The cone construction via intersection theory

TL;DR

This paper introduces a new method for constructing algebraic cycles using intersection theory, which provides a proof for the Lefschetz standard conjecture, advancing understanding in algebraic geometry.

Contribution

The paper presents a novel intersection-theoretic approach to construct algebraic cycles, leading to a proof of the Lefschetz standard conjecture.

Findings

Established a new construction method for algebraic cycles.

Provided a proof of the Lefschetz standard conjecture.

Enhanced techniques in intersection theory for algebraic geometry.

Abstract

We show a method in constructing algebraic cycles via intersection theory. It leads to a proof of the Lefschetz standard conjecture.

Equations

$$\begin{array}[]{ccccc}CH^{q-h}(X)&\stackrel{{\scriptstyle\mathbf{v}^{h}}}{{\longrightarrow}}&CH^{q}_{V^{h}}(X)&\stackrel{{\scriptstyle\mathbf{i}}}{{\longrightarrow}}&CH^{q}(X)\end{array}$$

$$\begin{array}[]{ccc}L^{h}:\displaystyle{\sum_{i=0}^{2n-2h}H^{i}(X)}&\rightarrow&\displaystyle{\sum_{i=0}^{2n-2h}H^{i+2h}(X)}\\ \alpha&\rightarrow&\alpha\cdot u^{h}.\end{array}$$

$$\begin{array}[]{ccc}L_{a}^{h}:A^{n-h\over 2}(X)&\rightarrow&A^{n+h\over 2}(X)\\ \alpha&\rightarrow&\alpha\cdot u^{h}.\end{array}$$

$$\begin{array}[]{ccccc}CH^{q-h}(X)&\stackrel{{\scriptstyle\mathbf{v}^{h}}}{{\longrightarrow}}&CH^{q}_{V^{h}}(X)&\stackrel{{\scriptstyle\mathbf{i}}}{{\longrightarrow}}&CH^{q}(X)\\ \downarrow&&\downarrow&&\downarrow\\ A^{q-h}(X)&\stackrel{{\scriptstyle v^{h}}}{{\longrightarrow}}&A^{q}_{V^{h}}(X)&\stackrel{{\scriptstyle i}}{{\longrightarrow}}&A^{q}(X)\end{array}$$

$$\quad\quad CH_{i+j-dim(Y)}\Biggr{(}P_{X}\biggl{(}(|x|\times|y|)\cap graph(f)\biggr{)}\Biggr{)}$$

$$C{H}_{i+j-dim(W)}(|x|{\times }_W|y|)\to C{H}_{i+j-dim(W)}(X{\times }_{W}Y)$$

$$\begin{array}[]{ccc}Con_{h}:CH^{q-h}(V^{h})&\rightarrow&CH^{q-h}(X),\end{array}$$

$$\begin{array}[]{ccc}CH^{q-h}(V^{h})&\rightarrow&CH^{q-h}(X)\end{array}$$

$$\mathbf{i}\circ {\mathbf{v}}^h\circ (Co{n}_h-\xi )=deg(X)\text{mathbbm}i.$$

$$l_{1}deg(X)[\delta ]+m^{\prime }{\mathbf{u}}^q$$

$$\begin{array}[]{ccccc}A^{q-h}(X)&\stackrel{{\scriptstyle v^{h}}}{{\longrightarrow}}&A^{q}_{V^{h}}(X)&\stackrel{{\scriptstyle i}}{{\longrightarrow}}&A^{q}(X)\end{array}$$

$${\mathbf{e}}_0,\cdots ,{\mathbf{e}}_{n+1}.$$

$$\begin{array}[]{c}k^{n+2-h}=span(\mathbf{e}_{0},\cdots,\mathbf{e}_{n+1-h}),\\ k^{h}=span(\mathbf{e}_{n+2-h},\cdots,\mathbf{e}_{n+1}).\end{array}$$

$$k^{n+2-h}\oplus k^h=k^{n+2}$$

$$k_z^h=span(z{\mathbf{e}}_{n+2-h}-{\mathbf{e}}_0,{\mathbf{e}}_{n+3-h},\cdots ,{\mathbf{e}}_{n+1}),forz\in k$$

$$k^{n+2}=k^{n+2-h}\oplus k_z^h.$$

$$\mathbf{x}=({\mathbf{x}}_1(z),{\mathbf{x}}_2(z)).$$

$$\begin{array}[]{ccc}k\times U\times(k^{n+2-h}\oplus k^{h}_{z})&\rightarrow&k^{n+2-h}\oplus k^{h}_{z}=k^{n+2}\\ (t,z,(\mathbf{x}_{1}(z),\mathbf{x}_{2}(z)))&\rightarrow&(\mathbf{x}_{1}(z),t\mathbf{x}_{2}(z)).\end{array}$$

$$\begin{array}[]{ccc}\kappa:\mathbf{P}^{1}\times\Upsilon\times\mathbf{P}^{n+1}&\dashrightarrow&\mathbf{P}^{n+1}\\ (t,z,[\mathbf{x}_{1}(z),\mathbf{x}_{2}(z)])&\dashrightarrow&[\mathbf{x}_{1}(z),t\mathbf{x}_{2}(z)],\end{array}$$

$$\begin{array}[]{cccc}g_{t}^{z}:&\mathbf{P}^{n+1}&\rightarrow&\mathbf{P}^{n+1}\\ &\mathbf{e}_{0}&\rightarrow&\mathbf{e}_{0}\\ &\vdots&\Rightarrow&\vdots\\ &\mathbf{e}_{n+1-h}&\rightarrow&\mathbf{e}_{n+1-h}\\ &\mathbf{e}_{n+2-h}&\rightarrow&t(z\mathbf{e}_{n+2-h}-\mathbf{e}_{0})\\ &\mathbf{e}_{n+3-h}&\rightarrow&t\mathbf{e}_{n+3-h}\\ &\vdots&\Rightarrow&\vdots\\ &\mathbf{e}_{n+1-h}&\rightarrow&t\mathbf{e}_{n+1}\end{array}$$

