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Here $f:A\to A\times A$ defined by $x\mapsto(x,-nx)$ is the fiber over $0\in
A$ of the trivial fibration $\sum\circ\beta:A\times A\to A$.

By [dCM02, Corollary 5.1.5],
$[Z^{\prime}]^{*}:H^{2n-1}(A^{[n+1]},\mathbb{Q})\to H^{1}(A\times
A,\mathbb{Q})$ is surjective. Furthermore, the restriction map
$f^{*}:H^{1}(A\times A,\mathbb{Q})\to H^{1}(A,\mathbb{Q})$ is surjective since
$f$ is the fiber of a trivial fibration. These imply that
$[Z]^{*}:H^{2n-1}(X,\mathbb{Q})\to H^{1}(A,\mathbb{Q})$ is surjective.
Finally, since the restriction map $i^{*}:H^{1}(A,\mathbb{Q})\to
H^{1}(C,\mathbb{Q})$ is injective by Lefschetz hyperplane theorem, the
composition map $i^{*}\circ[Z]^{*}:H^{2n-1}(X,\mathbb{Q})\to
H^{1}(C,\mathbb{Q})$ is nonzero. This implies that the restriction map
$H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(Z_{C},\mathbb{Q})$ is nonzero, as desired.
∎

## Chapter 3 On Some Birational Invariants of Hyper-Kähler Manifolds

We study in this chapter three birational invariants of projective varieties,
the degree of irrationality, the fibering gonality and the fibering genus. We
first improve the lower bound in a recent result of Voisin bounding from below
the fibering genus of a Mumford-Tate very general projective hyper-Kähler
manifold by a constant depending on its dimension and the second Betti number.
We also compare the relations between these birational invariants for
projective $K3$ surfaces of Picard number $1$ and study the asymptotic
behaviors of their degree of irrationality and fibering gonality.

This chapter presents the main result of [Bai24]. The structure is as follows.
In Section 3.1, we give a panorama of known results in this area and a brief
introduction to the main results of this chapter. Section 3.2, Section 3.3 and
Section 3.4 are the content of the article [Bai24]. In Section 3.2, we prove
Theorem 3.1.7. In Section 3.3, we prove Theorem 3.1.14. In Section 3.4, we
prove Propositions 3.1.15 and 3.1.16. In a final Section 3.5, we present some
related results and questions.

###  3.1 Background and Introduction

####  3.1.1 Some measures of irrationality

Let $X$ be a projective variety over $\mathbb{C}$ of dimension $n$. The
following invariant, known as the _degree of irrationality_ , was initially
defined and studied by Hisao Yoshihara and others before being further
explored in [BDELU17].

######  Definition 3.1.1.

The _degree of irrationality_ of $X$, denoted as $\operatorname{Irr}(X)$, is
the minimal degree of dominant rational maps
$\varphi:X\dashrightarrow\mathbb{P}^{n}$. Formally,

$\operatorname{Irr}(X):=\min\\{\deg\varphi:\text{there is a dominant rational
map }\varphi:X\dashrightarrow\mathbb{P}^{n}\\}.$

In their important work, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery
studied $\operatorname{Irr}(X)$ for very general hypersurfaces $X$ of large
degree in [BDELU17]. Note that the following Theorem C from [BDELU17],
regarding the degree of irrationality for hypersurfaces, has also appeared in
[BCD] and was proven earlier in the case of surfacees in the thesis by Renza
Cortini.

######  Theorem 3.1.2 (=Theorem C in [BDELU17]).

Let $X\subset\mathbb{P}^{n+1}$ be a very general smooth hypersurface of
dimension $n$ and degree $d\geq 2n+1$. Then $\operatorname{Irr}(X)=d-1$.
Furthermore, if $d\geq 2n+2$, then any rational mapping
$f:X\dashrightarrow\mathbb{P}^{n}$ with $\deg(f)=d-1$ is given by the
projection from a point of $X$.

In an attempt to determine the degree of irrationality and to better
understand the geometry of the variety, the following two measures of
irrationality are proposed and studied by Voisin in [Voi21].

