Let $\left(X,h\right)$ be a compact Kähler manifold of dimension $n$ with a Kähler metric $h$ and the associated fundamental 2-form Ω. Let $\ell =\left[\Omega \right]$ be the cohomology class of the 2-form Ω.

(2.1.1) In the holomorphic coordinates $\left(z_1,\cdots ,z_n\right)$ of $X$, we can write them as follows:

$h={\displaystyle \underset{\mu ,\nu }{\sum }{h}_{\mu \nu }d{z}_{\mu }\otimes d{\bar{z}}_{\nu }}=g+\left(-2i\right)\Omega ,h_{\mu \nu }={\bar{h}}_{\nu \mu },$

${}_{}{}^t\bar{u}\left(h_{\mu \nu }\right)u>0\text{if}u\ne 0,$

$\Omega =\frac{i}{2}{\displaystyle \underset{\mu ,\nu }{\sum }{h}_{\mu \nu }d{z}_{\mu }\wedge d{\bar{z}}_{\nu }},$

and $d\Omega =0$, Ω is real and type (1,1).

(2.1.2) If $\Delta =d{d}^{\text{*}}+d^{\text{*}}d$, $\square =\partial {\partial }^{\text{*}}+{\partial }^{\text{*}}$, $\bar{\square }=\bar{\partial }{\bar{\partial }}^{\text{*}}+{\bar{\partial }}^{\text{*}}\bar{\partial }$ are Laplacians, then

$\Delta =2=2\bar{\square },\ast \Delta =\Delta \ast ,\ast \bar{\square }=\bar{\square }\ast ,$

where $\ast$ is the Hodge star operator. Every cohomology class is represented by a unique harmonic form.

(2.1.3) The cohomologies of $X$ have the Hodge decompositions:

$H^k\left(X,ℂ\right)=\underset{p+q=k}{\oplus }{H}^{p,q}\left(X\right),{\bar{H}}^{p,q}\left(X\right)=H^{q,p}\left(X\right).$

(2.1.4) The Lefschetz decomposition:

$H^k\left(X,ℂ\right)=\underset{2r\le k}{\oplus }{\ell }^r\cup H_0^{k-2r}\left(X,ℂ\right),$

where $H_0^{k-2r}\left(X,ℂ\right)=\text{Ker}\left[{\ell }^{n-k+2r+1}\cup :H^{k-2r}\left(X,ℂ\right)\to H^{2n-k+2r+2}\left(X,ℂ\right)\right]$ is the primitive cohomology of X. Also,

$H_0^k\left(X,ℂ\right)=\underset{p+q=k}{\oplus }{H}_0^{p,q}\left(X\right),$

where $H_0^{p,q}\left(X\right)=\text{Ker}\left[{\ell }^{n-k+1}\cup :H^{p,q}\left(X\right)\to H^{n-k+p+1,n-k+q+1}\left(X\right)\right]$.

(2.1.5) There is a polarization $Q$ on $H^k\left(X,ℂ\right)$:

$Q:H^k\left(X,ℂ\right)\times H^k\left(X,ℂ\right)\to ℂ$

defined by

$Q\left(\xi ,\eta \right)=\underset{2r\le k}{{\displaystyle \sum }}{\left(-1\right)}^{\frac{k\left(k+1\right)}{2}+r}{\displaystyle {\int }_X{\ell }^{n-k+2r}\cup {\xi }_r\cup {\eta }_r},$

where $\xi ={\displaystyle {\sum }_r{\ell }^r\cup {\xi }_r}$, $\eta ={\displaystyle {\sum }_r{\ell }^r\cup {\eta }_r}\in H^k\left(X,ℂ\right)$. Then the polarization $Q$ on $H^k\left(X,ℂ\right)$ is orthogonal, nondegenerate, and positive definite on the Lefschetz decomposition, and satisfies the Hodge-Riemann bilinear relations [21] .