$$\Omega =graph(\kappa )\subset {\mathbf{P}}^1\times {\rm Y}\times {\mathbf{P}}^{n+1}\times {\mathbf{P}}^{n+1}$$

$$\mu :X\to {\mathbf{P}}^{n+1}$$

$$\begin{array}[]{ccc}\tau:\mathbf{P}^{1}\times\Upsilon\times X\times X&\rightarrow&\mathbf{P}^{1}\times\Upsilon\times\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}.\end{array}$$

$$\Sigma ={\tau }^{-1}(\Omega )$$

$${\Sigma }_c=[{\mathbf{P}}^1\times {\rm Y}\times X\times X]{\cdot }_{\tau }[\Omega ]\in C{H}_{n+1}({\mathbf{P}}^1\times {\rm Y}\times X\times X)$$

$$\{1\}\times {\rm Y}\times {\Delta }_X.$$

$${\mu }^{-1}(\Omega ).$$

$$\{1\}\times {\rm Y}\times {\Delta }_X$$

$${\Theta }_c$$

$${\Theta }_c={\Sigma }_c-a[\{1\}\times {\rm Y}\times {\Delta }_X]$$

Full text

It leads to a proof of the Lefschetz standard conjecture.

00footnotetext: Key words: Intersection theory, Chow group, Lefschetz standard conjecture, 00footnotetext: *2000 Mathematics subject classification *: 32S50, 14C30, 14C17, 14C25

1. 1 Introduction

2. 1.1 Statements

3. 1.2 Outline of the proof

4. 2 Cone family

5. 2.1 Cone family of cycle classes

6. 2.2 End cycles

7. 3 Cone operator

8. 3.1 Construction

9. 3.2 Intersection

1 Introduction

1.1 Statements

We present a construction of algebraic cycles, showing the Chow groups of cycles with mid-range dimensions could be accessible.

Let $X$ be a smooth projective variety over an algebraically closed field $k$, of dimension $n\geq 2$. Let $CH(X)$ denote the total Chow group tensed with $\mathbb{Q}$, i.e. $CH(X)=\sum_{i}CH_{i}(X)\otimes\mathbb{Q}$. Let the superscript of it denote the dimension of the cycles, and subscript of it the codimension. For a natural number $h\leq n$, let

$V^{h}$ be a smooth $h$-codimensional plane section of $X$,

$\mathbbm{i}:CH(V^{h})\rightarrow CH(X)$ be the inclusion map with $CH_{V_{h}}(X)=image(\mathbbm{i})$,

$\mathbf{i}:CH_{V_{h}}(X)\hookrightarrow CH(X)$ be the embedding.

For a natural number $q\geq h$, we consider the sequence of homomorphsims

[TABLE]

where $\mathbf{v}^{h}$ is the intersection map with the subvariety $V^{h}$. Our main result is

Theorem 1.1**.**

(Main theorem)

(1) If $h\leq q\leq n$, $\mathbf{v}^{h}$ is surjective,

(2) If $h<q<n$, $\mathbf{i}$ is surjective.

A general result like Main theorem 1.1 will have implications for the Chow groups in the mid-range dimensions. However our first application turns away from the Chow groups, and shows its motivic consequence is the Lefschetz standard conjecture proposed by Grothendieck in [2]. Originally the conjecture addresses a smooth projective variety $X$ of dimension $n\geq 3$ over an algebraically closed field $k$, equipped with étal cohomology denoted by $H^{i}(X)$. Let $u$ be the hyperplane section class in the cohomology $H^{2}(X)$. For a whole number $h\leq n$, let $L^{h}$ denote the homomorphism,

[TABLE]

The hard Lefschetz theorem asserts the $L^{h}$ is restricted to an isomorphism between $H^{n-h}(X)$ and $H^{n+h}(X)$. Grothendieck proposed

Conjecture 1.2**.**

(Lefschetz)

Let $A^{i}(X)\subset H^{2i}(X)$ be the image of the cycle map for the cycles of codimension $i$. If $n+h$ is even, then $L^{h}$ is restricted to an isomorphism $L_{a}^{h}$,

[TABLE]

Conjecture 1.2 is known as the $A$-conjecture or the Lefschetz standard conjecture which addresses the surjectivity of $L_{a}^{h}$. Main theorem shows a stronger surjectivity with a greater extent that covers the Lefschetz standard conjecture. To state it clearly, we use Kleiman’s axiomatic approach ([3]).

Definition 1.3**.**

(Chow-motivic). Let $X$ be a smooth projective variety over an algebraically closed field, of dimension $n\geq 2$. In [3], Kleiman formally listed all axioms for a Weil cohomology theory on $X$, with the motif in algebraic cycles. In [4], Jannsen formally listed all axioms for a Weil cohomology theory on $X$, with the motif in Chow groups. In this paper, we use the following commutative diagram following from the Jannsen’s Chow-motif in [4]:

[TABLE]

where $v^{h}=L^{h}|_{A^{q-h}(X)}$, all downward arrows are restrictions of cycle maps $cl$ and the second row consists of the images of $cl$. To stress the important role of Chow groups in the diagram (1.4), we say a Weil cohomology is Chow-motivic, and $v^{h},i$ are the cohomological descends of $\mathbf{v}^{h},\mathbf{i}$.

