{i},\,i\geq 2$. Then the pull-backs of the
equations (39) on $\widetilde{M}$ are

$z_{1}^{2}(z^{\prime}_{5}+z^{\prime}_{6}z^{\prime}_{2}),\,\,z_{1}^{2}(z^{\prime}_{5}z^{\prime}_{3}+z^{\prime}_{6}z^{\prime}_{4}).$

The proper transform of $Q_{1}$ is thus defined by the equations

$z^{\prime}_{5}+z^{\prime}_{6}z^{\prime}_{2}=0,\,\,z^{\prime}_{5}z^{\prime}_{3}+z^{\prime}_{6}z^{\prime}_{4}=0.$

It is clear that the closed analytic subset defined by these equations has two
smooth irreducible components of dimension $4$ meeting along a smooth
hypersurface, namely $\\{z^{\prime}_{5}=z^{\prime}_{6}=0\\}$ and
$\\{z^{\prime}_{5}+z^{\prime}_{6}z^{\prime}_{2}=0,\,-z^{\prime}_{2}z^{\prime}_{3}+z^{\prime}_{4}=0\\}$.

\- In the Zariski open set where $Z_{5}\not=0$, we have coordinates
$z^{\prime}_{1},\ldots,\,z^{\prime}_{4},\,z_{5},\,z^{\prime}_{6}$ on the
proper transform $\widetilde{M}$, the morphism $\tau$ being given by
$\tau^{*}z_{i}=z_{5}z^{\prime}_{i},\,i\not=5$. Then the pull-backs of the
equations (39) on $\widetilde{M}$ are

$z_{5}^{2}(z^{\prime}_{1}+z^{\prime}_{6}z^{\prime}_{2}),\,\,z_{5}^{2}(z^{\prime}_{3}+z^{\prime}_{6}z^{\prime}_{4}).$

The proper transform of $Q_{1}$ is thus defined by the equations

$z^{\prime}_{1}+z^{\prime}_{6}z^{\prime}_{2}=0,\,\,z^{\prime}_{3}+z^{\prime}_{6}z^{\prime}_{4}=0.$

It is clear that the closed analytic subset defined by these equations is
smooth.

(ii) Recall that $Q_{1}$ is the zero-set of a section $\sigma$ of a vector
bundle $\mathcal{F}$ of rank $m-4$ on $Y$. As we saw in the computation above,
at each point $y_{i}$, the differential $d\sigma:T_{Y_{\sharp}\mid
Q_{1}}\rightarrow\mathcal{F}_{\mid Q_{1}}$ has rank only $m-6$. We thus have
$2$-dimensional quotients

$q_{i}:\mathcal{F}_{\mid y_{i}}\rightarrow B_{i}.$

Denoting $E_{y_{i}}$ the exceptional divisor over $y_{i}$, these quotient maps
provide a surjective morphism of sheaves on $\widetilde{Y}$

$q:\tau^{*}\mathcal{F}(-\sum_{i}E_{y_{i}})\rightarrow\oplus_{i}B_{i}\otimes\mathcal{O}_{E_{y_{i}}}(-E_{y_{i}}),$

whose kernel is a vector bundle $\mathcal{F}^{\prime}$ on $\widetilde{Y}$. It
is clear from the explicit equations above that the section $\sigma$ lifts to
a section $\tilde{\sigma}$ of $\mathcal{F}^{\prime}$, and that the proper
transform $\widetilde{Q}_{1}$ is the zero-set of $\sigma^{\prime}$. ∎