(2.2.1) A closed form $\phi$ on $X$ is said to be integral if the cohomology class $\left[\phi \right]$ is in the image of the natural map $H^{\text{*}}\left(X,\mathbb{Z}\right)\to H^{\text{*}}\left(X,ℂ\right)$. Let $\left(X,h\right)$ be a compact Kähler manifold and Ω be its associated fundamental form. If Ω is an integral form, then Ω is called a Hodge form of $X$ and $h$ is called a Hodge metric on $X$, and $\left(X,h\right)$ is called a Hodge manifold.

(2.2.2) Kodaira Embedding Theorem [22] : A compact Kähler manifold is Hodge if and only if it is a complex projective algebraic manifold.

i) Every complex submanifold of a complex projective space is Hodge.

ii) Let $\left\{w_1,\cdots ,w_{2n}\right\}$ be a $2n$ independent periods in the complex space ${ℂ}^n$ of dimension $n$. The complex torus $X:={ℂ}^n/\left[w_1,\cdots ,w_{2n}\right]$ is a compact Kähler manifold. However, $X$ is a Hodge manifold if and only if the $n\times 2n$ matrix $\left[w_1,\cdots ,w_{2n}\right]$ is a Riemann matrix [21] .

(2.2.3) A compact Kähler manifold $X$ admits a positive line bundle $L\to X$ if and only if $X$ is Hodge. Here a line bundle $L\to X$ is positive means that, if

$h$ is a metric on $L$, $\Omega =\bar{\partial }\partial \log h$ and its first Chern class $c_1\left(L\right)=\frac{i}{2\pi }\left[\Omega \right]$, then

Ω is a positive definite Hermitian symmetric matrix.

(2.2.4) Let $X$ be a Hodge manifold and $L\to X$ be a positive holomorphic line bundle. If for each point $x\in X$ there is a section $s\in \text{Γ}\left(X,L\right)$ such that $s\left(x\right)\ne 0$, then $X$ is embedded in the projective space ${\mathbb{P}}^N$, where $N:=\dim \left(\Gamma \left(X,L\right)\right)-1$. In this case $L\to X$ is called very ample. A line bundle $L\to X$ is called ample if there is an $m\ge 1$ such that $L^{\otimes m}\to X$ is very ample. Thus every Hodge manifold admits an ample line bundle.

(2.2.5) Let $X$ be an algebraic variety defined by the zero locus of $k$ homogeneous polynomials on a projective space ${\mathbb{P}}^N$. Then $X$ is a Hodge manifold of codimension $k$ in ${\mathbb{P}}^N$. In particular, if $N=4$, $k=1$, and the homogeneous polynomial has degree 5, then the quintic threefold $X$ is Hodge and Calabi-Yau since $c_1\left(TM\right)=0$. The quintic threefold $X$ plays a crucial role in mirror symmetry [11] and the Hodge conjecture holds on $X$ by Subsection 3.5.

The cup product map

${\ell }^{n-k}\cup :H^k\left(X;\mathbb{Q}\right)\to H^{2n-k}\left(X,\mathbb{Q}\right)$

is an isomorphism of Hodge structures, and the pairing

${\left(\alpha ,\beta \right)}_{\ell }={\displaystyle {\int }_X{\ell }^{n-k}\cup \alpha \cup \beta },\alpha ,\beta \in H^k\left(X,\mathbb{Q}\right)$

is nondegenerate. On the Lefschetz decomposition:

$\begin{array}{l}{H}^{2k}\left(X,Q\right)=\underset{r=0}{\stackrel{k}{\oplus }}{\ell }^r\cup H_0^{2k-2r}\left(X,Q\right),\\ H_0^{k,k}\left(X\right)=\underset{r=0}{\stackrel{k}{\oplus }}{\ell }^r\cup H_0^{k-r,k-r}\left(X\right)\end{array}$

the pairing ${\left(,\right)}_{\ell }$ is orthogonal and nondegenerate.