Remark It is known in [4], all existing cohomology theories are Chow-motivic. In particular, the étal cohomology in the conjecture 1.2 is so.

Theorem 1.1 implies that

Theorem 1.4**.**

The Lefschetz standard conjecture is correct for any Chow-motivic Weil cohomology. In particular, the Grothendieck’s proposal 1.2 is correct.

Notation 1.5**.**

(1) $Z(\cdot)$=the Abelian group freely generated by reduced, irreducible

subvarieties of all dimensions with rational coefficients;

$CH(\cdot)$=the total Chow group with rational coefficients;

$H(\cdot)$=the total $l$-adic cohomology group.

$A(\cdot)$=image of the cycle map $Z(\cdot)\to H(\cdot)$;

A cycle=an element of $Z(\cdot)$;

A cycle class= an element of $CH(\cdot)$;

A cohomology class= an element of $H(\cdot)$;

Homogeneous = of the same dimensional components;

On all these groups and their subgroups, the superscript denotes the codi-

mension of cycles or classes, and subscript denotes the dimension of

them.

(2) $\Delta_{V}$ denotes the diagonal scheme in the product $V\times V$.

(3) A plane section is a complete intersection by hyperplane sections with

the same cohomology class.

(4) A plane section class is a class represented by a plane section.

(5) Let $[\cdot]$ denote the rational equivalence class in a Chow group.

(a) For $x\in Z_{i}(X),y\in Z_{j}(X)$, let $[x]\cdot_{X}[y]$ ( or $[x\cdot_{X}y]$) denote the

intersection class in the Chow group, $CH_{i+j-n}(|x|\cap|y|)$. To abuse

the notation its inclusion images in $CH_{i+j-n}(|x|)$ or $CH_{i+j-n}(|y|)$

or $CH_{i+j-n}(X)$ are also denoted by $[x]\cdot_{X}[y]$ ( or $[x\cdot_{X}y]$).

(b) If $f:X\to Y$ is a morphism as in chapter 8, [1], and $x\in Z_{i}(X)$,

$y\in Z_{j}(Y)$. Then the intersection class in

[TABLE]

for the projection $P_{X}:X\times Y\to X$, is denoted by $[x]\cdot_{f}[y]$ (or

$[x\cdot_{f}y]$), which also abusively denotes the intersection classes in

$CH_{i+j-dim(Y)}(|x|)$* or $CH_{i+j-dim(Y)}(f|_{|x|}^{-1}(|y|))$ or $CH_{i+j-dim(Y)}(X)$.*

(c) If there are two morphisms $g_{1}:X\to W$, $g_{2}:Y\to W$ for smooth

varieties and $x\in Z_{i}(X),y\in Z_{j}(Y)$, both intersection classes in

Chow groups

[TABLE]

are denoted by $[x]\cdot_{g_{1}g_{2}}[y]$ (or $[x\cdot_{g_{1}g_{2}}y]$).

1.2 Outline of the proof

The proof of Main theorem is based on the following construction, called cone construction.

1. $\mathbf{v}^{h}$ is surjective for $h\leq q\leq n$: Using intersection, we construct a homo-

morphism called the cone operator,

[TABLE]

such that there is a homomorphism $\xi$,

[TABLE]

with the intersection property:

[TABLE]

The formula (1.7) implies $\mathbf{v}^{h}$ is surjective. 111 However (1.7) does not imply the surjectivity of the other plane section map: $CH^{q-h}(X)\to CH^{q-h}(V_{h})$ because $\mathbbm{i}$ is not injective.

2. $\mathbf{i}$ is surjective for $h<q<n$: Assume $h<q<n$. For $\delta\in Z^{q}(X)$, using

intersection, we construct a family of cycle classes $\psi_{t}(\delta)\in CH^{q}(X)$,

$t\in\mathbf{P}^{1}$ called the cone family such that for the class $[\delta]$ of $CH^{q}(X)$,

one member of the family, $\psi_{0}(\delta)$ lies in $CH^{q}_{V^{h}}(X)$, and the other $\psi_{1}(\delta)$

is equal to the cycle class

[TABLE]

for an integer $m^{\prime}$ and natural number $l_{1}$, where $\mathbf{u}$ is the hyperplane

section class in $CH^{1}(X)$. Since $\psi_{0}(\delta),\psi_{1}(\delta)$ are equal, due to (1.8) $\mathbf{i}$ is

surjective.

Let (1.1) descend to the Weil cohomology to obtain the similar sequence with maps for $h\leq q$,

[TABLE]

where the maps with non-bold letters are the cohomological descends of those with bold letters in (1.1). In case $h<q<n$, (1.1) for Chow groups has the surjectivity which by the Chow-motivic (1.4) descends to (1.9) for the Weil cohomology. Now we come back to the index setting of Conjecture 1.2, i.e $q={n+h\over 2}$, for which there is the hard-Lefschetz-theorem axiom. In particular it asserts the composition $i\circ v^{h}=L_{a}^{h}$ is injective, therefore it is an isomorphism.

In the rest of the paper we give the rigorous argument on Chow groups. In section 2, we construct the cone family with a focus on the defining equations of the schemes. In section 3, based on the cone data we define the cone operator and prove an intersection formula that leads to Main theorem.

Acknowledgment Thanks are due to my wife Jessie Liu for creating the great environment.