Let $\tau:\widetilde{Y}\rightarrow Y_{\sharp}$ be the blow-up of a finite
reduced subset $W$ of $Y$ which contains all the points $y_{i}$ above, is the
zero set of a section of a vector bundle of rank $m$ and is generic otherwise
(in particular, the extra points are not in $Y^{\prime}$). By Proposition
3.14, the morphism $\tau_{*}:{\rm CH}(\widetilde{Y})\rightarrow{\rm
CH}(Y_{\sharp})$ maps ${\rm CH}(\widetilde{Y})_{\rm flpshCh}$ to ${\rm
CH}({Y_{\sharp}})_{\rm flpshCh}$. The smooth projective variety
$\widetilde{Y}$ contains the proper transforms $\widetilde{Z^{\prime}}$ and
$\widetilde{Y^{\prime}}$ of $Z^{\prime}$ and $Y^{\prime}$ respectively. By
Lemma 3.19(i), $\widetilde{Y^{\prime}}$ has ordinary quadratic singularities
along the proper transform $\widetilde{Q}$ of $Q$ and by Lemma 3.19(ii), it is
a generalized complete intersection in $\widetilde{Y}$. Finally, by Lemma 3.20
again, $\widetilde{Q}$ is an irreducible component of a 4-dimensional local
complete intersection $\widetilde{Q_{1}}$, which is a generalized complete
intersection on $\widetilde{Y}$ and has ordinary quadratic singularities along
a 3-dimensional smooth subvariety $T\subset\widetilde{Q}$. By Propositions
3.14 and 3.12, the morphism $j\circ{\rm
pr}_{1}\circ\tau:\widetilde{Y}\rightarrow X$ maps the proper transform
$Z^{\prime}$, which is smooth, birationally to $Z$, and satisfies

$(j\circ{\rm pr}_{1}\circ\tau)_{*}({\rm CH}(\widetilde{Y})_{\rm
flpshCH})\subset{\rm CH}(X)_{\rm flpshCH}.$

Finally, as we already mentioned, the previous proof applies to desingularize
the proper transform $\widetilde{Y^{\prime}}$ by successive blow-ups of smooth
generalized complete intersections starting from $\widetilde{Y}$, since
$\widetilde{Q}$ is a component of a 4-dimensional generalized complete
intersection in $\widetilde{Y}$ that has only ordinary quadratic
singularities. We thus conclude by mimicking the previous proof that the
proper transform $\widetilde{Z}$ of $Z$ in $\widetilde{Y}$ satisfies

$(d-6)![Z_{1}]\in{\rm CH}_{d}(\widetilde{Y})_{\rm flpshCh}.$

By Propositions 3.14 applied to $\tau$, it follows that
$(d-6)![Z^{\prime}]\in{\rm CH}_{d}(Y)_{\rm flpshCh}$, hence that

$(d-6)![Z]=(d-6)!j_{*}[Z^{\prime}]\in{\rm CH}_{d}({X})_{\rm flpshCh}.$

∎

##  4 The case of homogeneous varieties

###  4.1 Cycles on abelian varieties

We start with the proof of Theorem 1.13(3). The result in this case is
stronger than Conjecture 1.5 since it states that ${\rm CH}_{i}(A)={\rm
CH}_{i}(A)_{\rm smpshCh}$ for any abelian variety $A$.

###### Proof of Theorem 1.13(3).

Let $z\in{\rm CH}_{d}(A)$. We want to prove that

$\displaystyle z\in{\rm CH}_{d}(A)_{\rm smpshCh}.$ (40)

We can assume $z=[Z]$ for some subvariety $Z\subset A$ of dimension $d$. We
denote by $\tau:\widetilde{Z}\rightarrow A$ a desingularization of $Z$.
Consider the morphism

$\phi:A\times\widetilde{Z}\rightarrow A$ $(x,\tilde{z})\mapsto
x+\tau(\tilde{z}).$

Obviously $\phi$ is smooth, hence in particular flat. Furthermore, we have,
denoting $0_{A}\in A$ the origin

$\displaystyle z=\phi_{*}([0_{A}\times\widetilde{Z}]).$ (41)

