2.1 Notations

Let $k$ be an algebraically closed field and all varieties are assumed to be over $k$ . We shortly recall some notations (see [Reference Fulton6, Chapter 17] for detailed definitions). For any variety $X$ , denote its Chow group by $\mathrm {CH}_*(X)$ and Chow cohomology by $A^*(X)$ [Reference Fulton6, Definition 17.3]. In general, $\mathrm {CH}_*(X)$ may be not a ring for a singular variety. For a subvariety $V\subseteq X$ , write $[V]$ as the class of $V$ in $\mathrm {CH}_*(X)$ . There exists a cup product on $A^*(X)$ , denoted by $\cdot $ , which makes $A^*(X)$ a graded associative ring with unit $\mathbf {1}_X$ [Reference Fulton6, p. 324]. There exists a cap product $\cap :A^p(X)\otimes \mathrm {CH}_q(X)\rightarrow \mathrm {CH}_{q-p}(X)$ satisfying that $\mathbf {1}_X\cap \alpha =\alpha $ . For a morphism $f:X\rightarrow Y$ of varieties, there exists a ring homomorphism $f^*:A^{*}(Y)\rightarrow A^*(X)$ [Reference Fulton6, p. 324]. Set $n=\mathrm {dim}_kX-\mathrm {dim}_kY$ . If $f$ is a flat or a local complete intersection morphism [Reference Fulton6, Appendix B. 7.6], there exists a Gysin homomorphism $f^*:\mathrm {CH}_*(Y)\rightarrow \mathrm {CH}_{*+n}(X)$ [Reference Fulton6, p. 328 ( ${G}_1$ )]. Moreover, if $f$ is proper, there exists another Gysin homomorphism $f_*:A^*(X)\rightarrow A^{*-n}(Y)$ [Reference Fulton6, p. 328 ( ${G}_2$ )]. In particular, if $f$ is a closed embedding, set $\alpha |_X=f^*\alpha $ for $\alpha \in A^*(Y)$ or $\mathrm {CH}_*(Y)$ . In the category of smooth varieties,

(1) [Reference Fulton6, Corollary 17.4] $\bullet \cap [X]:A^{*}(X)\rightarrow \mathrm {CH}_{\mathrm {dim}_kX-*}(X)$ is an isomorphism, and $(\alpha \cap [X])\cdot (\beta \cap [X])=(\alpha \cdot \beta )\cap [X]$ for $\alpha $ , $\beta \in A^*(X)$ ;

(2) [Reference Fulton6, Example 17.4.3(c)] the Gysin homomorphism $f^*$ coincides with the usual pullback of the Chow groups [Reference Fulton6, p. 113].

On a complete variety $X$ , the degree of $\alpha \in \mathrm {CH}_0(X)$ is denoted by $\int _X\alpha $ [Reference Fulton6, Definition 1.4]. Let $f:X\rightarrow Y$ be a proper surjective smooth morphism of varieties. For any point $y\in Y(k)$ , the fiber $X_y=f^{-1}(y)$ is a complete variety with dimension $n=\mathrm {dim}_k X-\mathrm {dim}_k Y$ . Denote by $i:\{y\}\rightarrow Y$ the inclusion. For any $\alpha \in A^n(X)$ ,

$$ \begin{align*} i^*f_*(\alpha)=(f|_{X_y})_*(\alpha|_{X_y})=\left(\int_{X_y}\alpha|_{X_y}\cap [X_y]\right)\cdot \mathbf{1}_{\{y\}} \end{align*} $$

by [Reference Fulton6, Example 17.4.1(b)]. Since $A^0(Y)=\mathbb {Z}\cdot \mathbf {1}_Y$ , $i^*:A^0(Y)\rightarrow A^0(\{y\})$ is an isomorphism, so $f_*(\alpha )=\left (\int _{X_y}\alpha |_{X_y}\cap [X_y]\right )\cdot \mathbf {1}_Y$ .

2.2 The complex cases

Suppose that $X$ is a smooth variety over $\mathbb {C}$ .

Denote by $H^*(X,\mathbb {Z})$ (resp. $H^*(X,\mathbb {Q})$ ) the Betti cohomology of $X$ , that is, the singular cohomology of $X(\mathbb {C})$ under the analytic topology with integral (resp. rational) coefficients. Denote by $\mathrm {CH}^*(X)$ (= $\mathrm {CH}_{\mathrm {dim}_{\mathbb {C}}X-*}(X)$ ) the Chow ring of $X$ and set $\mathrm {CH}^*(X)_{\mathbb {Q}}=\mathrm {CH}^*(X)\otimes _{\mathbb {Z}}\mathbb {Q}$ .

Assume that $X$ is complete. Denote the cycle map by $\mathrm {cl}^*_X:\mathrm {CH}^*(X)\rightarrow H^*(X,\mathbb {Z})$ and set $\mathrm {cl}^*_{X,\mathbb {Q}}=\mathrm {cl}^*_X\otimes 1: \mathrm {CH}^*(X)_{\mathbb {Q}}\rightarrow H^*(X,\mathbb {Q})$ .