######  Definition 3.1.3 ([Voi21]).

(i) The fibering gonality of $X$, $\mathrm{Fibgon}(X)$, is the minimal number
$c$ such that there exists a rational dominant map $\pi:X\dashrightarrow B$
such that $\dim B=\dim X-1$ and the general fiber is a connected curve $C$ of
gonality $c$.
(ii) The fibering genus of $X$, $\mathrm{Fibgen}(X)$, is the minimal number
$c$ such that there exists a rational dominant map $\pi:X\dashrightarrow B$
such that $\dim B=\dim X-1$ and the general fiber is a connected curve $C$ of
geometric genus $c$.

Following [Voi21], the following inequalities hold between these two measures
of irrationality and the degree of irrationality.

######  Lemma 3.1.4.

For any smooth projective variety $X$, we have
(i) $\mathrm{Fibgon}(X)\leq\mathrm{Irr}(X)$.
(ii) $\mathrm{Fibgon}(X)\leq\frac{1}{2}(\mathrm{Fibgen}(X)-1)+2$.

####  3.1.2 Measures of irrationality of hyper-Kähler manifolds

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. The following
result is directly implied by the main result of [AP92].

######  Theorem 3.1.5 (Alzati-Pirola [AP92]).

The degree of irrationality of $X$ is greater than $n$.

A direct proof of Theorem 3.1.5 goes as follows.

###### Proof.

Let $X\dashrightarrow Y$ be a dominant generically finite rational map of
degree $d$ to a smooth variety $Y$ such that $H^{l,0}(Y)=0$ for any $l>0$. We
are going to show that $d>n$. First we desingularize the rational map as
$\phi:\tilde{X}\to Y$ and let $\sigma_{\tilde{X}}$ be the pull-back of the
holomorphic $2$-form of $X$ on $\tilde{X}$.

Consider the fibered product
$Z^{\prime}=\tilde{X}\times_{Y}\tilde{X}\times_{Y}\ldots\times_{Y}\tilde{X}\subset\tilde{X}\times\ldots\times\tilde{X}$
with $d$ copies of $\tilde{X}$. Let $\Delta_{ij}$ be the pull back of the
diagonal under the natural projection map
$pr_{ij}:Z^{\prime}\to\tilde{X}\times\tilde{X}$. Let $Z$ be the Zariski
closure of $Z^{\prime}-\bigcup_{i,j}\Delta_{ij}$ in
$\tilde{X}\times\ldots\times\tilde{X}$. Since $\phi:\tilde{X}\to Y$ is of
degree $d$, the natural projection map $pr_{i}:Z\to\tilde{X}$ is dominant and
generically finite for any $i$. Hence, we may choose an irreducible component
of $Z$ such that the projection maps are dominant and generically finite.
Denote $X^{\prime}$ a desingularisation of such an irreducible component of
$Z$. Let $p_{i}:X^{\prime}\to\tilde{X}$ be the natural projection maps and let
$f:X^{\prime}\to Y$ be the structural map of the fibered product. Let $I\in
CH(X^{\prime}\times\tilde{X})$ be the incidence cycle such that
$I_{*}:CH_{0}(X^{\prime})\to CH_{0}(\tilde{X})$ is given by
$x\mapsto\sum_{i=1}^{d}p_{i}(x)$ for general $x\in X^{\prime}$. Then,
$I_{*}=\phi^{*}f_{*}:CH_{0}(X^{\prime})\to CH_{0}(\tilde{X})$. By Mumford-type
theorems [Voi14, 1.1.3],
$I^{*}=f^{*}\phi_{*}:H^{0}(\tilde{X},\Omega_{\tilde{X}}^{l})\to
H^{0}(X^{\prime},\Omega_{X^{\prime}}^{l})$ for $l>0$. Since $H^{l,0}(Y)=0$ for
$l>0$, and since $I^{*}$ factorizes through $H^{l,0}(Y)$, we have
$I^{*}=0:H^{0}(\tilde{X},\Omega_{\tilde{X}}^{l})\to
H^{0}(X^{\prime},\Omega_{X^{\prime}}^{l})$. Hence, for any $l>0$, we have

$\sum_{i=1}^{d}p_{i}^{*}\sigma_{\tilde{X}}^{l}=0$

By the theory of symmetric polynomials, any symmetric polynomial without
constant terms in
$p_{1}^{*}\sigma_{\tilde{X}},\ldots,p_{d}^{*}\sigma_{\tilde{X}}$ is zero.
Hence,

$\prod_{i=1}^{d}(T-p_{i}^{*}\sigma_{\tilde{X}})=T^{d}.$

Taking $T=p_{1}^{*}\sigma_{\tilde{X}}$, one gets
$p_{1}^{*}\sigma_{\tilde{X}}^{d}=0$. This implies in particular that the rank
of $\sigma_{X}$ is less than $d$. Hence, $d>n$. ∎

For the fibering genus of hyper-Kähler manifolds, Voisin proves the following
result in [Voi21].