If $[0_{A}]\in{\rm CH}_{0}(A)$ belongs to the subring of ${\rm CH}^{*}(A)$
generated by Chern classes of coherent sheaves on $A$, so does ${\rm
pr}_{1}^{*}([0_{A}])=[0_{A}\times\widetilde{Z}]\in{\rm
CH}(A\times\widetilde{Z})$, hence (41) implies (40) in this case. It is proved
however in [4] that for a very general abelian variety $A$ with sufficiently
divisible polarization degree and high dimension, the class of a point does
not belong to this subring of ${\rm CH}^{*}(A)$, so we cannot apply the
argument directly to $A$. Nevertheless, Debarre also proves in loc. cit. that,
if $J$ is the Jacobian of a curve $C$ of genus $g$, then for any point $x$ of
$J$, there exists a rank $g$ vector bundle on $J$ with a section whose zero
locus is $\\{x\\}$ (with its reduced structure). In particular, the class
$[x]$ belongs to the subring of ${\rm CH}^{*}(J)$ generated by Chern classes
of coherent sheaves on $J$. Let now $j:C\hookrightarrow A$ be the inclusion of
a smooth curve of genus $g$ which is a complete intersection of ample
hypersurfaces in $A$. Then by Lefschetz theorem on hyperplane sections, we
have a surjective (hence smooth) morphism $\psi=j_{*}:J=JC\rightarrow A$ of
abelian varieties, and $\psi_{*}([0_{J}])=[0_{A}]$. Let

$\phi_{J}:J\times\widetilde{Z}\rightarrow A$

be the composite $\phi\circ(\psi,Id)$. Then $\phi_{J}$ is smooth and we have

$\displaystyle z=\phi_{J*}([0_{J}\times\widetilde{Z}]).$ (42)

As $[0_{J}]$ belongs to the subring of ${\rm CH}^{*}(J)$ generated by Chern
classes of coherent sheaves on $J$, $[0_{J}\times\widetilde{Z}]$ belongs to
the subring of ${\rm CH}^{*}(J\times\widetilde{Z})$ generated by Chern classes
of coherent sheaves on $J\times\widetilde{Z}$, so (40) follows from formula
(42). ∎

###  4.2 More general homogeneous varieties

We prove in this section Theorem 1.13(1), which is the following statement

######  Theorem 4.1.

(1) Let $X$ be a projective variety which is homogeneous under a group $G$.
Then

$\displaystyle{\rm CH}_{i}(X)_{\rm flpshCh}={\rm CH}_{i}(X)$ (43)

for all $i$.

(2) If furthermore there exists a smooth projective $G$-equivariant completion
$\overline{G}$ of $G$ which satisfies ${\rm CH}_{0}(\overline{G})={\rm
CH}_{0}(\overline{G})_{\rm smpshCh}$, then

$\displaystyle{\rm CH}_{i}(X)_{\rm smpshCh}={\rm CH}_{i}(X)$ (44)

for all $i$.

By (1), $X$ satisfies Conjecture 1.5. It follows that cycles in ${\rm
CH}_{d}(X)$ are smoothable for $2d<{\rm dim}\,X$ by Proposition 2.9.

###### Proof of Theorem 4.1.

Let $z\in{\rm CH}_{d}(X)$ be the class of a subvariety $Z\subset X$ and let
$\tau:\widetilde{Z}\rightarrow X$ be a desingularization of $Z$. We consider
the morphism

$f:G\times\widetilde{Z}\rightarrow X,$ $(g,\tilde{z})\mapsto
g\cdot\tau(\tilde{z}).$

The morphism $f$ is obviously smooth, since it is $G$-equivariant so its
restriction to any $G\times\tilde{z}$ is smooth. It is however not proper, but
it provides a $G$-equivariant rational map

$\displaystyle F:\overline{G}\times\widetilde{Z}\dashrightarrow X,$ (45)

where $\overline{G}$ is a smooth projective completion of $G$ on which $G$
acts (which exists by $G$-equivariant resolution of singularities [13]). The
action of $G$ on the left hand side of (45) is via its action on
$\overline{G}$. By $G$-equivariant resolution of indeterminacies [16], there
exist a smooth projective variety $Y$ on which $G$ acts, a $G$-equivariant
birational morphism $\eta:Y\rightarrow\overline{G}\times\widetilde{Z}$, and a
morphism

$\widetilde{F}:Y\rightarrow X,$

such that $F\circ\eta=\widetilde{F}$ as rational maps to $X$. The morphism
$\widetilde{F}$ is proper since $Y$ is projective, and it is again smooth
because it is $G$-equivariant. A fortiori it is flat. Furthermore we have