Assume that $X$ is projective. Set $Hdg^{p}(X,\mathbb {Z})=H^{2p}(X,\mathbb {Z})\cap H^{p,p}(X)\subseteq H^{2p}(X,\mathbb {Z})$ and $Hdg^{*}(X,\mathbb {Z})=\bigoplus _{p\geq 0} Hdg^p(X,\mathbb {Z})$ , where $H^{p,p}(X)$ is the $(p,p)$ -th Dolbeault cohomology of $X(\mathbb {C})$ . Using $\mathbb {Q}$ instead of $\mathbb {Z}$ , we can define $Hdg^{*}(X,\mathbb {Q})$ similarly. There is a natural isomorphism $Hdg^{*}(X,\mathbb {Z})\otimes _{\mathbb {Z}}\mathbb {Q}\cong Hdg^{*}(X,\mathbb {Q})$ . Furthermore, denote the refined cycle map by $\lambda ^*_X:\mathrm {CH}^*(X)\rightarrow Hdg^*(X,\mathbb {Z})$ and set $\lambda ^*_{X,\mathbb {Q}}=\lambda ^*_X\otimes 1: \mathrm {CH}^*(X)_{\mathbb {Q}}\rightarrow Hdg^*(X,\mathbb {Q})$ . For a smooth complex projective variety, the cycle map and the refined cycle map are the same map up to a change of codomain. Let $V$ be a closed subvariety of $X$ . To avoid confusion, $[V]^{top}$ and $[V]^{alg}$ mean the classes of $V$ in $H^*(X,\mathbb {Z})$ and $\mathrm {CH}^*(X),$ respectively. Clearly, $\mathrm {cl}^*_X([V]^{alg})=[V]^{top}$ . Set $H_{alg}^*(X,\mathbb {Z})=\mathrm {Im}\, \mathrm {cl}^*_{X}$ (resp. $H_{alg}^*(X,\mathbb {Q})=\mathrm {Im}\, \mathrm {cl}^*_{X,\mathbb {Q}}$ ). It is well-known that $H_{alg}^*(X,\mathbb {Z})\subseteq Hdg^{*}(X,\mathbb {Z})$ (resp. $H_{alg}^*(X,\mathbb {Q})\subseteq Hdg^{*}(X,\mathbb {Q})$ ). A class in $H_{alg}^*(X,\mathbb {Z})$ (resp. $H_{alg}^*(X,\mathbb {Q})$ ) is said to be an algebraic class and a subset $A\subseteq H_{alg}^*(X,\mathbb {Z})$ (resp. $H_{alg}^*(X,\mathbb {Q})$ ) is said to be algebraic. The integral Hodge conjecture $\mathrm {Hodge}(X,\mathbb {Z})$ (resp. Hodge conjecture $\mathrm {Hodge}(X,\mathbb {Q})$ ) means that $Hdg^{*}(X,\mathbb {Z})$ (resp. $Hdg^{*}(X,\mathbb {Q})$ ) is algebraic, that is, $H_{alg}^*(X,\mathbb {Z})= Hdg^{*}(X,\mathbb {Z})$ (resp. $H_{alg}^*(X,\mathbb {Q})= Hdg^{*}(X,\mathbb {Q})$ ), or equivalently, $\lambda ^*_{X}$ (resp. $\lambda ^*_{X,\mathbb {Q}}$ ) is surjective.

3.1 The suitable conditions

By analogy with the situation in topology, the Leray–Hirsch theorem should be stated as the following form.

Let $\pi :E\rightarrow X$ be an algebraic fiber bundle with canonical fiber F. Assume that E satisfies $\textbf {{(L-H)}}$ and F is a smooth complete variety satisfying $\textbf {{(C)}}$ (here, $\textbf {{(C)}}$ is an undetermined condition). Then

$$ \begin{align*} \sum_{i=1}^{r}e_i\cap\pi^*(\bullet):\bigoplus_{i=1}^{r}\mathrm{CH}_{*+u_i-n}(X)\rightarrow \mathrm{CH}_{*}(E) \end{align*} $$

is an isomorphism, where $e_i\in A^{u_i}(E)$ for $1\leq i\leq r$ .

Now, we attempt to find the assumption $\textbf {(C)}$ as weak as possible for such a version of Leray–Hirsch theorem. Let $X$ be any variety and $F$ a smooth complete variety. Suppose that $F$ satisfies $\textbf {(C)}$ and $A^*(F)$ is a free abelian group. Let $pr_1$ and $pr_2$ be the two projections from $X\times F$ onto $X$ and $F,$ respectively. Suppose that $f_1$ , $\ldots $ , $f_r$ is a homogeneous basis of $A^*(F)$ , where $f_i\in A^{u_i}(F)$ for $1\leq i\leq r$ . Then $pr_2^*f_1$ , $\ldots $ , $pr_2^*f_r$ satisfies $\textbf {(L-H)}$ . Hence, the trivial fiber bundle $pr_1:X\times F\rightarrow X$ satisfies the assumptions of Leray–Hirsch theorem. Then

$$ \begin{align*} \sum_{i=1}^{r}(pr_2^*f_i)\cap pr_1^*(\bullet):\bigoplus_{i=1}^{r}\mathrm{CH}_{*+u_i-n}(X)\rightarrow \mathrm{CH}_{*}(X\times F) \end{align*} $$

should be an isomorphism. By [Reference Fulton6, Corollary 17.4(b)], we have

$$ \begin{align*} \sum_{i=1}^{r}(pr_2^*f_i)\cap pr_1^*(\bullet)=\sum_{i=1}^{r} \bullet\times f_i. \end{align*} $$