2 Cone family

2.1 Cone family of cycle classes

Step 2 relies on a family of cycle classes in $X$. The family will be parametrized by the projective space $\mathbf{P}^{1}$, and named as the cone family. Let $k^{n+2}$ be a linear space over $k$ with a standard basis

[TABLE]

Let $h$ be a natural number $\leq n$. Consider two subspaces

[TABLE]

Then

[TABLE]

Next we consider a variation of $k^{h}$. Let $k\cup\{\propto\}\simeq\mathbf{P}^{1}$ be the parameter space of the variation, denoted by $\Upsilon$, where $\propto$ is the infinity point of $\mathbf{P}^{1}$. The variation is defined to be

[TABLE]

and $k^{h}_{\propto}$ is the original $k^{h}$ which is the limit of subspaces $k^{h}_{z}$ in Grassmannian as $z\to\propto$. Let $U=k^{\ast}\cup\{\propto\}$ be the affine open set that parametrizes those $k^{h}_{z}$ satisfying

[TABLE]

The only point $z=0$ not in $U$ corresponds to the plane $k^{h}_{0}$ that fails the decomposition (2.5). We call $z=0$ the unsteady point, others steady points. Let $\mathbf{x}$ be the linear coordinates for $k^{n+2}$ under the basis $\mathbf{e}_{i}$. Therefore for each steady point $z\in U$, we have the unique decomposition (2.5)

[TABLE]

The decomposition gives a regular map

[TABLE]

which yields a rational map of the projective variety

[TABLE]

where $t,z$ are points in the affine open sets $k,U$. To view it intrinsically, we just set up a family of linear transformations $g_{t}^{z}$ on $\mathbf{P}^{n+1}$ as the restriction of $\kappa$ to each $t\in k^{\ast},z\in U$:

[TABLE]

Let

[TABLE]

where the graph of a rational map is defined to be the closure of the graph at the regular locus ( the same for images and preimages of rational maps).

Now we consider the smooth projective variety $X$ of dimension $n$, equipped with the polarization $\mathbf{u}$ as in (1.8). Let

[TABLE]

be a birational morphism to a hypersurface of $\mathbf{P}^{n+1}$ in a general position in the following sense: $\mu(X)$ is in a general position as a subvariety, in particular its first order deformation in $g_{t}^{z}(\mu(X))$ along $t$ varies with $z\in U$; $X$ has a very ample line bundle $\mu^{\ast}(\mathcal{O}_{\mathbf{P}^{n+1}}(1))$ such that $\mathbf{u}=c_{1}(\mu^{\ast}(\mathcal{O}_{\mathbf{P}^{n+1}}(1)))$ is the original polarization. The collection of above spaces $k^{n+2}$, $k^{n+2},k^{h}_{z}$ and $\mu$ is called cone data. The cone data is extrinsic in a sense that it can be obtained through any embedding $X\subset\mathbf{P}^{N}$, by taking a projection to a generic subspace: $\mathbf{X}\to\mathbf{P}^{n+1}$.

Let $\tau=(id,id,\mu,\mu)$ be the map

[TABLE]

Let

[TABLE]

be the intersection scheme. Also let

[TABLE]

(“c” stands for “cycle class”.)

Proposition 2.1**.**

$\Sigma_{c}$* is represented by a cycle whose support contains*

[TABLE]

Proof.

In intersection theory, $\Sigma_{c}$ has dimension $n+1$ and is represented by an intersection cycle $\mathcal{I}$ whose support is contained in the subscheme

[TABLE]

In a neighborhood of the fibre of $\mathbf{P}^{1}\times\Upsilon\times X\times X\to\mathbf{P}^{1}$ over $1$,

[TABLE]

is the only algebraic set of an irreducible component of the scheme $\mu^{-1}(\Omega)$. Hence it is a component of the support of $\mathcal{I}$. ∎

Proposition 2.1 gives a definition

Definition 2.2**.**

(a) We define

[TABLE]

to be the rest of $\Sigma_{c}$, i.e.

[TABLE]

where $a$ is the intersection multiplicity of (2.11) at $\{1\}\times\Upsilon\times\Delta_{X}$.

Similarly we define the scheme

[TABLE]

where $<\{1\}\times\Upsilon\times\Delta_{X}>$ is the irreducible component of $\mu^{-1}(\Omega)$ whose

reduced subvariety is $\{1\}\times\Upsilon\times\Delta_{X}$.

(b) For a homogeneous $\delta\in Z(X)$, we define a family of classes in $CH(X)$

to be

[TABLE]

where

[TABLE]

[TABLE]

is the projection, and $(\cdot)_{\mathbf{P}^{1}}$ stands for those supports dominating $\mathbf{P}^{1}$.

So we obtain a family of cycle classes $\psi_{t}(\delta),t\in\mathbf{P}^{1}$ in $CH(X)$, and call

it cone family. We call the member $\psi_{t}(\delta)$ of the family at the point

$t\in\mathbf{P}^{1}$, the $t$-end cycle.

Remark.

(1) It can be proved, but requires a longer argument that cycle classes $\Sigma_{c},\Theta_{c}$

are actually represented by the reduced, irreducible subschemes $\Sigma,\Theta$.

(2) In Fulton’s definition ([1]), $\psi_{t}(\delta)$ lifted to $\{t\}\times\Upsilon\times X\times X$, is the family

of cycle classes determined by the class $(\Theta_{c}\cdot[\mathbf{P}^{1}\times\Upsilon\times\delta\times X])_{\mathbf{P}^{1}}$ which

however is not equal to $\Theta_{c}\cdot[\mathbf{P}^{1}\times\Upsilon\times\delta\times X]$, i.e. in the support of the

class $\Theta_{c}\cdot[\mathbf{P}^{1}\times\Upsilon\times\delta\times X]$, there could be components not dominating

$\mathbf{P}^{1}$. We’ll show this, indeed, is the case for a [math]-cycle $\delta$.