######  Theorem 3.1.6 (Voisin [Voi21]).

Let $X$ be a Mumford-Tate very general hyper-Kähler manifold of dimension $2n$
with $n\geq 3$ and $b_{2,\mathrm{tr}}(X)\geq 5$. Then

$\mathrm{Fibgen}(X)\geq\min\\{n+2,2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}\\}.$

The reader is referred to Section 1.4 and Section 1.4.5 for details about
Mumford-Tate groups. In this chapter, we will present a sharper bound.

######  Theorem 3.1.7.

Let $X$ be a Mumford-Tate very general hyper-Kähler manifold of dimension $2n$
and assume $b_{2,\mathrm{tr}}(X)\geq 5$. Then

$\mathrm{Fibgen}(X)\geq\min\left\\{n+\left\lceil\frac{-1+\sqrt{8n-7}}{2}\right\rceil,2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}\right\\}.$

####  3.1.3 Measures of irrationality of K3 surfaces

We first recollect some known results on the assymptotic behaviors of the
measures of irrationality of K3 surfaces.

##### Degree of irrationality

In [BDELU17], the following conjecture on the degree of irrationality of K3
surfaces is proposed.

######  Conjecture 3.1.8 ([BDELU17]).

Let $\\{(S_{d},L_{d})\\}_{d\in\mathbb{N}}$ be very general polarized $K3$
surfaces such that $L_{d}^{2}=2d-2$. Then

$\limsup_{d\to\infty}\mathrm{Irr}(S_{d})=+\infty.$

In the litterature, the $d$ appearing in Conjecture 3.1.8 is called the
_genus_ of the K3 surface. To justify the name, for any smooth curve $C\subset
S$ in the linear sytem $|L_{d}|$, it follows from tne adjunction formula that
the arithemetic genus of $C$ is $d$.

######  Remark 3.1.9.

We can calculate the degree of irrationality of $S_{d}$ when the genus $d$ is
small.

  * •

When $d=2$, the very general K3 surface $S_{2}$ is a double cover of
$\mathbb{P}^{2}$ ramified along a sextic curve. Since K3 surfaces are not
rational, the degree of irrationality of $S_{2}$ is $2$.

  * •

When $d=3$, the very general K3 surface $S_{3}$ is a quartic surface in
$\mathbb{P}^{3}$ and is not birational to a double cover of $\mathbb{P}^{2}$
since $S_{3}$ does not admit a rational involution. Projecting
$S_{3}\subset\mathbb{P}^{3}$ via a general point on $S_{3}$ gives a rational
map of degree $3$ of $S_{3}$ to $\mathbb{P}^{2}$. Hence, the degree of
irrationality of $S_{3}$ is $3$.

Despite the above examples, it is in general quite hard to determine the
degree of irrationality of K3 surfaces.

One general result we know in the spirit of this conjecture is the following
theorem proved in Stapleton’s thesis [Stap17].

######  Theorem 3.1.10 (Stapleton [Stap17]).

There exists a constant $C>0$ such that $\mathrm{Irr}(S_{d})\leq C\sqrt{d}$.

##### Fibering genus

The fibering genus of surfaces has already been studied by Ein and Lazarsfeld
in [EL20], under the name _Konno invariant_. They give a good estimate of the
asymptotic behavior of fibering genus of K3 surfaces as the genus tends to
infinity.

######  Theorem 3.1.11 (Ein–Lazarsfeld [EL20]).

Let $S_{d}$ be a polarized K3 surface of Picard rank $1$ and genus $d$. Then

$\mathrm{Fibgen}(S_{d})=\Theta(\sqrt{d}).$

More precisely,

$\sqrt{\frac{d}{2}}\leq\mathrm{Fibgen}(S_{d})\leq 2\sqrt{2d}.$

######  Corollary 3.1.12.

We have

$\lim_{d\to\infty}\mathrm{Fibgen}(S_{d})=+\infty.$

##### Fibering gonality

We do not know much about the fibering gonality of K3 surfaces. In this
chapter, we will prove the following relation between the degree of
irrationality and the fibering gonality of K3 surfaces.