So

(3.1) $$ \begin{align} \times:\mathrm{CH}_{*}(X)\otimes_{\mathbb{Z}}\mathrm{CH}_*(F)\rightarrow \mathrm{CH}_{*}(X\times F) \end{align} $$

should be an isomorphism. Hence, a necessary condition that the Leray–Hirsch theorem for Chow groups holds is that (3.1) is an isomorphism for any variety X. In fact, this condition and $\textbf {(L-H)}$ are enough for the Leray–Hirsch theorem for Chow groups (see Theorem 1.4 and Proposition 3.6). With the same argument, a necessary condition that the Leray–Hirsch theorem for Chow groups with rational coefficients holds is that

$$ \begin{align*} \times:\mathrm{CH}_{*}(X)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathrm{CH}_*(F)_{\mathbb{Q}}\rightarrow \mathrm{CH}_{*}(X\times F)_{\mathbb{Q}} \end{align*} $$

is an isomorphism for any variety $X$ .

3.2 Künneth property

All smooth complete linear schemes (in the sense of Totaro [Reference Totaro16, Section 3], e.g., cellular varieties) satisfy the Künneth property by [Reference Totaro16, Proposition 1]. Let $B$ be a Borel subgroup of a reductive group $G$ over the ground field $\mathbb {C}$ , then the quotient $G/B$ is a cellular variety (see [Reference Borel2, p. 193, Theorem (b) and Remark (3)]). Projective spaces, Grassmann varieties, and flag varieties are all examples of this type.

Proof We only prove the integral case. For any variety $Z$ , we have the commutative diagram

(3.2)

Since $X$ and $Y$ satisfy the Künneth property, the lower row and the left vertical map in (3.2) are both surjective. So the right vertical map is also surjective.

Let $X$ and $Y$ be varieties over $k$ . The class $\alpha \in \mathrm {CH}_*(X\times Y)$ (resp. $\mathrm {CH}_*(X\times Y)_{\mathbb {Q}}$ ) is called decomposable, if $\alpha $ in the image of

$$ \begin{align*} \times:\mathrm{CH}_*(X)\otimes_{\mathbb{Z}}\mathrm{CH}_*(Y)\rightarrow \mathrm{CH}_*(X\times Y) \end{align*} $$

(resp.

$$ \begin{align*} \times:\mathrm{CH}_*(X)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathrm{CH}_*(Y)_{\mathbb{Q}}\rightarrow \mathrm{CH}_*(X\times Y)_{\mathbb{Q}}). \end{align*} $$

Denote by $\mathbb {L}=(\mathrm {Spec}k, \mathrm {id}, -1)$ . Then $\mathfrak {h}(\mathbb {P}^1)=\unicode{x1D7D9}\oplus \mathbb {L}$ , where $\unicode{x1D7D9}=(\mathrm {Spec}k, \mathrm {id}, 0)$ . We call $\mathbb {L}^{\otimes n}$ ( $n\in \mathbb {N}$ ) the Lefschetz motives. For complex smooth projective varieties, the $\mathbb {Q}$ -Künneth property admits the following equivalent formulations.

Proposition 3.3 holds more generally for smooth projective varieties over a universal domain (see [Reference Kimura10, Reference Vial17, Reference Vial18]). In fact, the arguments in [Reference Kimura10, Reference Vial17, Reference Vial18] show that it holds more generally for smooth complete varieties over a universal domain. Hence, we easily obtain the following result. For the convenience for readers, we give a direct proof.

Proof Suppose that $\Delta =\sum _{i=1}^r c_i\cdot \alpha _i\times \beta _i$ , where $\alpha _i$ , $\beta _i\in \mathrm {CH}_*(X)_{\mathbb {Q}}$ and $c_i\in \mathbb {Q}$ for $1\leq i\leq r$ . For any $\gamma \in \mathrm {CH}_*(X)_{\mathbb {Q}}$ ,

(3.3) $$ \begin{align} \gamma=pr_{1*}(\Delta\cdot pr_2^*\gamma)=\sum_{i=1}^r c_i\left(\int_X \beta_i\cdot\gamma\right)\cdot \alpha_i. \end{align} $$

Assume that $\mathrm {cl}^*_{X,\mathbb {Q}}(\gamma )=0$ . Set $\gamma =\sum _{i=1}^l p_i [Z_i]^{alg}$ , where $p_1$ , $\ldots $ , $p_l\, \in \mathbb {Q}$ and $Z_1$ , $\ldots $ , $Z_l$ are subvarieties of $X$ . Then there exists $n\in \mathbb {Z}$ such that $np_1$ , $\ldots $ , $np_l\, \in \mathbb {Z}$ and $\sum _{i=1}^l np_i Z_i$ is integral homological to zero, hence is numerical to zero. So

$$ \begin{align*} \int_X \beta_i\cdot\gamma= \frac{1}{n}\int_X \beta_i\cdot\left(\sum_{i=1}^l np_i Z_i\right)=0. \end{align*} $$

By (3.3), $\gamma =0$ . We proved that $\mathrm {cl}^*_{X,\mathbb {Q}}$ is injective.