2.2 End cycles

We’ll approach cycles in the scheme-theoretical point of view. The principle idea of the analysis follows two points: the schemes related to cycles over steady points $z\neq 0$ are the isomorphic linear transformations, but over the unsteady point $z=0$, they are exceptional, so the understanding requires the detailed defining equations.

So we set up coordinates for the defining equations of schemes. Let $x_{0},\cdots,x_{n+1}$ be the coefficients of the basis $\mathbf{e}_{0},\cdots,\mathbf{e}_{n+1}$ for $k^{n+2}$. Then $x_{0},\cdots,x_{n+1}$ are homogeneous coordinates for $\mathbf{P}^{n+1}$. Recall $z\neq 0$ are parametrizing the affine neighborhood of $\Upsilon$. Then the homogeneous coordinates for $\mathbf{P}^{n+1-h}=\mathbf{P}(k^{n+2-h})$, $\mathbf{P}^{h-1}_{z}=\mathbf{P}(k^{h}_{z})$ in the basis $\{\mathbf{e}_{i}\}$ as in the decomposition (2.6) are

[TABLE]

In the product space

[TABLE]

the coordinates for the third component $\mathbf{P}^{n+1}$ has $x$-coordinates as above. The coordinates for the last component $\mathbf{P}^{n+1}$ will be denoted by the letter $y$,

[TABLE]

Applying the coordinates to the graph in (2.9), we obtain the scheme

[TABLE]

is explicitly defined by the following equations.

[TABLE]

(lots of equations!). We denote its fibres over $t\in\mathbf{P}^{1},z\in\Upsilon$ by $\Omega^{z},\Omega_{t},\Omega_{t}^{z}$. Similar notations $\Theta^{z},\Theta_{t},\Theta_{t}^{z}$ for $\Theta$ will also be used. Also their projections to their bases in the fibration will be denoted by adding the tilde $\tilde{\cdot}$ to the parameters $t,z$: such as $\Omega^{\tilde{z}}$ is a subscheme of $\mathbf{P}^{1}\times\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$, etc.

We define a special type of irreducible subvarieties of $\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$. They will be used to describe the end cycles.

Definition 2.3**.**

For a whole number $r<n$, let

[TABLE]

be the projection from $\mathbf{e}_{0}$. So $k^{n+1-r}$ is a relative scheme over $k^{n-r}$ (the trivial line bundle over $k^{n-r}$). Let $E_{r}$ be the closure of the fibre product

[TABLE]

in the projective variety $\mathbf{P}^{n+1-r}\times\mathbf{P}^{n+1-r}$. So $dim(E_{r})=n+2-r$. We are interested in $E_{0},E_{h}$ that have coordinates expressions:

$E_{0}$* is defined by*

[TABLE]

$E_{h}$* is defined by*

[TABLE]

$\bullet$  $1$-end cycle.

Proposition 2.4**.**

Denote $(\mu,\mu):X\times X\to\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$ by $\mu^{2}$. Let

[TABLE]

be the class in $CH^{n}(X\times X)$ for a whole number $l_{2}$. Then if $\delta$ is homogeneous and $dim(\delta)\neq 0$,

[TABLE]

where $l_{1}$ is a natural number.

Remark If $dim(\delta)=0$, the formula (2.23) does not hold due to the fact the first term on the right hand side, $l_{1}deg(X)[\delta]$ could vanish.

Proof.

In Definition 2.2 for a family of classes, the difficulty is the selection of suitable components in the intersection schemes. So our strategy is first to investigate the triple intersection in $CH(\mathbf{P}^{1}\times\Upsilon\times X\times X)$,

[TABLE]

which by associativity and commutativity is equal to

[TABLE]

then to select the suitable components of (2.24).

So let’s first focus on $\Theta_{c}\cdot_{Y_{1}}[\{1\}\times\Upsilon\times X\times X]$ which is supported on the intersection scheme $\Theta_{1}$. We consider $\Theta_{1}$ lying on two subvarieties:

(1) projective $\{1\}\times\{0\}\times X\times X$ (over the unsteady point),

(2) quasi-projective $\{1\}\times U\times X\times X.$ (over the steady points)

(1) Over the unsteady point $z=0$: When $z=0$, the defining equations of $\Omega_{1}^{0}$ according to (2.18) are

[TABLE]

Notice the scheme $\Omega_{\tilde{1}}^{\tilde{0}}$ (i.e. the projection of $\Omega_{\tilde{1}}^{\tilde{0}}$ to $\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$ ) is $E_{0}$. Hence the part of

[TABLE]

lying in $\{1\}\times\{0\}\times X\times X$ is supported on the support of

[TABLE]

Notice that the general position of $X$ implies that $[X\times X]\cdot_{\mu^{2}}[E_{0}]$ is represented by a prime cycle (i.e. fundamental cycle of a reduced, irreducible scheme.). Hence the part of

[TABLE]

lying on $\{1\}\times\{0\}\times X\times X$ is

[TABLE]

where $l_{2}$ is a multiplicity. Denote $l_{2}[X\times X]\cdot_{\mu^{2}}[E_{0}]$ by $\omega$ to obtain the part supported over $z=0$ is

[TABLE]

(2) We discuss the part of the class dominating $\Upsilon$. So we focus on the scheme

[TABLE]

where $t$ is near $1$. Let $\mu(X)$ be the hypersurface of $\mathbf{P}^{n+1}$. Assume $\mu(X)$ is defined by a polynomial $f$. Then $\mu^{2}(X\times X)$ is a complete intersection defined by two polynomials $f(\mathbf{x}),f(\mathbf{y})$ in

[TABLE]

for $(\mathbf{x},\mathbf{y})\in\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$. Then