######  Theorem 3.1.13.

Let $\\{S_{d}\\}_{d\in\mathbb{N}}$ be projective K3 surfaces such that the
Picard group of $S_{d}$ is generated by a line bundle with self intersection
number $2d-2$. Then

$\limsup_{d\to\infty}\mathrm{Irr}(S_{d})=+\infty\iff\limsup_{d\to\infty}\mathrm{Fibgon}(S_{d})=+\infty.$

The proof of Theorem 3.1.13 relies on Theorem 3.1.11 and the following
theorem.

######  Theorem 3.1.14.

Let $S$ be a projective K3 surface whose Picard number is $1$. Then one of the
following two cases holds:
(a) $\mathrm{Irr}(S)=\mathrm{Fibgon}(S)$;
(b) $\mathrm{Fibgen}(S)^{2}\leq\mathrm{Fibgon}(S)^{21}$.

###### Proof of Theorem 3.1.13 assuming Theorem 3.1.14.

The implication $\Leftarrow$ is clear since
$\mathrm{Fibgon}(S_{d})\leq\mathrm{Irr}(S_{d})$ for any $d$. Now let us prove
the implication $\Rightarrow$. Theorem 3.1.11 ([EL20, Theorem B]) shows that
$\lim_{d}\mathrm{Fibgen}(S_{d})=+\infty$. If
$\limsup_{d}\mathrm{Fibgon}(S_{d})\neq+\infty$, then there exist constants $C$
and $D$ such that for every $d>D$ we have
$\mathrm{Fibgon}(S_{d})^{21/2}<C<\mathrm{Fibgen}(S_{d})$. Hence for $d>D$, we
must have $\mathrm{Irr}(S_{d})=\mathrm{Fibgon}(S_{d})<C^{2/21}$ by Theorem
3.1.14, which contradicts the assumption that
$\limsup_{d}\mathrm{Irr}(S_{d})=+\infty$. ∎

Conjecture 3.1.8 predicts that $\limsup_{d}\mathrm{Irr}(S_{d})=+\infty$, so it
should be expected that $\limsup_{d}\mathrm{Fibgon}(S_{d})=+\infty$ as well.

####  3.1.4 Some lattice theoretic results about hyper-Kähler manifolds

In the course of the proof of Theorem 3.1.14, we obtain the following
inequalities about the discriminant of Picard lattices of projective hyper-
Kähler manifolds. Although we only use this proposition in the case of $K3$
surfaces and we only use one side of the inequalities, the following general
form is of independent interest. Here $\mathrm{disc}(\mathrm{Pic}(X))$ denotes
the discriminant of the Picard lattice of $X$ with respect to the Beauville-
Bogolov-Fujiki form and similarly for
$\mathrm{disc}(\mathrm{Pic}(X^{\prime}))$.

######  Proposition 3.1.15.

Let $X$ and $X^{\prime}$ be deformation equivalent projective hyper-Kähler
manifolds of dimension $2n$. Let $\phi\colon X\dashrightarrow X^{\prime}$ be a
dominant rational map. Then

$(\deg\phi)^{(\frac{1}{n}-2)b_{2,\mathrm{tr}}(X)}\leq\left|\frac{\mathrm{disc}(\mathrm{Pic}(X))}{\mathrm{disc}(\mathrm{Pic}(X^{\prime}))}\right|\leq(\deg\phi)^{\frac{b_{2,\mathrm{tr}}(X)}{n}}.$

If in the previous proposition $X$ and $X^{\prime}$ are $K3$ surfaces, we can
get a slightly better lower bound.