Denote by $\tilde {\Delta }$ , $\tilde {\alpha }_i$ , and $\tilde {\beta }_i$ the corresponding classes in $H^* (X, \mathbb {Q})$ of $\Delta $ , $\alpha _i$ , and $\beta _i$ , respectively. Then $\tilde {\Delta }=\sum _{i=1}^r c_i\cdot \tilde {\alpha }_i\times \tilde {\beta }_i$ . Analog to (3.3),

$$ \begin{align*} u=\sum_{i=1}^r c_i\left(\int_X \tilde{\beta}_i\cup u\right)\cdot \tilde{\alpha}_i=\mathrm{cl}^*_{X,\mathbb{Q}}\left(\sum_{i=1}^r c_i\left(\int_X \tilde{\beta}_i\cup u\right)\cdot \alpha_i\right) \end{align*} $$

for any $u\in H^*(X,\mathbb {Q})$ , which implies that $\mathrm {cl}^*_{X,\mathbb {Q}}$ is surjective.

Proof We only prove the integral case. Since $F$ satisfies the Künneth property, the class of the diagonal $F\times F$ is decomposable. By [Reference Ellingsrud and Strømme5, Theorem 2.1(ii)], $\mathrm {CH}_*(F)$ is a free abelian group. Assume that $\{\,\gamma _i\,|\,1\leq i\leq r\,\}$ is a homogeneous $\mathbb {Z}$ -basis of $\mathrm {CH}_*(F)$ . Let $V\subseteq X$ be an irreducible open neighborhood of $x_0$ such that $E_V\cong V\times F$ . Since $F$ satisfies the Künneth property, we have $\alpha |_{E_V}=\sum _{i=1}^r \beta _{i}\times \gamma _i$ for some $\beta _i\in \mathrm {CH}_{p_i}(V)$ ( $1\leq i\leq r$ ). Set $n=\mathrm {dim}_kX$ . For $p_i=n$ , there exists $n_i\in \mathbb {Z}$ such that $\beta _i=n_i[V]$ . Then

$$ \begin{align*} 0=\alpha|_{E_{x_0}}=\sum_{i=1}^r \beta_{i}|_{\{x_0\}}\times \gamma_i=\sum\limits_{\substack{1\leq i\leq r\\p_i=n}} \beta_{i}|_{\{x_0\}}\times \gamma_i=\sum\limits_{\substack{1\leq i\leq r\\p_i=n}} n_{i}\cdot [x_0]\times \gamma_i, \end{align*} $$

which implies that $\beta _{i}=0$ if $p_i=n$ . Set $Z=\bigcup _{\substack {1\leq i\leq r\\p_i<n}}\mathrm {supp} \beta _i$ and $U=V-Z$ . Clearly, $Z$ is a subvariety of $V$ with dimension $<n$ and hence $U$ is a nonempty open subset in $X$ . Moreover, if $p_i<n$ , $\beta _i|_U=0$ in $\mathrm {CH}_*(U)$ . So

$$ \begin{align*} \alpha|_{E_V}=\sum\limits_{\substack{1\leq i\leq r\\p_i<n}} \beta_{i}\times \gamma_i \end{align*} $$

restricts to zero in $\mathrm {CH}_*(E_{U})$ .

3.3 Proof of Theorem 1.4

We only give a proof for the integral case and the rational case can be proved similarly. Set $E_C=\pi ^{-1}(C)$ for an open or a closed subset $C$ of $X$ . Let $i_C:C\rightarrow X$ and $i_{E_C}:E_C\rightarrow E$ be the inclusions. Set

$$ \begin{align*} \Phi_C=\sum_{i=1}^{r}(i_{E_C}^*e_i)\cap(\pi|_{E_C})^*(\bullet):\bigoplus_{i=1}^{r}\mathrm{CH}_{*+u_i-n}(C)\rightarrow \mathrm{CH}_{*}(E_C). \end{align*} $$

We aim to prove that $\Phi _X$ is an isomorphism. Fix a point $x_0\in X(k)$ .

Denote by $\Delta \in \mathrm {CH}_*(E_{x_0}\times E_{x_0})$ the cycle class of the diagonal of $E_{x_0}$ . Since $E_{x_0}\cong F$ satisfies the Künneth property, there exist $\alpha _i\in A^*(E_{x_0})$ for $1\leq i\leq r$ such that

$$ \begin{align*} \Delta=\sum_{i=1}^r(\alpha_i\cap [E_{x_0}])\times (e_i|_{E_{x_0}}\cap [E_{x_0}]). \end{align*} $$

Let $pr_1$ and $pr_2$ be the two projections of $E_{x_0}\times E_{x_0}$ onto $E_{x_0}$ , respectively. Then

$$ \begin{align*} \begin{aligned} e_i|_{E_{x_0}}\cap [E_{x_0}]&=pr_{2*}\left(\Delta\cdot pr_1^*(e_i|_{E_{x_0}}\cap [E_{x_0}])\right)\\ &= \sum_{j=1}^rpr_{2*}\left[\left((\alpha_j\cap [E_{x_0}])\cdot (e_i|_{E_{x_0}}\cap [E_{x_0}])\right)\times (e_j|_{x_0}\cap [E_{x_0}])\right]\\ &= \sum_{j=1}^r\left(\int_{E_{x_0}}(\alpha_j\cdot e_i|_{E_{x_0}})\cap [E_{x_0}]\right)\cdot (e_j|_{E_{x_0}}\cap [E_{x_0}]). \end{aligned} \end{align*} $$