[TABLE]

is the subvariety of

[TABLE]

defined by two hypersurfaces $f(\mathbf{x}),f(\mathbf{y})$. Let $[\beta_{0}],[\beta_{1}^{z}]$ be the points in the decomposition

[TABLE]

where $\mathbf{P}^{h-1}_{z}=\mathbf{P}(k_{z}^{h})$. At $z=\infty$, we denote $\beta_{1}=\beta_{1}^{\infty}$. Since $t$ is near $1$, it can not be [math] or $\infty$. Then $\Omega_{\tilde{t}}^{\tilde{z}}$ is a graph isomorphic to $\mathbf{P}^{n+1}$ expressed as the graph

[TABLE]

Then $\mu^{2}(\Theta_{\tilde{t}}^{\tilde{z}})$ is a complete intersection explicitly defined by

[TABLE]

inside of $\Omega_{\tilde{t}}^{\tilde{z}}\simeq\mathbf{P}^{n+1}$. Consider the expansion along $t-1$,

[TABLE]

where $g_{r}^{z}(\beta_{0},\beta_{1})$ is a hypersurface in $\{(\beta_{0},\beta_{1}):all\ \beta_{0},\beta_{1}\}=\Delta_{\mathbf{P}^{n+1}}$ dependent of $z$. By the assumption on the cone data, $r=1$ and $\{g_{r}^{z}(\beta_{0},\beta_{1})=0\}$ is a varied hypersurface with $z\in U$. Then the specialization $\mu^{2}(\Theta_{1}^{z})$ at $t=1$ in $\Delta_{\mathbf{P}^{n+1}}$ is defined by two polynomials

[TABLE]

Therefore the specialization $\Theta^{\tilde{z}}_{\tilde{1}}$ is birational to the hypersurface $\{g_{r}^{z}(\beta_{0},\beta_{1})=0\}$ in $\mu^{2}(\Delta_{X})\subset\Delta_{\mathbf{P}^{n+1}}$, of degree $deg(X)$. Thus $\Theta^{\tilde{z}}_{\tilde{1}}$ is a non-constant hypersurface of the diagonal $\Delta_{X}$ parametrized by $z\in U$. Let $W$ be the closure of the algebraic set

[TABLE]

Because $deg(\{g_{r}^{z}=0\})=deg(X)$, $W\to\Delta_{X}$ is a covering map of degree $deg(X)$. Combining part (1), (2), we have

[TABLE]

Converting the expression to use the cycle $\Theta_{c}$, we obtain that $\psi_{1}(\delta)$ is the part of the cycle class

[TABLE]

for some intersection multiplicity $l_{1}>0$, where $Y_{2}=\Upsilon\times X\times X,P_{3}:Y_{2}\to X(3rd\ component)$.

To find out which part of (2.30) belongs to $\psi_{1}(\delta)$, we need to consider $\psi_{1}(\delta)$ in another format, as a part the projection of the triple

[TABLE]

By Definition 2.2, we only select the components of $\Theta_{c}\cdot_{Y_{1}}[\mathbf{P}^{1}\times\Upsilon\times\delta\times X]$ that dominate $\mathbf{P}^{1}$. It suffices to consider the case $z\neq 0$, i.e. we observe the cycle

[TABLE]

(without the unsteady point $z=0$). Set-theoretically if $\delta$ is homogeneous and $dim(\delta)\neq 0$, then over

[TABLE]

the algebraic set of the scheme

[TABLE]

is non-empty because its projection to the last component $X$ is the intersection in $\mathbf{P}^{n+1}$ between a linear isomorphism of $\mu(|\delta|)$ (dimension $\neq 0$) and the hypersurface $\mu(X)$. This proves that every components $\Theta_{c}\cdot_{Y_{1}}[\mathbf{P}^{1}\times U\times\delta\times X]$ dominates $\mathbf{P}^{1}$. Hence every components of

[TABLE]

that does not entirely lie over $z=0$ will be in the support of $\psi_{1}(\delta)$. Let’s apply this criterion to $W$. Since $[W]$ does not lie entirely over $z=0$, neither does

[TABLE]

Hence the projection (to $X$) of it should be the part of $\psi_{1}(\delta)$. Notice the $W$ is a $deg(X)$-covering of $\Delta_{X}$. Applying a projection formula to (2.35), we obtain the projection

[TABLE]

where $l_{1}$ is a natural number and coefficient $l_{2}$ hidden in $\omega$ (as in part (1)) is extended to [math] to include the case $\omega_{\ast}([\delta])$ is not selected in $\psi_{1}(\delta)$.

∎

Proposition 2.5**.**

For a whole number $r$, the correspondence

[TABLE]

for $\delta\in Z^{r}(X)$ is a multiple of the plane section class in $CH^{r}(X)$.

Proof.

We recall $\omega$ is a multiple of the intersection cycle class $[X\times X]\cdot_{\mu^{2}}[E_{0}]$, where

[TABLE]

is the subvariety of dimension $n+2$ defined by

[TABLE]

for the homogeneous coordinates $x_{0},\cdots,x_{n+1}$ and $y_{0},\cdots,y_{n+1}$ of the first and second copies of $\mathbf{P}^{n+1}$ in $\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$ respectively. By the associativity of Fulton’s intersection 222 Both rows of (2.38) use the Fulton’s formulation of intersection in chapter 8, [1], which is the intersection through morphisms. In our case, it is equivalent to regard $X$ as a subvariety of $\mathbf{P}^{n+1}$. in ambient space $\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$,

[TABLE]

Notice

[TABLE]

So the upper row of (2.38) is

[TABLE]

whose projection to $X$ is $\omega_{\ast}([\delta])$.