######  Proposition 3.1.16.

Let $S$ and $S^{\prime}$ be projective $K3$ surfaces and let
$\lambda(S)=\min\\{\rho(S),b_{2,\mathrm{tr}}(S)\\}$ where $\rho(S)$ is the
Picard number of $S$. Let $\phi\colon S\dashrightarrow S^{\prime}$ be a
dominant rational map. Then

$(\deg\phi)^{-\lambda(S)}\leq\left|\frac{\mathrm{disc}(\mathrm{Pic}(S))}{\mathrm{disc}(\mathrm{Pic}(S^{\prime}))}\right|\leq(\deg\phi)^{b_{2,\mathrm{tr}}(S)}.$

###  3.2 Fibering genus of very general hyper-Kähler manifolds

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. Let $f\colon
X\dashrightarrow B$ be a fibration into curves and let $\tau\colon\tilde{X}\to
X$ and $\tilde{f}\colon\tilde{X}\to B$ be a resolution of indeterminacy
points. Let $\tilde{X}_{b}$ be a smooth fiber of $\tilde{f}$ over a general
point $b\in B$. The exact sequence of vector bundles on $\tilde{X}_{b}$

$0\to
N_{\tilde{X}_{b}/\tilde{X}}^{*}\to\Omega_{\tilde{X}|\tilde{X}_{b}}\to\Omega_{\tilde{X}_{b}}\to
0$

induces an exact sequence

$0\to
N_{\tilde{X}_{b}/\tilde{X}}^{*}\wedge\Omega_{\tilde{X}|\tilde{X}_{b}}\to\Omega_{\tilde{X}|\tilde{X}_{b}}^{2}\to\Omega_{\tilde{X}_{b}}^{2}\to
0.$

Since $\tilde{X}_{b}$ is a curve, we have $\Omega_{\tilde{X}_{b}}^{2}=0$.
Therefore, $\Omega_{\tilde{X}|\tilde{X}_{b}}^{2}\cong
N_{\tilde{X}_{b}/\tilde{X}}^{*}\wedge\Omega_{\tilde{X}|\tilde{X}_{b}}$ as
vector bundles on $\tilde{X}_{b}$. Let us view
$N_{\tilde{X}_{b}/\tilde{X}}^{*}\wedge\Omega_{\tilde{X}|\tilde{X}_{b}}$ as a
subbundle of
$N_{\tilde{X}_{b}/\tilde{X}}^{*}\otimes\Omega_{\tilde{X}|\tilde{X}_{b}}$. Then
$(\tau^{*}(\sigma_{X}))|_{\tilde{X}_{b}}\in
H^{0}(\tilde{X}_{b},\Omega_{\tilde{X}|\tilde{X}_{b}}^{2})$ can be viewed as an
element in
$H^{0}(\tilde{X}_{b},N_{\tilde{X}_{b}/\tilde{X}}^{*}\wedge\Omega_{\tilde{X}|\tilde{X}_{b}})=\mathrm{Hom}_{\tilde{X}_{b}}(N_{\tilde{X}_{b}/\tilde{X}},\Omega_{\tilde{X}|\tilde{X}_{b}})$.
Therefore, $(\tau^{*}(\sigma_{X}))|_{\tilde{X}_{b}}$ induces a morphism
$\phi_{b}\colon H^{0}(\tilde{X}_{b},N_{\tilde{X}_{b}/\tilde{X}})\to
H^{0}(\tilde{X}_{b},\Omega_{\tilde{X}_{b}})$. Therefore, we have a morphism

$\sigma_{b}\colon T_{B,b}\to
H^{0}(\tilde{X}_{b},\Omega_{\tilde{X}_{b}})=H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$

defined as the composition of the natural morphisms $T_{B,b}\to
H^{0}(\tilde{X}_{b},N_{\tilde{X}_{b}/\tilde{X}})$ and $\phi_{b}\colon
H^{0}(\tilde{X}_{b},N_{\tilde{X}_{b}/\tilde{X}})\to
H^{0}(\tilde{X}_{b},\Omega_{\tilde{X}_{b}})$. Here,
$H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\cong\mathbb{C}^{g}$ where
$g=g(\tilde{X}_{b})$ is the genus of the curve $\tilde{X}_{b}$. Let
$\rho\colon T_{B,b}\to H^{1}(\tilde{X}_{b},T_{\tilde{X}_{b}})$ be the Kodaira-
Spencer map. In [Voi21], Voisin proves the following

######  Proposition 3.2.1 (Voisin [Voi21]).

(i) If $n\geq 2$, then $\tilde{X}_{b}$ is _not_ hyperelliptic.
(ii) The rank of $\sigma_{b}$ is $\geq n$.
(iii) Let $I_{b}\subset H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}^{\otimes 2})$ be
image of $\mathrm{Im}\sigma_{b}\otimes H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$
under the multiplication map