Since $\{\,e_i|_{E_{x_0}}\,|\,i=1,\ldots ,r\,\}$ is a $\mathbb {Z}$ -basis of $A^*(E_{x_0})$ ,

(3.4) $$ \begin{align} \int_{E_{x_0}}(\alpha_j\cdot e_i|_{E_{x_0}})\cap [E_{x_0}]=\delta_{ij}, \end{align} $$

where $\delta _{ij}$ is Kronecker’s delta. There exist $f_i\in A^*(E)$ such that $f_i|_{E_{x_0}}=\alpha _i$ for $1\leq i\leq r$ , since the restriction $A^*(E)\rightarrow A^*(E_{x_0})$ is surjective. By (3.4), we have

(3.5) $$ \begin{align} \pi_*(f_j\cdot e_i)=\left(\int_{E_{x_0}}(f_j\cdot e_i)|_{E_{x_0}}\cap [E_{x_0}]\right)\cdot\mathbf{1}_X=\delta_{ij}\cdot\mathbf{1}_X, \end{align} $$

where $\mathbf {1}_X\in A^0(X)$ is the unit of $A^*(X)$ . Assume that $\Phi _X(\beta _1,\ldots , \beta _r)=0$ , that is, $\sum _{i=1}^{r}e_i\cap \pi ^*(\beta _i)=0$ in $\mathrm {CH}_{*}(E)$ . By [Reference Fulton and MacPherson7, p. 26, (G₄)(ii)] and (3.5),

$$ \begin{align*} 0=\pi_*\left(f_j\cap\left(\sum_{i=1}^{r}e_i\cap\pi^*(\beta_i)\right)\right)=\sum_{i=1}^{r}\pi_*(f_j\cdot e_i)\cap\beta_i=\beta_j. \end{align*} $$

So $\Phi _X$ is injective.

Now, we prove that $\Phi _X$ is surjective by induction for the dimension of $X$ . Clearly, $\Phi _X$ is surjective for $\mathrm {dim}_k X=0$ . Assume that $\mathrm {dim}_k X>0$ .

Case $1$ : Suppose that $X$ is irreducible.

For any $\alpha \in \mathrm {CH}_{*}(E)$ , $\alpha |_{E_{x_0}}=\gamma \cap [E_{x_0}]$ for some $\gamma \in A^{n-*}(E_{x_0})$ , since $E_{x_0}$ is a smooth variety. By the assumption of the theorem, $\gamma =\sum _{i=1}^rn_ie_i|_{E_{x_0}}$ for some integers $n_1$ , $\ldots $ , $n_r$ . Set

$$ \begin{align*} \beta=(n_1[X], \ldots, n_r[X])\in \bigoplus_{i=1}^{r}\mathrm{CH}_{*+u_i-n}(X). \end{align*} $$

Then $\alpha |_{E_{x_0}}=\Phi _X(\beta )|_{E_{x_0}}$ , that is, $(\alpha -\Phi _X(\beta ))|_{E_{x_0}}=0$ . By Lemma 3.5, $(\alpha -\Phi _X(\beta ))|_{E_U}=0$ for a nonempty open subvariety $U$ of $X$ . Set $Z=X-U$ . Notice that $i_{E_Z*}(\pi |_{E_Z})^*=\pi ^*i_{Z*}$ on $\mathrm {CH}_{*}(Z)$ [Reference Fulton6, Example 17.4.1(a)] and $i_{E_U}^*(\xi \cap \eta )=i_{E_U}^*(\xi )\cap i_{E_U}^*(\eta )$ for $\xi \in A^*(E)$ and $\eta \in \mathrm {CH}_*(E)$ . We have the commutative diagram

of localization sequences. So $\alpha -\Phi _X(\beta )=i_{E_Z*}(\zeta )$ for some $\zeta \in \mathrm {CH}_{*}(E_Z)$ . Since $\mathrm {dim}_k Z<\mathrm {dim}_k X$ , $\Phi _Z$ is surjective by the inductive hypothesis. There exists $\omega \in \bigoplus _{i=1}^{r}\mathrm {CH}_{*+u_i-n}(Z)$ such that $\zeta =\Phi _Z(\omega )$ . Hence,

$$ \begin{align*} \alpha=\Phi_X(\beta)+i_{E_Z*}(\Phi_Z(\omega))=\Phi_X(\beta+i_{Z*}(\omega)). \end{align*} $$

Case $2$ : Suppose that $X$ is reducible.