Let $P_{2}:\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}\to\mathbf{P}^{n+1}(2nd\ copy)$ be the projection. For the lower row of (2.38), by the projection formula,

[TABLE]

where $(P_{2})_{\ast}([\delta\times\mathbf{P}^{n+1}]\cdot[E_{0}])$ lies in $CH^{r}(\mathbf{P}^{n+1})\simeq\mathbb{Q}$ generated by the hyperplane section class. Thus

[TABLE]

is a multiple of a plane section class $\in CH^{r}(X)$.

∎

$\bullet$  [math]-end cycle.

Let

[TABLE]

[TABLE]

be subspaces of dimensions $n+1-h,h$ respectively. Correspondingly, let

[TABLE]

and

[TABLE]

be the $n-h$ and $h-1$ dimensional, smooth, irreducible plane sections of $X$ respectively.

Proposition 2.6**.**

Let $\delta\in Z(X)$ be homogeneous. If $dim(\delta)<n-h$,

[TABLE]

is a class lying in $V^{h}$ (i.e. a representative lying in $V^{h}$).

Proof.

Notice through the equations (2.18), $\Omega_{\tilde{0}}$ is defined by

[TABLE]

By observing the third set of equations

[TABLE]

we can see that there are two types of components for cycles. One lies in

[TABLE]

the other lies in

[TABLE]

Using $\mu^{2}$, we pull them back to $\{0\}\times\Upsilon\times X\times X$ to have two types of components of $\Theta_{0}$. One, $\Theta_{0}^{\prime}$ lies in

[TABLE]

the other, $\Theta_{0}^{\prime\prime}$ lies in

[TABLE]

Now we consider the intersection

[TABLE]

Since $dim(\delta)<n-h$, we can apply the moving lemma to deform the cycle $\delta$ inside of $V^{h}$ to a general position in $\mathbf{P}^{n+1}$. Then the intersection

[TABLE]

is empty. So there is no component of (2.43) can lie in

[TABLE]

Thus the scheme (2.43) lies in

[TABLE]

The projection to the last component lies in $V^{h}$. We complete the proof.

∎

We conclude it with the proof of part (2) of Main theorem.

Theorem 2.7**.**

For natural numbers $q,h,n$ satisfying $h<q<n$,

[TABLE]

Proof.

Since the parameter space is a rational curve $\mathbf{P}^{1}$, $\psi_{0}(\delta)=\psi_{1}(\delta)$. By the computation of Propositions 2.4, 2.5 for $\psi_{1}(\delta)$ and computation of Proposition 2.6 for $\psi_{0}(\delta)$, $\mathbf{i}$ is surjective. Since

[TABLE]

is an embedding, it must be the identity. ∎

3 Cone operator

3.1 Construction

The cone operator is originated from the $\infty$-end cycle in the cone family. But it is more concise to study it without this attachment.

The decomposition (2.5) has a natural projection,

[TABLE]

which yields the rational map,

[TABLE]

where $\mathbf{x}=\mathbf{x}_{1}(z)\oplus\mathbf{x}_{2}(z)$ is the unique decomposition (2.6). Let $G$ be its graph. Let $I_{h}$ be the intersection cycle class

[TABLE]

where $\tilde{\mu}=\mu|_{V^{h}}$.

Definition 3.1**.**

Let $I_{h}\in CH_{n}(\Upsilon\times X\times V^{h})$ define a correspondence

[TABLE]

Then we define a map

[TABLE]

Example 3.2**.**

Let $X$ be a smooth 3-fold. Let $\mu:X\to\mathbf{P}^{4}$ be a birational morphism in a general position as in the cone data. Let $V^{1}\subset X$ be a hyperplane section by a hyperplane $\mathbf{P}^{3}\subset\mathbf{P}^{4}$. Let $\Upsilon\subset\mathbf{P}^{4}$ be a line and $U\subset\Upsilon$ be the affine open set whose points lie outside of $\mathbf{P}^{3}$. The collection of the morphism $X\to\mathbf{P}^{4},\mathbf{P}^{3},\Upsilon$ is called cone data. The cone operator

[TABLE]

is a homomorphism dependent of cone data. It can be described as follows. Let

[TABLE]

be the subvariety whose fibre $\Omega_{\tilde{\infty}}^{\tilde{z}}$ over each $z\in U$ is identical to the subvariety $\subset\mathbf{P}^{4}\times\mathbf{P}^{4}$ defined as

[TABLE]

(where $\#$ is the join operator in the projective space). Let

[TABLE]

By the definition, $\Theta_{\tilde{\infty}}$ is in $CH_{3}(\Upsilon\times X\times V^{1})$. For any curve $\delta\subset V^{1}$, let

[TABLE]

where $\mathcal{S}=\Upsilon\times X\times V^{1}$, $P:\mathcal{S}\to X$ is the projection

The class $Con_{1}([\delta])$ is represented by a cycle $B_{1}$ in $Z_{2}(\mu^{-1}(\Omega_{\tilde{\infty}}))$ through a proper intersection of (3.4) with a suitable $\delta$ in the same class. To grasp its meaning, we observe the representative $B_{2}$ of the class $[X]\cdot_{\mathbf{P}^{4}}[\mu(\delta)\#\Upsilon]$ where the intersection is proper. We note that $B_{1}$ and $B_{2}$ have the same support that equals to the algebraic set of the scheme $\mu^{-1}(\Omega_{\tilde{\infty}})$. Nonetheless they are distinct cycles in $Z_{2}(\mu^{-1}(\Omega_{\tilde{\infty}}))$ and further distinct in $Z_{2}(X)$. Their classes $Con_{1}(\delta)$ and $[X]\cdot_{\mathbf{P}^{4}}[\mu(\delta)\#\Upsilon]$ in $CH_{2}(X)$ are also distinct. So both $Con_{1}(\delta)$ and $[X]\cdot_{\mathbf{P}^{4}}[\mu(\delta)\#\Upsilon]$ are obtained by adding the multiplicities to each component of $Z_{2}(\mu^{-1}(\Omega_{\tilde{\infty}}))$. For instance $Con_{1}(\delta)$ could be a multiple of $[X]\cdot_{\mathbf{P}^{4}}[\mu(\delta)\#\Upsilon]$. This indeed is the case when $\mu$ is an embedding.