$\displaystyle z=\widetilde{F}_{*}(\eta^{*}([e\times\widetilde{Z}]))\,\,{\rm
in}\,\,{\rm CH}_{d}(X),$ (46)

where $e\in G\subset\overline{G}$ is the neutral element. We now recall from
Theorem 1.6 that $0$-cycles on a smooth projective variety $T$ belong to ${\rm
CH}_{0}(T)_{\rm flpshCh}$. For $T=\overline{G}$, this provides us with a
smooth projective (nonnecessarily connected but that we can assume
equidimensional) variety $W$ and a flat proper morphism
$\phi:W\rightarrow\overline{G}$ such that $e$ can be written as

$\displaystyle e=\phi_{*}(D_{1}\ldots D_{N})\,\,\in\,\,{\rm
CH}_{0}(\overline{G}),\,\,N={\rm dim}\,W,$ (47)

for some divisors $D_{l}\in{\rm CH}^{1}(W)$. Let
$Y^{\prime}:=W\times_{\overline{G}}Y$, with first projection
$q:Y^{\prime}\rightarrow W$ and second projection $p:Y^{\prime}\rightarrow Y$.
Then $p$ is flat and proper and denoting
$\widetilde{F}^{\prime}:=\widetilde{F}\circ p:Y^{\prime}\rightarrow X$,
$\widetilde{F}^{\prime}$ is also flat and proper. Moreover, by (46) and (47),
$z$ can be written as

$\displaystyle z=\widetilde{F}^{\prime}_{*}(q^{*}D_{1}\ldots
q^{*}D_{N})\in\,\,{\rm CH}_{d}(X).$ (48)

If ${\rm CH}_{0}(\overline{G})={\rm CH}_{0}(\overline{G})_{\rm smpshCh}$, then
we can assume that the morphism $\phi:W\rightarrow\overline{G}$ is smooth,
hence $Y^{\prime}$ is smooth and $p:Y^{\prime}\rightarrow Y$ is smooth, so
that the morphism $\widetilde{F}^{\prime}:Y^{\prime}\rightarrow X$ is smooth.
In this case (48) shows that $z\in{\rm CH}_{i}(X)_{\rm smpshCh}$. This proves
statement (2).

In general, in order to conclude from (48) that $z\in{\rm CH}_{d}(X)_{\rm
flpshCh}$, that is, statement (1), we only need to know that we can arrange so
that $Y^{\prime}$ is smooth. This is the contents of Proposition 4.2 below,
which refines Theorem 1.6 for $0$-cycles. The proof of the Theorem is
finished. ∎

######  Proposition 4.2.

Let $T$ be any smooth projective variety, and let $f:Y\rightarrow T$ be a
morphism. Let $t\in T$ be any point. Then there exist a smooth projective
variety $W$ and a flat morphism $\phi:W\rightarrow T$ such that

$t=\phi_{*}t^{\prime}\,\,{\rm in}\,\,{\rm CH}_{0}(T),$

with $t^{\prime}\in{\rm CH}_{0}(W)$ a combination of products of divisor
classes on $W$, and

$T^{\prime}:=W\times_{T}Y$

is smooth.

###### Proof.

We look at the proof of Theorem 1.6 for $0$-cycles. It relies directly on
Lemma 3.11. By Chow moving lemma, it suffices to prove the statement for a
general point $t$ of $T$. We choose a sequence of inclusion

$\\{t\\}\subset T_{1}\subset T_{2}\subset\ldots\subset T_{n-1}\subset
T_{n}=T,$

where each $T_{i}$ is a general complete intersection of dimension $i$
containing $T_{i-1}$. Lemma 3.11 tells us that the blow-up
$\tau_{i}:M_{i}\rightarrow T_{i-1}\times T_{i}$ of the graph
$\Gamma_{i}\subset T_{i-1}\times T_{i}$ of the inclusion
$j_{i-1}:T_{i-1}\hookrightarrow T_{i}$ has the property that the morphism
$p_{i}:={\rm pr}_{1}\circ\tau_{i}:M_{i}\rightarrow T_{i-1}$ is smooth and the
morphism $q_{i}:={\rm pr}_{2}\circ\tau_{i}:M_{i}\rightarrow T_{i}$ is flat.
Following the proof of Proposition 3.9, the variety $W$ will be obtained by
taking the successive fibred product of the $M_{i}$ for $i\geq 2$, or
equivalently