For $0\le k\le n$, the group

${\text{Hdg}}^{2k}\left(X\right):=H^{2k}\left(X,\mathbb{Q}\right)\cap H^{k,k}\left(X\right)$

is called the Hodge group of Hodge classes with degree $2k$, and the group

$H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}:=\left\{{\displaystyle \sum {\alpha }_i\left[Z_i\right]}:\text{a}\text{finite}\text{sum}|\begin{array}{l}{\alpha }_i\in \mathbb{Q},Z_i\text{is}\text{a}\text{subvariety}\text{of}\hfill \\ \text{codimension}k\text{in}X,\text{for}\text{each}i\hfill \end{array}\right\}$

is called the algebraic group of $X$ of algebraic classes with degree $2k$.

Let $X$ be a Hodge manifold of dimension $n$. Then for $0\le k\le n$, the Hodge decomposition is

$H^{2k}\left(X,ℂ\right)=\underset{p+q=2k}{\oplus }{H}^{p,q}\left(X\right)\supset H^{k,k}\left(X\right)$

For each subvariety $Z$ of codimension $k$ in $X$, if $\alpha \notin H^{n-k,n-k}\left(X\right)$, then ${\displaystyle {\int }_Z\alpha }=0$ by type reason. Thus, the algebraic class $\left[Z\right]$ is in ${\text{Hdg}}^{2k}\left(X\right)$. Therefore

$H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}\subset {\text{Hdg}}^{2k}\left(X\right).$

Let $X$ be a Hodge manifold of dimension $n$. Then

${\text{Hdg}}^{2k}\left(X\right)=H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}},0\le k\le n$.

That is, ${\text{Hdg}}^{2k}\left(X\right)\subset H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}$, $0\le k\le n$.

Let $X$ be a connected Hodge manifold of dimension $n$. Then

1) $H^0\left(X,\mathbb{Q}\right)={\text{Hdg}}^0\left(X\right)=H^0{\left(X,\mathbb{Q}\right)}_{\text{alg}}=\mathbb{Q}\left(\left[X\right]\right)$,

2) $H^{2n}\left(X,\mathbb{Q}\right)={\text{Hdg}}^{2n}\left(X\right)=H^{2n}\left(X,\mathbb{Q}\right)=\mathbb{Q}\left(\left[\text{pt}\right]\right)$,

3) (Lefschetz)

${\text{Hdg}}^2\left(X,\mathbb{Z}\right)=H^2{\left(X,\mathbb{Z}\right)}_{\text{alg}},$

${\text{Hdg}}^{2n-2}\left(X\right)=H^{2n-2}{\left(X,\mathbb{Q}\right)}_{\text{alg}}.$

Here (3), (4) can be proved by the exponential exact sequence

$0\to \mathbb{Z}\to {\Theta }_X\to {\Theta }_X^{*}\to 1,$

and Subsection 3.1 the hard Lefschetz theorem.

By Subsection 3.5 we have, for $k=0,1,n-1,n$

${\text{Hdg}}^{2k}\left(X\right)=H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}},$

and by Subsection 3.3, for $0\le k\le n$

$H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}\subset {\text{Hdg}}^{2k}\left(X\right).$

We want to prove that for $2\le k\le n-2$,

${\text{Hdg}}^{2k}\left(X\right)\subset H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}$

by the induction.

The Hodge and Lefschetz decompositions are

$\begin{array}{l}{H}^{2k}\left(X,C\right)=\underset{p+q=2k}{\oplus }{H}^{p,q}\left(X\right)=\underset{0\le r\le k}{\oplus }{\ell }^r\cup H_0^{2k-2r}\left(X,C\right),\\ H^{k,k}\left(X\right)={\displaystyle \underset{r=0}{\overset{k}{\sum }}{\ell }^r\cup H_0^{k-r,k-r}}\left(X\right),\end{array}$

where

$H_0^{k-r,k-r}=\text{Ker}\left[{\ell }^{n-2k+2r+1}\cup :H^{k-r,k-r}\left(X\right)\to H^{n-k+r+1,n-k+r+1}\left(X\right)\right]$

is the primitive cohomology group of type $\left(k-r,k-r\right)$ of $X$. By (2.1.4), there is a polarization $Q$ on $H^{2k}\left(X,ℂ\right)$ such that the decompositions are orthogonal and nondegenerate.