$\mu\colon H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\otimes
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\to
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}^{\otimes 2}).$

Then $\rho(\ker\sigma_{b})\subset H^{1}({\tilde{X}_{b}},T_{\tilde{X}_{b}})$ is
orthogonal to $I_{b}$ via the Serre pairing for $T_{\tilde{X}_{b}}$:

$H^{1}({\tilde{X}_{b}},T_{\tilde{X}_{b}})\otimes
H^{0}({\tilde{X}_{b}},K_{\tilde{X}_{b}}^{\otimes 2})\to
H^{1}({\tilde{X}_{b}},K_{\tilde{X}_{b}})\cong
H^{0}(\tilde{X}_{b},\mathcal{O}_{\tilde{X}_{b}})\cong\mathbb{C}.$

(iv) Assume that that $b_{2,\mathrm{tr}}(X)\geq 5$ and that the Mumford-Tate
group of $H^{2}(X,\mathbb{Q})_{tr}$ is maximal. If $\rho\colon T_{B,b}\to
H^{1}({\tilde{X}_{b}},T_{\tilde{X}_{b}})$ is _not_ injective, then $g\geq
2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}$.

The proofs of (i), (ii) and (iii) are explicitly written in [Voi21]. Although
(iv) is not stated in [Voi21] with this generality, it is essentially proved
there (see the proof of Lemma 2.13 and Lemma 2.15 in loc. cit.).

####  3.2.1 Proof of Theorem 3.1.7

For $n=1$, the inequality is obvious, since $K3$ surfaces cannot be fibered
into rational curves. From now on, let us assume $n\geq 2$ to be able to
assume $\tilde{X}_{b}$ is not hyperelliptic (see Proposition 3.2.1 (i)). Let
$k$ be the corank of $\sigma_{b}$, i.e., $\mathrm{rank}\sigma_{b}=g-k$. We now
prove

######  Lemma 3.2.2.

With notation as above, we have the following inequality

$\dim\ker\rho\geq 2n-1-(g-k)-\frac{k(k+1)}{2}.$

###### Proof.

We use the notation as in Proposition 3.2.1 (iii). By Serre duality, the
orthogonality result in Proposition 3.2.1 (iii) implies that
$\dim\rho(\ker\sigma_{b})+\dim I_{b}\leq\dim
H^{0}({\tilde{X}_{b}},K_{\tilde{X}_{b}}^{\otimes 2})$, that is,

$\mathrm{codim}(I_{b}\subset H^{0}({\tilde{X}_{b}},K_{\tilde{X}_{b}}^{\otimes
2}))\leq\dim\rho(\ker\sigma_{b}).$ (3.1)

The multiplication map $\mu\colon
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\otimes
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\to
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}^{\otimes 2})$ factors through the
symmetric product $\mathrm{Sym}^{2}H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$.
Let $\mathrm{Im}\,\sigma_{b}\cdot H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$
denote the image of $\mathrm{Im}\,\sigma_{b}\otimes
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$ under the canonical symmetrization map
$\mathrm{pr}\colon H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\otimes
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\to\mathrm{Sym}^{2}H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$.
Recall that $k$ is the corank of $\sigma_{b}$, that is,
$\mathrm{codim}(\mathrm{Im}\,\sigma_{b}\subset
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}))=k$. Then we have

$\mathrm{codim}(\mathrm{Im}\,\sigma_{b}\cdot
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}))\subset\mathrm{Sym}^{2}H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}))=\frac{k(k+1)}{2}.$
(3.2)

On the other hand, by Max Noether theorem (see [ACGH85, Chapter III, §2] or
[Voi03]), the multiplication map
$\mu^{\prime}\colon\mathrm{Sym}^{2}H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\to
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}^{\otimes 2})$ is surjective, since
$\tilde{X}_{b}$ is not hyperelliptic. Taking into account the fact that
$I_{b}$ is the image of $\mathrm{Im}\,\sigma_{b}\cdot
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})$ under the map $\mu^{\prime}$, we have
the following inequality