3.2 Intersection

Proposition 3.3**.**

For $\delta\in Z(V^{h})$, there is an intersection formula in $CH(X)$,

[TABLE]

where $\zeta_{\ast}([\delta])$ is a multiple of the plane section class.

Proof.

Let’s use the intersection rules in the Chow groups. Specifically we use projection formula and two distinct rules in associativity. By the projection formula (Proposition 8.1.1, [1]) for the projection

[TABLE]

[TABLE]

where $\mathcal{Y}=\Upsilon\times X\times V^{h}$. By the associativity (Proposition 8.1.1, [1]) of the intersection product in $CH(\mathcal{Y})$, we have

[TABLE]

By the associativity for multiplicities ( Example 7.1.8, [1]),

[TABLE]

where $\mathcal{P}=\Upsilon\times\mathbf{P}^{n+1}\times\mathbf{P}^{n+1-h}$. Next to focus on the intersection

[TABLE]

we use coordinates (2.17) to express the intersection scheme

[TABLE]

where $y_{0},\cdots,y_{n+1-h}$ are homogeneous coordinates for $\mathbf{P}^{n+1-h}$, $x_{0},\cdots,x_{n+1}$ for the $\mathbf{P}^{n+1}$. Then the scheme

[TABLE]

is defined by equations

[TABLE]

The first set of equations shows $\digamma$ has two reduced components of dimension $n+2-h$: $\digamma_{1}$ defined

[TABLE]

and $\digamma_{2}$ defined by

[TABLE]

So

[TABLE]

Hence the intersection in $\Upsilon\times\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$,

[TABLE]

is

[TABLE]

where $[\digamma_{1}]$ is onto $\Upsilon$, but $[\digamma_{2}]$ is not. Hence the intersection with

[TABLE]

which is $[\Upsilon\times V^{h}\times V^{h}]\cdot_{\mathcal{Y}}I_{h}$ has two parts classified by their support

[TABLE]

where the first one is an excess intersection and the second one is proper. Notice the projection of the first part to $V^{h}\times V^{h}$ is $m[\Delta_{V^{h}}]$ ($m$ is the multiplicity) and the projection of the other is $[V^{h}\times V^{h}]\cdot_{\mu^{2}}E_{h}$, denoted by $\omega^{h}$. Then after intersecting with $X\times\delta$, followed by the projection formula for the projection $X\times V^{h}\to X$, we obtain

[TABLE]

where $\omega^{h}$ is regarded the correspondence $CH(V^{h})\to CH(V^{h})$. We denote $\mathbbm{i}(\omega^{h}_{\ast}([\delta]))$ by $\zeta_{\ast}([\delta])$. Applying Proposition 2.5 with replacement of $X$ by $V^{h}$, we obtain that $\omega^{h}_{\ast}([\delta])$ is a multiple of the plane section class of $V^{h}$. Since $V^{h}$ is a plane section of $X$, $\zeta_{\ast}([\delta])$ is a multiple of the plane section class of $X$.

At last we need to determine the multiplicity $m$. First we notice the excess intersection in the formula (3.14) is

[TABLE]

in $\Upsilon\times\mathbf{P}^{n+1}\times\mathbf{P}^{n+1}$. For this excess intersection, we can use the same type of the linear deformation $g_{t}^{z}$ (but applied to $V^{h}$). As in the argument part (2) of Proposition 2.4, we obtain the same multiplicity $m=deg(V^{h})=deg(X)$. This completes the proof. ∎

The computation for $Con_{h}$ shows the part (1) of Main theorem.

Theorem 3.4**.**

For natural numbers $q,h,n$ satisfying $h\leq q\leq n$,

[TABLE]

is surjective.

Proof.

Let $\delta\in Z^{q}(V^{h})$. In Proposition 3.3, we choose a multiple of the plane section $\xi([\delta])\in CH^{q-h}(X)$ such that $\mathbf{v}^{h}(\xi([\delta]))$ has the rational equivalence class $\zeta_{\ast}([\delta])\in CH^{q}(X)$. Then formula (3.5) becomes

[TABLE]

By our choice, $\zeta_{\ast}([\delta])-\mathbf{v}^{h}(\xi([\delta]))=0\in CH^{q}(X)$. Hence $\mathbf{v}^{h}$ is surjective. ∎

Let’s see Main theorem implies Theorem 1.4.

Proof.

of Theorem 1.4: Observe the Chow-motivic diagram,

[TABLE]

In the setting of Conjecture 1.2, $h$ may be assumed to be a natural number less than $n$. Then the condition $h<q<n$ is satisfied. So all right arrows in the first row and cycle maps in the columns are surjective. Hence the maps $v^{h},i$ are also surjective. This implies that the composition $i\circ v^{h}=L_{a}^{h}$ is surjective. Main theorem has the further index setting $q={n+h\over 2}$ for the axiom of the hard-Lefschetz-theorem. In particular $L^{h}_{a}$ is injective. So it is an isomorphism. Therefore

[TABLE]

∎