Let $\alpha \in {\text{Hdg}}^{2k}\left(X\right)=H^{2k}\left(X,\mathbb{Q}\right)\cap H^{k,k}\left(X\right)$ be a Hodge class with degree $2k$ of $X$. Then $\alpha \in H^{k,k}\left(X\right)$ has a type $\left(k,k\right)$ with coefficients in $\mathbb{Q}$. Since

$H^{k,k}\left(X\right)={\ell }^0\cup H_0^{k,k}\left(X\right)\oplus {\ell }^1\cup H_0^{k-1,k-1}\left(X\right)\oplus \cdots \oplus {\ell }^k\cup H_0^{0,0}\left(X\right)$

is orthogonal and nondegenerate, $\alpha \in H^{k,k}\left(X\right)$ is uniquely written as

$\alpha ={\displaystyle \underset{r=0}{\overset{k}{\sum }}{\ell }^r\cup {\alpha }_{k-r}},{\alpha }_{k-r}\in H_0^{k-r,k-r}\left(X\right).$

Choose generic sections ${\sigma }_i$ of the ample bundle $L\to X$, $i=1,2,\cdots ,k$, and let $L_r={\displaystyle {\cap }_{i=1}^r{\sigma }_i^{-1}\left(0\right)}$, and $L_0=X$. Then $L_r\subset X$ is a subvariety of codimension $r$ in $X$ and the cycle class $\left[L_r\right]={\ell }^r\in H^{2r}\left(X,\mathbb{Q}\right)\cap H^{r,r}\left(X\right)$.

Find a codimension $k-r$ subvariety $A_{k-r}$ of $X$ such that its cohomology cycle class $\left[A_{k-r}\right]={\alpha }_{k-r}\in H_0^{k-r,k-r}\left(X\right)$. Then the Hodge class

${\ell }^r\cup {\alpha }_{k-r}=\left[L_r\right]\cup \left[A_{k-r}\right]=\left[L_r\cap A_{k-r}\right]\in {\ell }^r\cup H_0^{k-r,k-r}\left(X\right)\subset H^{k,k}\left(X\right)$

is algebraic. Prove this by the mathematical induction on $k-r$. By Subsection 3.5 there are $A_0=X$ and $A_1\subset X$ such that

${\alpha }_0=\left[X\right],{\alpha }_1=\left[A_1\right]$

and

$\begin{array}{l}{\ell }^k\cup {\alpha }_0=\left[L_k\cap X\right]\in {\ell }^k\cup H_0^{0,0}\left(X\right),\\ {\ell }^{k-1}\cup {\alpha }_1=\left[L_{k-1}\cap A_1\right]\in {\ell }^{k-1}\cup H_0^{1,1}\left(X\right).\end{array}$

Assume that there are subvarieties $A_0,A_1,\cdots ,A_{r-1}$ in $X$ such that

$\left[A_i\right]={\alpha }_i,\left[L_{k-i}\cap A_i\right]={\ell }^{k-i}\cup {\alpha }_i\in {\ell }^{k-i}\cup H_0^{i,i}\left(X\right).$

Since ${\alpha }_{k-r}\in H_0^{k-r,k-r}\left(X\right)$, the cup map

${\ell }^{n-2k+2r+1}\cup :H^{k-r,k-r}\left(X\right)\to H^{n-k+r+1,n-k+r+1}\left(X\right)$

satisfies ${\ell }^{n-2k+2r+1}\cup {\alpha }_{k-r}=0$. Locally we can write

${\ell }^{n-2k+2r+1}:=\left[{\displaystyle \underset{\left|J\right|=n-2k+2r+1}{\sum }{c}_{J}d{z}_J\wedge d{\bar{z}}_J}\right],$

and

${\alpha }_{k-r}:=\left[{\displaystyle \underset{\left|I\right|=k-r}{\sum }{d}_{I}d{z}_I\wedge d{\bar{z}}_I}\right].$