$\displaystyle
W:=M_{2}\times_{T_{2}}M_{3}\times_{T_{3}}M_{4}\times\ldots\times_{T_{n}}M_{n}.$
(49)

We denote by $p:W\rightarrow T_{1}$ the first projection and by
$q:W\rightarrow T_{n}=T$ the last projection. Using the divisor $t$ in $T_{1}$
and the exceptional divisors of each $M_{i}$, we conclude by combining
formulas as in (18) that the variety $W$ of (49) works. It remains to prove
that the variety $W$ constructed above has the property that
$T^{\prime}:=W\times_{T}Y$ is smooth. Let $Y_{n-1}:=f^{-1}(T_{n-1})$. Then by
Bertini, $Y_{n-1}$ is a smooth hypersurface in $Y$. This fact also says that
the image ${\rm Im}\,f_{*,y}$ of the differential

$f_{*,y}:T_{Y,y}\rightarrow T_{T,f(y)}$

satisfies

$T_{T_{n-1},y}+{\rm Im}\,f_{*,y}=T_{T,y}$

at any point $y\in Y_{n-1}$. It follows that $W_{n}:=Y\times_{T}M_{n}$ is
smooth. Indeed the morphism $q_{n}:M_{n}\rightarrow T$ is smooth above
$T\setminus T_{n-1}$ and, at any point $m$ over $t\in T_{n-1}$, the image
${\rm Im}\,q_{n*,m}$ of its differential contains $T_{T_{n-1},t}$. As ${\rm
Im}\,f_{*,y}+{\rm Im}\,q_{n*,m}=T_{T,t}$, $f(y)=t$, one finds that $W_{n}$ is
smooth. We now continue inductively with the morphism $W_{n}\rightarrow
T_{n-1}$ instead of $Y\rightarrow T$ to conclude finally that $T^{\prime}$ is
smooth. ∎

##  5 Further results

###  5.1 A further stability result for flat push-forwards of Chern classes

We establish in this section the following complement to Theorem 1.9.

######  Proposition 5.1.

Let $X$ be smooth and projective and let $j_{Z}:Z\hookrightarrow X$ be the
inclusion of a smooth codimension $2$ subvariety of $X$. Then $j_{Z*}({\rm
CH}(Z)_{\rm flpshCh})\subset{\rm CH}(X)_{\rm flpshCh}$.

###### Proof.

Let $\tau:\widetilde{X}\rightarrow X$ be the blow-up of $Z$ and
$E_{Z}\subset\widetilde{X}$ be the exceptional divisor. We choose a very ample
line bundle $H$ on $X$ and denote $H^{\prime}:=\tau^{*}H\in{\rm
Pic}\,\widetilde{X}$. We have the following

######  Lemma 5.2.

Let $b\geq{\rm dim}\,Z$ be an integer. Then for $a>>0$, the inclusion
$\tilde{j}:Y\hookrightarrow\widetilde{X}$ of a general divisor in the linear
system $|aH^{\prime}-bE|$ on $\widetilde{X}$ has the property that the
composite morphism $j:=\tau\circ\tilde{j}:Y\rightarrow X$ is a finite
morphism.