Let $\{\,X_{\alpha }\,|\, \alpha \in \Lambda \,\}$ be all irreducible components of $X$ . Set $X_{\alpha \beta }=X_\alpha \cap X_\beta $ for $\alpha \neq \beta $ and $Y=\bigcup _{\substack {\alpha ,\beta \in \Lambda \\ \alpha \neq \beta }}X_{\alpha \beta }$ . Then $Y$ is a variety of dimension $< \mathrm {dim}_k X$ and $X-Y=\bigsqcup _{\alpha \in \Lambda } X^\prime _{\alpha }$ is a disjoint union of irreducible varieties, where $X^\prime _{\alpha }=X_{\alpha }-\bigcup _{\substack {\beta \in \Lambda \\ \beta \neq \alpha }}X_{\alpha \beta }$ . Notice that $\Phi _{X-Y}$ is just

$$ \begin{align*} \bigoplus_{\alpha\in\Lambda}\Phi_{X^\prime_{\alpha}}:\bigoplus_{\alpha\in\Lambda}\bigoplus_{i=1}^{r}\mathrm{CH}_{*+u_i-n}(X^\prime_{\alpha})\rightarrow \bigoplus_{\alpha\in\Lambda}\mathrm{CH}_{*}(E_{X^\prime_{\alpha}}). \end{align*} $$

Since $X^\prime _{\alpha }$ is irreducible, $\Phi _{X^\prime _{\alpha }}$ is surjective by Case $1$ . So $\Phi _{X-Y}$ is surjective. We easily prove that $\Phi _X$ is surjective by the inductive hypothesis and the commutative diagram of localization sequences for $(Y,X,X-Y)$ .

This completes the proof.

3.4 Several applications of Theorem 1.4

Proposition 3.6 Let $X$ be a smooth complete variety satisfying the Künneth property (resp. $\mathbb {Q}$ -Künneth property). Then

$$ \begin{align*} \times:\mathrm{CH}_*(X)\otimes_{\mathbb{Z}}\mathrm{CH}_*(Y)\rightarrow \mathrm{CH}_*(X\times Y) \end{align*} $$

(resp.

$$ \begin{align*} \times:\mathrm{CH}_*(X)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathrm{CH}_*(Y)_{\mathbb{Q}}\rightarrow \mathrm{CH}_*(X\times Y)_{\mathbb{Q}}) \end{align*} $$

is an isomorphism for any variety $Y$ .

Proof We only prove the integral case. By [Reference Ellingsrud and Strømme5, Theorem 2.1(ii)], $\mathrm {CH}_*(X)$ is a free abelian group. Let $\{\,\gamma _i\in \mathrm {CH}_{p_i}(X)\,|\,1\leq i\leq r\,\}$ be a homogeneous $\mathbb {Z}$ -basis of $\mathrm {CH}_*(X)$ . Since $X$ is a smooth variety, there exists $e_i\in A^{\mathrm {dim}_kX-p_i}(X)$ such that $e_i\cap [X]=\gamma _i$ for $1\leq i\leq r$ . For any variety $Y$ , let $pr_1:X\times Y\rightarrow X$ and $pr_2:X\times Y\rightarrow Y$ be the projections from $X\times Y$ onto $X$ and $Y,$ respectively. For any $y\in Y(k)$ , $(pr_1^*e_i)|_{X\times \{y\}}$ , $\ldots $ , $(pr_1^*e_r)|_{X\times \{y\}}$ is a basis of $\mathrm {CH}_*(X\times \{y\})$ . We view $pr_2:X\times Y\rightarrow Y$ as a fiber bundle over $Y$ . By Theorem 1.4,

$$ \begin{align*} \sum_{i=1}^{r}\gamma_i \times \bullet=\sum_{i=1}^{r}pr_1^*e_i\cap pr_2^*(\bullet):\bigoplus_{i=1}^{r}\mathrm{CH}_{*-p_i}(Y)\rightarrow \mathrm{CH}_{*}(X\times Y) \end{align*} $$

is an isomorphism, which implies the conclusion.

We may derive from Theorem 1.4 the Leray–Hirsch theorem for the algebraic part of the cohomology.

Proposition 3.7 Let $E$ be a complex projective variety and let $\pi :E\rightarrow X$ be an algebraic fiber bundle of $\mathbb {Q}$ -Leray–Hirsch type on a complex smooth projective variety $X$ . Assume that the cycle classes $e_1$ , $\ldots $ , $e_r$ in $\mathrm {CH}^*(E)_{\mathbb {Q}}$ are of pure degrees $u_1$ , $\ldots $ , $u_r$ , respectively, such that their restrictions $e_1|_{E_x}$ , $\ldots $ , $e_r|_{E_x}$ form a basis of $\mathrm {CH}^*(E_x)_{\mathbb {Q}}$ for every point $x\in X(\mathbb {C})$ . Then

$$ \begin{align*} \sum\limits_{i=1}^{r}\mathrm{cl}^{u_i}_{E,\mathbb{Q}}(e_i)\cup \pi^*(\bullet): \bigoplus_{i=1}^{r}H_{alg}^{*-2u_i}(X,\mathbb{Q}) \tilde{\rightarrow} H_{alg}^{*}(E,\mathbb{Q}). \end{align*} $$

is an isomorphism.

Proof Consider the commutative diagram

(3.6)

For any $x\in X(\mathbb {C})$ , since $E_x$ satisfies the $\mathbb {Q}$ -Künneth property, $\mathrm {cl}_{E_x}^*:\mathrm {CH}^*(E_x)_{\mathbb {Q}}\rightarrow H^*(E_x,\mathbb {Q})$ is an isomorphism by Proposition 3.4, hence $\mathrm {cl}^{u_1}_{E,\mathbb {Q}}(e_1)|_{E_x}$ , $\ldots $ , $\mathrm {cl}^{u_r}_{E,\mathbb {Q}}(e_r)|_{E_x}$ is a basis of $H^*(E_x,\mathbb {Q})$ . By Theorem 1.1, the lower row in (3.6) is an isomorphism. By Theorem 1.4, the upper row in (3.6) is an isomorphism. We easily conclude the result.