Then ${\ell }^{n-2k+2r+1}\cup {\alpha }_{k-r}=0$ if and only if for each $I,J$, $I\cap J\ne \varnothing$, $I,J\subset \left\{1,2,\cdots ,n\right\}$.

While $\ell$ is locally represented, in holomorphic coordinates

$\ell =\left[\frac{i}{2}{\displaystyle \underset{\mu ,\nu }{\sum }{h}_{\mu \nu }d{z}_{\mu }\wedge d{\bar{z}}_{\nu }}\right]$

and $\ell$ is hermitian symmetric. Since $I\cap J\ne \varnothing$ for each $I,J$ in ${\ell }^{n-2k+2r+1}$ and ${\alpha }_{k-r}$ there is a 2-form ${\ell }^{\prime }$ in ${\ell }^{n-2k+2r+1}$ and ${\alpha }_{k-r}$ if necessary, by the change of coordinates. By the analytic continuation [24] , the 2-form is globally defined on $X$ such that ${\ell }^{n-2k+2r}\cup \left[{\ell }^{\prime }\right]={\ell }^{n-2k+2r+1}$ and $\left[{\ell }^{\prime }\right]=\ell$. A generic section $s:X\to L$ has the zero locus $s^{-1}\left(0\right)$, and its cycle class

$\left[s^{-1}\left(0\right)\right]=\left[L_1\right]=\left[{\ell }^{\prime }\right]=\ell =c_1\left(L\right).$

The Lefschetz isomorphism

${\ell }^{n-2\left(k-r\right)}\cup :H^{2\left(k-r\right)}\left(X\right)\to H^{2n-2\left(k-r\right)}\left(X\right)$

implies that ${\ell }^{n-2\left(k-r\right)}\cup {\alpha }_{k-r}\ne 0$ if ${\alpha }_{k-r}\ne 0$. However ${\ell }^{n-2\left(k-r\right)+1}\cup {\alpha }_{k-r}=\left({\ell }^{n-2\left(k-r\right)}\cup {\alpha }_{k-r}\right)\cup \ell =0$.

The 2-form $\ell \in H^{1,1}\left(X\right)$, the orientation, the metric on $X$ and the polarization $Q$ on $H^{2k}\left(X,ℂ\right)$ define the interior and exterior products, for the definitions see [8] . Let

$\begin{array}{l}i:H^{k-r,k-r}\left(X\right)\to H^{k-r-1,k-r-1}\left(X\right),\\ e:H^{k-r-1,k-r-1}\left(X\right)\to H^{k-r,k-r}\left(X\right)\end{array}$

be the interior and exterior products.

Then for each $\ell \in H^{1,1}\left(X\right)$, $\alpha \in H^{k-r,k-r}\left(X\right)$, $\beta \in H^{k-r-1,k-r-1}\left(X\right)$,

$\begin{array}{c}Q\left(i\left(l\right)\alpha ,\beta \right)={\left(-1\right)}_{}^{\left[\frac{\left(2k-2r-2\right)\left(2k-2r-1\right)}{2}+\left(r+1\right)\right]}{\displaystyle {\int }_X{\ell }^{n-2k+2r+2}\cup i\left(\alpha \right)\cup \beta }\\ ={\left(-1\right)}^k{\displaystyle {\int }_X{\ell }^{n-2k+2r+1}\cup \alpha \cup \beta }\\ ={\left(-1\right)}^k{\displaystyle {\int }_X{\ell }^{n-2k+2r}\cup \alpha \cup e\left(l\right)\beta }\\ ={\left(-1\right)}_{}^{\left[\frac{\left(2k-2r\right)\left(2k-2r+1\right)}{2}+r\right]}{\displaystyle {\int }_X{\ell }^{n-2k+2r}\cup \alpha \cup e\left(l\right)\beta }\\ =Q\left(\alpha ,e\left(l\right)\beta \right).\end{array}$