$\mathrm{codim}(I_{b}\subset H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}}^{\otimes
2}))\leq\mathrm{codim}(\mathrm{Im}\,\sigma_{b}\cdot
H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})\subset\mathrm{Sym}^{2}H^{0}(\tilde{X}_{b},K_{\tilde{X}_{b}})).$
(3.3)

Combining (3.1), (3.2) and (3.3), we get
$\dim\rho(\ker\sigma_{b})\leq\frac{k(k+1)}{2}$, from which we deduce that

$\dim\ker\rho\geq\dim\ker\sigma_{b}-\frac{k(k+1)}{2}=2n-1-(g-k)-\frac{k(k+1)}{2}.$

∎

###### Proof of Theorem 3.1.7.

Assuming that
$g(\tilde{X}_{b})<2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}$, we have
to prove $g\geq n+\lceil\frac{-1+\sqrt{8n-7}}{2}\rceil$. By Proposition 3.2.1
(ii), (iv) and Lemma 3.2.2, we have the following constraints on $g$ and $k$:

$\left\\{\begin{array}[]{lr}k\geq 0\\\ &\cr g-k-n\geq 0\\\ &\cr
2n-1-(g-k)-\frac{k(k+1)}{2}\leq 0.\end{array}\right.$

In order to find the minimal possible value of $g$ under these constraints, we
make the following discussion according to the values of $k$.

  * •

When $0\leq k\leq\frac{-1+\sqrt{8n-7}}{2}$, we have
$k-\frac{k(k+1)}{2}+2n-1\geq n+k$. Hence, the minimal possible value of $g$ in
this domain is the minimum of $k-\frac{k(k+1)}{2}+2n-1$ with $0\leq
k\leq\frac{-1+\sqrt{8n-7}}{2}$, which is $n+\frac{-1+\sqrt{8n-7}}{2}$.

  * •

When $k\geq\frac{-1+\sqrt{8n-7}}{2}$, we have $k-\frac{k(k+1)}{2}+2n-1\leq
n+k$. Hence, the minimal possible value of $g$ in this domain is the minimum
of $n+k$ with $k\geq\frac{-1+\sqrt{8n-7}}{2}$, which is
$n+\frac{-1+\sqrt{8n-7}}{2}$.

Since $g$ and $k$ are integers, we find $g\geq
n+\lceil\frac{-1+\sqrt{8n-7}}{2}\rceil$, as desired.

∎

######  Remark 3.2.3.

Our proof relies on the inequality (3.3) which only uses the surjectivity of
the multiplication map $\mu^{\prime}$. With more information on the geometry
of the canonical embedding, and in particular, on the gonality of the fibers,
we could get a better estimate in Theorem 3.1.7.

###  3.3 Relations between birational invariants of $K3$ surfaces

In this section, we are going to prove Theorem 3.1.14 that relates the three
birational invariants, namely, the degree of irrationality, the fibering
gonality and the fibering genus, of projective $K3$ surfaces of Picard number
$1$.

####  3.3.1 A factorization

Let $S$ be a smooth projective surface and let $f\colon S\dashrightarrow B$ be
a fibration into curves over a smooth base $B$. After a resolution of
inderminacies of $f$ and replacing $S$ by another birational model, we may
assume $f\colon S\to B$ is a morphism. Let $d$ be the gonality of the general
fiber of $f$, so the general fiber $C$ admits a degree $d$ morphism from $C$
to $\mathbb{P}^{1}$. A standard argument shows that we can spread this
morphism into a family up to a generically finite base change.

######  Lemma 3.3.1.

There is a generically finite morphism $\pi\colon B^{\prime}\to B$ and a
degree $d$ dominant rational map $\psi\colon
S\times_{B}B^{\prime}\dashrightarrow\mathbb{P}^{1}\times B^{\prime}$ over
$B^{\prime}$.