For a complex smooth projective variety $X$ , set $h^p(X)=\mathrm {dim}_{\mathbb {Q}}H_{alg}^{2p}(X,\mathbb {Q})$ for every $p\in \mathbb {Z}$ . Clearly, $h^p(X)=0$ for $p< 0$ or $> \mathrm {dim}_{\mathbb {C}}X$ .

Corollary 3.8 Suppose that the complex projective variety $E$ is an algebraic fiber bundle of $\mathbb {Q}$ -Leray–Hirsch type with the fiber $F$ of dimension $m$ over a complex smooth projective variety $X$ . Then

$$ \begin{align*} h^p(E)=\sum_{j=0}^mb_{2j}(F)h^{p-j}(X) \end{align*} $$

for any $p\in \mathbb {Z}$ .

Proof Let $e_1$ , $\ldots $ , $e_r\in \mathrm {CH}^*(E)_{\mathbb {Q}}$ be of pure degrees $u_1$ , $\ldots $ , $u_r$ , respectively, such that their restrictions $e_1|_{E_x}$ , $\ldots $ , $e_r|_{E_x}$ form a basis of $\mathrm {CH}^*(E_x)_{\mathbb {Q}}$ for every point $x\in X(\mathbb {C})$ . By Proposition 3.7, $h^p(E)=\sum _{i=1}^rh^{p-u_i}(X)$ for any $p\in \mathbb {N}$ . By Proposition 3.4, $\mathrm {dim}_{\mathbb {Q}}\mathrm {CH}^j(E_x)_{\mathbb {Q}}=b_{2j}(E_x)=b_{2j}(F)$ , where $b_i(\bullet )$ is the $i$ -th Betti number. Since $e_1|_{E_x}$ , $\ldots $ , $e_r|_{E_x}$ is a basis of $\mathrm {CH}^*(E_x)_{\mathbb {Q}}$ , the cardinality of $\{\,i\,|\,u_i=j,\,1\leq i\leq r\,\}$ is $b_{2j}(F)$ and $0\leq u_i\leq m$ for $1\leq i\leq r$ . Hence,

$$ \begin{align*} h^p(E)=\sum_{j=0}^n\sum\limits_{\substack{1\leq i\leq r\\u_i=j}}h^{p-u_i}(X)=\sum_{j=0}^mb_{2j}(F)h^{p-j}(X) \end{align*} $$

for any $p\in \mathbb {Z}$ .

Now, we give a proof of Corollary 1.5.

Proof Let $\pi :E\rightarrow X$ be the projection of the algebraic fiber bundle. Obviously, $E$ is smooth and its canonical fiber is projective. By Theorem 1.4,

$$ \begin{align*} \sum_{i=1}^{r}e_i\cdot\pi^*(\bullet):\bigoplus_{i=1}^{r}\mathrm{CH}^{*-u_i}(X)_{\mathbb{Q}}\rightarrow \mathrm{CH}^{*}(E)_{\mathbb{Q}} \end{align*} $$

is an isomorphism. By Manin’s identity principle, we have an isomorphism

$$ \begin{align*} \bigoplus_{i=1}^{r}\mathfrak{h}(X)(u_i)\overset{\sim}{\to} \mathfrak{h}(E) \end{align*} $$

in $\mathscr {M}(k,\mathbb {Q})$ .

4.1 Examples

Assume that $X$ is a complex smooth projective variety. Now, we list some examples of algebraic fiber bundles of Leray–Hirsch type as follows.

$\bullet $ The trivial fiber bundle $X\times F$ over X, where F is a complex smooth complete variety satisfying the Künneth property. Obviously, its canonical fiber is $F$ . Let $pr_1:X\times F\rightarrow X$ and $pr_2:X\times F\rightarrow F$ be projections from $X\times F$ onto $X$ and $F,$ respectively. Assume that $\{\,f_i\,|\,1\leq i\leq r\,\}$ is a homogeneous $\mathbb {Z}$ -basis of $\mathrm {CH}_*(F)$ . For any $x\in X(\mathbb {C})$ , $(pr_2^*f_1)|_{\{x\}\times F}$ , $\dots $ , $(pr_2^*f_r)|_{\{x\}\times F}$ is a basis of $\mathrm {CH}^*(\{x\}\times F)$ .

$\bullet $ A quadric bundle associated with a vector bundle with a nondegenerate hyperbolic quadric form over X. Let $V$ be a vector bundle of rank $N$ ( $=2n$ or $2n+1$ ) with a nondegenerate hyperbolic quadric form over a smooth projective variety $X$ and let $Q\subseteq \mathbb {P}(V)$ be the associated quadric bundle. The fiber over any point of $X(\mathbb {C})$ is a hyperbolic quadric in $\mathbb {C}P^{N-1}$ , which has a cellular decomposition. Set $h=c_1({\mathcal {O}}_{\mathbb {P}(V)}(1))|_{Q}\in \mathrm {CH}^1(Q)$ and denote by $\gamma \in \mathrm {CH}^n(Q)$ the class of a maximal isotropic subbundle. Then $1$ , $h$ , $h^2$ , $\ldots $ , $h^{n-1}$ , $\gamma $ , $h\gamma $ , …, $h^{n-1}\gamma $ is a basis of $\mathrm {CH}^*(Q)$ as a $\mathrm {CH}^*(X)$ -module [Reference Edidin and Graham4, Theorem 7], whose restrictions freely generate $\mathrm {CH}^*(Q_x)$ for any $x\in X(\mathbb {C})$ [Reference Edidin and Graham4, Lemma 1].