Thus, we have

$Q\left(i\left(\ell \right)\alpha ,\beta \right)=Q\left(\alpha ,e\left(\ell \right)\beta \right).$

Then there is a unique class ${\beta }_{k-r-1}\in H^{k-r-1,k-r-1}\left(X\right)$ such that

$i\left(\ell \right)\cdot {\alpha }_{k-r}={\beta }_{k-r-1}\text{and}{\alpha }_{k-r}=e\left(\ell \right){\beta }_{k-r-1}.$

Also

$Q\left({\beta }_{k-r-1},{\beta }_{k-r-1}\right)=Q\left(i\left(\ell \right){\alpha }_{k-r},i\left(\ell \right){\alpha }_{k-r}\right)=Q\left({\alpha }_{k-r},e\left(\ell \right)i\left(\ell \right){\alpha }_{k-r}\right)=Q\left({\alpha }_{k-r},{\alpha }_{k-r}\right)\ne 0$,

since ${\alpha }_{k-r}\ne 0$. Thus the ${\beta }_{k-r-1}\ne 0$. Moreover,

$\begin{array}{c}{\ell }^{n-2k+2\left(r+1\right)+1}\cup {\beta }_{k-r-1}={\ell }^{n-2k+2r+1}\cup \left(l\cup {\beta }_{k-r-1}\right)\cup l\\ =\left({\ell }^{n-2k+2r+1}\cup {\alpha }_{k-r}\right)\cup l\\ =0\cup l=0.\end{array}$

Thus ${\beta }_{k-r-1}\in H_0^{k-r-1,k-r-1}\left(X\right)$ is primitive. While the degree of ${\beta }_{k-r-1}$ is

$\text{deg}\left({\beta }_{k-r-1}\right)=2\left(k-r-1\right)<2\left(k-r\right)$

for $0\le r\le k$. By our induction hypothesis of the degree $k$, there is a subvariety $B_{k-r-1}$ of codimension $k-r-1$ in $X$ such that $\left[B_{k-r-1}\right]={\beta }_{k-r-1}$.

Let $A_{k-r}=B_{k-r-1}\cap L_1$. Then $A_{k-r}$ is a subvariety of codimension $k-r$ in $X$. The corresponding cycle class is

$\left[A_{k-r}\right]=\left[B_{k-r-1}\cap L_1\right]=\left[B_{k-r-1}\right]\cup \left[L_1\right]={\beta }_{k-r-1}\cup \ell ={\alpha }_{k-r}.$

The Lefschetz components of the Hodge class $\alpha$

${\ell }^r\cup {\alpha }_{k-r}=\left[L_r\cap A_{k-r}\right]\in H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}$

are algebraic cycle classes. Thus, the Hodge class

$\alpha ={\displaystyle \underset{r=0}{\overset{k}{\sum }}{\ell }^r\cup {\alpha }_{k-r}}={\displaystyle \underset{r=0}{\overset{k}{\sum }}\left[L_r\cap A_{k-r}\right]}\in H^{2k}{\left(X,Q\right)}_{\text{alg}}$

is an algebraic cycle class, that is,

${\text{Hdg}}^{2k}\left(X\right)\subset H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}.$

Let $X$ be a Hodge manifold of dimension $n$. Then the Hodge group ${\text{Hdg}}^{2k}\left(X\right)$ is equal to the algebraic group $H^{2k}{\left(X,\mathbb{Q}\right)}_{\text{alg}}$ for $k=0,1,\cdots ,n$. ◼

Let $H_1$ and $H_2$ be Hodge structures of weight $k_1$ and $k_2$, respectively and $k_2-k_1=2r$. Then the morphisms $\varphi :H_1\to H_2$ are Hodge classes in ${\text{Hdg}}^{2r}\left(\text{Hom}\left(H_1,H_2\right)\right)$. For details see [9] .