###### Proof.

It suffices to know that $Y$ contains no fiber of $\tau_{E}:E\rightarrow Z$.
Equivalently, it suffices to know that the section

$\displaystyle\sigma_{\mid E}\in H^{0}(E,(aH^{\prime}-bE)_{\mid
E})=H^{0}(Z,\tau_{E*}((aH^{\prime}-bE)_{\mid E})=H^{0}(Z,{\rm
Sym}^{b}N^{*}_{Z/X}(aH))$ (50)

vanishes nowhere on $Z$, where $\tau_{E}:=\tau_{\mid E}:E\rightarrow Z$. The
vector bundle ${\rm Sym}^{b}N^{*}_{Z/X}(aH)$ has rank $b+1$ and is generated
by sections for $a$ large enough, hence a general section of ${\rm
Sym}^{b}N^{*}_{Z/X}(aH)$ vanishes at no point of $Z$ since $b\geq{\rm
dim}\,Z$. Furthermore, the restriction morphism
$H^{0}(\widetilde{X},(aH^{\prime}-bE))\rightarrow
H^{0}(E,(aH^{\prime}-bE)_{\mid E})$ is surjective for $a>>0$ by Serre
vanishing. The lemma follows. ∎

By Lemma 3.4, $\tau_{E}^{*}({\rm CH}(Z)_{\rm flpshCh})\subset{\rm CH}(E)_{\rm
flpshCh}$, hence by Proposition 3.9,

$\displaystyle j_{E*}(\tau_{E}^{*}({\rm CH}(Z)_{\rm flpshCh})\subset{\rm
CH}(\widetilde{X})_{\rm flpshCh},$ (51)

where $j_{E}:E\rightarrow\widetilde{X}$ is the inclusion. We now prove the
following

######  Lemma 5.3.

Let $M$ be a smooth projective variety and let $L$ be a very ample line bundle
on $M$. Then if

$k:N\hookrightarrow M$

is the inclusion of a very general member of the linear system $|L|$, we have

$k^{*}({\rm CH}(M)_{\rm flpshCh})\subset{\rm CH}(D)_{\rm flpshCh}(D).$

###### Proof.

Let $f:P\rightarrow M$ be a flat projective morphism, where $P$ is smooth, and
$D_{i}\in{\rm CH}^{1}(P)$, $i=1,\ldots,\,r$ be divisors. Then for any
subvariety $N\subset M$, one gets by taking the Cartesian product a flat
morphism $f_{N}:P_{N}:=P\times_{M}N\rightarrow N$, and we have by [7,
Proposition 2.3],

$\displaystyle(f_{*}(D_{1}\ldots D_{r}))_{\mid N}=f_{N*}(D_{1\mid P_{N}}\ldots
D_{r\mid P_{N}})\,\,{\rm in}\,\,{\rm CH}(N).$ (52)

If now $N$ is a general member of a linear system $|L|$, with $L$ globally
generated, $P_{N}$ is smooth by Bertini, so (52) implies that
$(f_{*}(D_{1}\ldots D_{r}))_{\mid N}\in{\rm CH}(N)_{\rm flpshCh}$ for general
$N$. Here general may depend on $z:=f_{*}(D_{1}\ldots D_{r})\in{\rm
CH}(M)_{\rm flpshCh}$ but by Lemma 3.3(d), one needs to check Lemma 5.3 only
for countably many elements of ${\rm CH}(M)_{\rm flpshCh}$. A very general $N$
will satisfy the property above for countably many choices of $f,\,D_{i}$
generating ${\rm CH}(M)_{\rm flpshCh}$ modulo algebraic equivalence, so the
proof is finished. ∎

Let $b,\,a$ and $\tilde{j}:Y\hookrightarrow\widetilde{X}$ be very general as
in Lemmas 5.2 and 5.3. Then by (51) and Lemma 5.3, for any $z\in{\rm
CH}(Z)_{\rm flpshCh}$, we have

$\displaystyle(j_{E*}(\tau_{E}^{*}z))_{\mid Y}\in{\rm CH}(Y)_{\rm flpshCh}.$
(53)

By Lemma 5.2, $j:\tau\circ\tilde{j}:Y\rightarrow X$ is finite, hence by
Proposition 3.9, (53) implies

$\displaystyle{j}_{*}((j_{E*}(\tau_{E}^{*}z))_{\mid Y})\in{\rm CH}(X)_{\rm
flpshCh}.$ (54)

We have

${j}_{*}((j_{E*}(\tau_{E}^{*}z))_{\mid Y})=bj_{Z*}(z)\,\,{\rm in}\,\,{\rm
CH}(X),$

hence we conclude that $bj_{Z*}(z)\in{\rm CH}(X)_{\rm flpshCh}$. It follows as
well that $(b+1)j_{Z*}(z)\in{\rm CH}(X)_{\rm flpshCh}$, hence
$j_{Z*}(z)\in{\rm CH}(X)_{\rm flpshCh}$. ∎

###  5.2 A weaker criterion for smoothability

The general principle underlying the proof of Theorem 1.13 is that movable
cycles (in a precise sense) are flat pushforwards of Chern classes because
$0$-cycles are, hence they are smoothable in the Whitney range. The same idea
leads to the following criterion for smoothability.