$\bullet $ A flag bundle over X. The canonical fiber of a flag bundle is a flag variety, which is a cellular variety. Suppose that $V$ is an algebraic vector bundle with rank $n$ over $X$ and $n_1$ , $\ldots $ , $n_r$ is a sequence of positive integers with $\sum _{i=1}^rn_i=n$ , such that the fiber $E_x$ of $E$ over $x\in X(\mathbb {C})$ is the flag variety $Fl(n_1,\ldots ,n_r)(V_x)=$

$$ \begin{align*} \begin{aligned} \{(V_1,\ldots,V_{r-1})|&0=V_0\subsetneq V_1\subsetneq V_2\subsetneq \cdots \subsetneq V_{r-1}\subsetneq V_r=V_x, \mathrm{ where } V_i \mathrm{ is a } \\ & \mathrm{ complex vector space with dimension } \sum_{j=1}^{i}n_j \mathrm{ for } 1\leq i\leq r\}. \end{aligned} \end{align*} $$

For $0\leq i\leq r$ , let $E_i$ be the universal subbundle on $E$ whose fiber over the point $(V_1,\ldots ,V_{r-1})$ is $V_i$ . Notice that $E_0=0$ and $E_r=\pi ^*V$ , where $\pi $ is the projection from $E$ onto $X$ . Denote by $E^{(i)}=E_i/E_{i-1}$ the successive universal quotient bundles and by $c_j^{(i)}=c_j(E^{(i)})\in \mathrm {CH}^*(E)$ the $j$ -th Chern class of $E^{(i)}$ for $1\leq i\leq r$ . As we know, $\mathrm {CH}^*(E)$ is a $\mathrm {CH}^*(X)$ -module via $\pi ^*$ freely generated by the monomials $e_1$ , $\ldots $ , $e_k$ on $c_1^{(1)}$ , $\ldots $ , $c_{n_1}^{(1)}$ , $c_1^{(2)}$ , $\ldots $ , $c_{n_2}^{(2)}$ , $\ldots $ , $c_1^{(r)}$ , $\ldots $ , $c_{n_r}^{(r)}$ . For every $x\in X$ , the restrictions $c_{j_i}^{(i)}|_{E_x}$ ( $1\leq i\leq r$ , $1\leq j_i\leq n_i$ ) to $E_x$ are exactly the Chern classes of the successive universal quotient bundles of the flag variety $E_x$ . So $e_1|_{E_x}$ , $\ldots $ , $e_k|_{E_x}$ freely generate the Chow ring $\mathrm {CH}^*(E_x)$ as an abelian group.

4.2 Proof of Theorem 1.6

By Corollary 1.5, there exists an isomorphism

$$ \begin{align*} \bigoplus_{i=1}^{r}\mathfrak{h}(X)(u_i)\overset{\sim}{\to} \mathfrak{h}(E) \end{align*} $$

in $\mathscr {M}(k,\mathbb {Q})$ , which implies that $S(E) \Leftrightarrow S(X)$ immediately.

4.3 Proof of Theorem 1.7

We only give a proof for the integral case and the rational case can be proved similarly. Assume that $\pi :E\rightarrow X$ is the projection of the algebraic fiber bundle and $F$ is its canonical fiber. Set $n=\mathrm {dim}_{\mathbb {C}}X$ and $m=\mathrm {dim}_{\mathbb {C}}F$ . Suppose that $V_1$ , $\ldots $ , $V_r$ are algebraic cycles with integral coefficients on $E$ such that their restrictions $[V_1]^{alg}|_{E_x}$ , $\dots $ , $[V_r]^{alg}|_{E_x}$ freely generate $\mathrm {CH}^*(E_x)$ for every point $x\in X(\mathbb {C})$ . Set $\mathrm {codim}_{\mathbb {C}}V_i=u_i$ for $1\leq i\leq r$ . Consider the commutative diagram

For any $x\in X(\mathbb {C})$ , since $E_x$ satisfies the Künneth property, $\mathrm {cl}_{E_x}^*:\mathrm {CH}^*(E_x)\rightarrow H^*(E_x,\mathbb {Z})$ is an isomorphism by [Reference Ellingsrud and Strømme5, Theorem 2.1(iii)], hence $[V_1]^{top}|_{E_x}$ , $\dots $ , $[V_r]^{top}|_{E_x}$ freely generate $H^*(E_x,\mathbb {Z})$ . The lower row is an isomorphism by Theorem 1.1 and the Hodge decompositions. By Theorem 1.4, the upper row is an isomorphism. Hence, $\lambda ^*_X$ is surjective if and only if $\lambda ^*_E$ is surjective.