Let $X$ be a Hodge manifold of dimension $n$ and $\ell$ be the first Chern class of an ample line bundle $L\to X$. Then the Lefschetz theorem states that the cup map

${\ell }^{n-k}\cup :H^k\left(X,\mathbb{Q}\right)\to H^{2n-k}\left(X,\mathbb{Q}\right)$

is an isomorphism of Hodge structures for each $k$. By Subsection 4.10,

$\begin{array}{c}\left[{\ell }^{n-k}\cup \right]\in {\text{Hdg}}^{\text{**}}\left[Hom\left(H^k\left(X,Q\right),H^{2n-k}\left(X,Q\right)\right)\right]\\ ={\text{Hdg}}^{\text{**}}\left[H^k{\left(X,\mathbb{Q}\right)}^{\text{*}}\otimes H^{2n-k}\left(X,Q\right)\right]\\ ={\text{Hdg}}^{\text{**}}\left[H^{2n-k}\left(X,Q\right)\otimes H^{2n-k}\left(X,Q\right)\right]\\ \subset {\text{Hdg}}^{\text{**}}\left[H^{4n-2k}\left(X\times X,Q\right)\right],\end{array}$

where $\ast \ast =\left(2n-k\right)+\left(2n-k\right)=4n-2k$. By (4.9), the Hodge class $\left[{\ell }^{n-k}\cup \right]$ is algebraic in $X\times X$.

In fact, let ${\sigma }_i:X\to L$, $i=1,2,\cdots ,n-k$, be generic sections, $Z={\displaystyle {\cap }_{i=1}^{n-k}{\sigma }_i^{-1}\left(0\right)}\subset X$, and $W=i_{\Delta }\left(Z\right)$ where $i_{\Delta }:X\to X\times X$ is the diagonal map, then $Z$ has codimension $n-k$ in $X$, and $W$ has codimension $2n-k$ in $X\times X$. Thus $\left[{\ell }^{n-k}\cup \right]=\left[W\right]\in H^{4n-2k}\left(X\times X,\mathbb{Q}\right)$ is algebraic.

The inverse

${\left({\ell }^{n-k}\cup \right)}^{-1}:H^{2n-k}\left(X,\mathbb{Q}\right)\to H^k\left(X,\mathbb{Q}\right)$

of the Lefschetz isomorphism is an isomorphism of Hodge structures. As in Subsection 4.11,

$\begin{array}{c}\left[{\left({\ell }^{n-k}\cup \right)}^{-1}\right]\in {\text{Hdg}}^{2k}\left(Hom\left(H^{2n-k}\left(X,Q\right),H^k\left(X,Q\right)\right)\right)\\ ={\text{Hdg}}^{2k}\left(H^k\left(X,\mathbb{Q}\right)\otimes H^k\left(X,\mathbb{Q}\right)\right)\subset {\text{Hdg}}^{2k}\left(H^{2k}\left(X\times X,\mathbb{Q}\right)\right)\end{array}$

is Hodge and algebraic by Subsection 4.9 since the product space $X\times X$ is a Hodge manifold of dimension $2n$.

These contents are called the Lefschetz standard conjecture.

Let $X$ be a Hodge manifold of dimension $n$ and $\ell$ be the first Chern class of an ample line bundle $L\to X$. Then there is a codimension $k$ subvariety $Z$ in $X\times X$ such that

$\left[Z\right]=\left[{\left({\ell }^{n-k}\cup \right)}^{-1}\right]\in H^{2k}\left(X\times X,\mathbb{Q}\right),$

that is, the inverse of the Lefschetz isomorphism is algebraic. ◼