######  Theorem 5.4.

Let $p:\mathcal{Z}\rightarrow X$ be a projective morphism, where $X$ is smooth
projective. Let $q:\mathcal{Z}\rightarrow B$ be a surjective morphism of
relative dimension $d$, where $B$ is smooth projective. Assume that the locus
of points $z\in\mathcal{Z}$ where either $\mathcal{Z}$ is not smooth, or $p$
is not flat has codimension $>d$ in $\mathcal{Z}$. Then for any point $b\in
B$, the cycle is smoothable if $2d<{\rm dim}\,X$. Moreover, if $\mathcal{Z}$
is smooth and $p$ is flat, $p_{*}\mathcal{Z}_{b}$ belongs to ${\rm
CH}_{d}(X)_{\rm flpshCh}$.

###### Proof.

Assume first that $\mathcal{Z}$ is smooth and $p:\mathcal{Z}\rightarrow X$ is
flat. Let $q:\mathcal{Z}\rightarrow B$ be a surjective morphism of relative
dimension $d$, where $B$ is smooth projective. By Proposition 4.2 applied to
the morphism $q:\mathcal{Z}\rightarrow B$ (in place of $Y\rightarrow T$), for
any $b\in B$, there exist a smooth projective variety $W$ and a flat morphism
$\phi:W\rightarrow B$ such that

$b=\phi_{*}b^{\prime}\,\,{\rm in}\,\,{\rm CH}_{0}(B),$

with $b^{\prime}\in{\rm CH}_{0}(W)$ a combination of products of divisor
classes on $W$, and $\mathcal{Z}_{W}:=W\times_{B}\mathcal{Z}$ is smooth. We
now consider the morphisms

$p_{W}:\mathcal{Z}_{W}\rightarrow X,\,\,q_{W}:\mathcal{Z}_{W}\rightarrow W.$

The morphism $p_{W}$ is flat, as the composition of the flat morphisms
$\mathcal{Z}_{W}\rightarrow\mathcal{Z}$ and $\mathcal{Z}\rightarrow X$.
Furthermore we have

$\displaystyle p_{*}(\mathcal{Z}_{b})=p_{W*}(q_{W}^{*}(b^{\prime}))\,\,{\rm
in}\,\,{\rm CH}_{d}(X).$ (55)

Hence $p_{*}(\mathcal{Z}_{b})\in{\rm CH}_{d}(X)_{\rm flpshCh}$ since
$b^{\prime}\in{\rm CH}_{0}(W)_{\rm Ch}$, so $q_{W}^{*}(b^{\prime})\in{\rm
CH}_{d}(\mathcal{Z}_{W})_{\rm Ch}$. This proves the second statement of the
theorem. If, in the situation above, we only assume $B$ smooth and $p,\,q$
projective, formula (55) is still true. If furthermore $\mathcal{Z}$ is smooth
in codimension $d$, so is $\mathcal{Z}_{W}$, by Proposition 4.2 applied to the
smooth locus of $\mathcal{Z}$ and by flatness of $\phi$. If $p$ is flat in
codimension $d$, so is $p_{W}:\mathcal{Z}_{W}\rightarrow X$, again by flatness
of $\phi$. We are thus in this case in the situation of Proposition 2.9(b),
which tells us that $p_{*}(\mathcal{Z}_{b})$ is smoothable if $2d<{\rm
dim}\,X$. ∎

###  5.3 A generalization of the diagonal property

In the paper [15], the following “diagonal property” is considered, for any
smooth variety $X$ of dimension $n$:

(*) There exist a rank $n$ vector bundle $E$ on $X\times X$ and a section
$\sigma$ of $E$ whose (schematic) zero set is the diagonal $\Delta_{X}\subset
X\times X$.

Property (*) implies in particular that $\Delta_{X}=c_{n}(E)$ in ${\rm
CH}(X\times X)$ for some vector bundle $E$ on $X\times X$, so that
$\Delta_{X}\in{\rm CH}(X\times X)_{\rm Ch}$. This last property is not
satisfied by general abelian varieties with sufficiently divisible
polarization degree (see [4]).

Let us instead consider the following weaker property:

(**) The class $\Delta_{X}\in{\rm CH}(X\times X)$ of the diagonal
$\Delta_{X}\subset X\times X$ belongs to ${\rm CH}(X\times X)_{\rm smpshCh}$.

By Proposition 2.4, this property is satisfied by $X$ if $\Delta_{X}\in{\rm
CH}(X\times X)_{\rm Ch}$. Condition (**) is satisfied by abelian varieties by
Theorem 1.13 applied to $A\times A$.

######  Proposition 5.5.

Assume $X$ satisfies property (**). Then ${\rm CH}(X)={\rm CH}(X)_{\rm
smpshCh}$. That is, any cycle $z\in{\rm CH}(X)$ is of the form $f_{*}\beta$,
for some smooth proper morphism $f:Y^{\prime}\rightarrow X$ and class $\beta$
in the subring of ${\rm CH}(Y^{\prime})$ generated by Chern classes of
coherent sheaves on $Y^{\prime}$.

In particular, any cycle $z\in{\rm CH}_{d}(X)$ is smoothable if $2d<n={\rm
dim}\,X$.

###### Proof.

The second statement follows from the first by Proposition 2.9. To prove the
first statement, it suffices to consider the case of the class $[Z]$ of a
subvariety $Z\subset X$. Let $\tau:\widetilde{Z}\rightarrow X$ be a
desingularization of $Z$. Then we have the following relation in ${\rm CH}(X)$

$\displaystyle[Z]={\rm pr}_{2*}((\tau,Id_{X})^{*}\Delta_{X}),$ (56)

where ${\rm pr}_{2}:\widetilde{Z}\rightarrow X\rightarrow X$ is the second
projection. Our assumption is that $\Delta_{X}=\pi_{*}\alpha$, where $Y$ is
smooth, $\pi:Y\rightarrow X\times X$ is a smooth proper morphism, and
$\alpha\in{\rm CH}(Y)$ belongs to the subring generated by Chern classes of
coherent sheaves on $Y$. We thus have

$\displaystyle(\tau,Id_{X})^{*}\Delta_{X}=\pi^{\prime}_{*}\alpha^{\prime}\,\,{\rm
in}\,\,{\rm CH}(\widetilde{Z}\times X),$ (57)

where

$\displaystyle Y^{\prime}=(\widetilde{Z}\times X)\times_{X\times X}Y,$ (58)

$\pi^{\prime}:Y^{\prime}\rightarrow\widetilde{Z}\times X$ is the first
projection and $\alpha^{\prime}$ is the pull-back of $\alpha$ to $Y^{\prime}$
via the second projection $Y^{\prime}\rightarrow Y$. As the morphism $\pi$ is
smooth, the morphism $\pi^{\prime}$ is also smooth (hence in particular
$Y^{\prime}$ is smooth). Furthermore, $\alpha^{\prime}$ belongs to the subring
of ${\rm CH}(Y^{\prime})$ generated by Chern classes of coherent sheaves on
$Y^{\prime}$, hence the result follows from (56), (57). ∎

######  Remark 5.6.

We could weaken property (**) by asking that $\Delta_{X}$ belongs to ${\rm
CH}(X\times X)_{\rm flpshCh}$. However, this does not imply as above that
${\rm CH}(X)={\rm CH}(X)_{\rm flpshCh}$. The reason is that, in the proof
above, if we only assume that $\pi$ is flat, the variety $Y^{\prime}$ of (58)
is not necessarily smooth.

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