###### Proof.

Let $B_{0}$ be the smooth locus of $f\colon S\to B$ and let $f_{0}\colon
S_{0}\to B_{0}$ be the restriction of $f$ on the smooth locus. Let
$p\colon\mathrm{Pic}^{d}(S_{0}/B_{0})\to B_{0}$ be the degree $d$ relative
Picard variety of $f_{0}\colon S_{0}\to B_{0}$. By the assumption on the
general fiber of $f\colon S\to B$, the restriction of the map
$p\colon\mathrm{Pic}^{d}(S_{0}/B_{0})\to B_{0}$ to the Brill-Noether locus in
$\mathrm{Pic}^{d}(S_{0}/B_{0})$ of the linear systems of degree $d$ and
dimension $1$ is dominant. Let $B_{0}^{\prime}$ be a general reduced
irreducible subscheme of $\mathrm{Pic}^{d}(S_{0}/B_{0})$ that is dominant and
generically finite over $B_{0}$ by $p$. Let us take $B^{\prime}$ to be a
completion of $B_{0}^{\prime}$. Then by construction, the universal line
bundle restricted to $S_{0}\times_{B}B_{0}^{\prime}$ gives a dominant rational
map $\psi\colon
S\times_{B}B^{\prime}\dashrightarrow\mathbb{P}_{B^{\prime}}^{1}$ of degree
$d$, as desired. ∎

It is natural to ask if $\psi\colon
S\times_{B}B^{\prime}\dashrightarrow\mathbb{P}^{1}\times B^{\prime}$ over
$B^{\prime}$ descends to a rational map $\psi_{B}\colon
S\dashrightarrow\mathbb{P}^{1}\times B$ over $B$. A moment of thinking will
convince us that we are asking too much, because $\pi\colon B^{\prime}\to B$
is in general not a Galois cover. We make the following construction. Let $n$
be the degree of the morphism $\pi\colon B^{\prime}\to B$. Consider the $n$-th
self fibred product of $B^{\prime}$ over $B$:
$B^{\prime}\times_{B}\cdots\times_{B}B^{\prime}$. define $B^{\prime\prime}$ to
be the closure in $B^{\prime}\times_{B}\cdots\times_{B}B^{\prime}$ of the set

$\\{(x_{1},\ldots,x_{n})\in
B^{\prime}\times_{B}\cdots\times_{B}B^{\prime}\colon
x_{1},\ldots,x_{n}\textrm{ are distinct }\\}.$

Then $\pi^{\prime}\colon B^{\prime\prime}\to B$ is of degree $n!$ and the
symmetric group $\mathfrak{S}_{n}$ permuting the components of
$B^{\prime}\times_{B}\cdots\times_{B}B^{\prime}$ acts on an open dense subset
of $B^{\prime\prime}$. The rational map $\psi\colon
S\times_{B}B^{\prime}\dashrightarrow\mathbb{P}^{1}\times B^{\prime}$ over
$B^{\prime}$ given in Lemma 3.3.1 can be extended to a rational map

$\psi^{\prime}\colon
S\times_{B}B^{\prime\prime}\dashrightarrow(\mathbb{P}^{1})^{n}\times
B^{\prime\prime}$ (3.4)

over $B^{\prime\prime}$ in a natural way: let $x\in S$ and let
$y=(y_{1},\ldots,y_{n})\in B^{\prime\prime}$ be general points, we define
$\psi^{\prime}(x,y)=(\psi(x,y_{1}),\ldots,\psi(x,y_{n}),y)$. Moreover, the
symmetric group $\mathfrak{S}_{n}$ acts canonically on both sides of (3.4) in
the following way. To define the action of $\mathfrak{S}_{n}$ on
$S\times_{B}B^{\prime\prime}$, we let $\mathfrak{S}_{n}$ act trivially on $S$
and act as permutations of components of $B^{\prime\prime}$; and to define the
action on $(\mathbb{P}^{1})^{n}\times B^{\prime\prime}$, we let
$\mathfrak{S}_{n}$ act as permutations of components for both
$(\mathbb{P}^{1})^{n}$ and $B^{\prime\prime}$. It is clear from the
construction that $\psi^{\prime}\colon
S\times_{B}B^{\prime\prime}\dashrightarrow(\mathbb{P}^{1})^{n}\times
B^{\prime\prime}$ is $\mathfrak{S}_{n}$-equivariant. Thus $\psi^{\prime}$
induces a rational map $S\dashrightarrow((\mathbb{P}^{1})^{n}\times
B^{\prime\prime})/\mathfrak{S}_{n}$ over $B$. Let $S^{\prime}$ be the image of
this map. Thus we get a dominant rational map

$\phi\colon S\dashrightarrow S^{\prime}.$

######  Proposition 3.3.2.

$S^{\prime}$ is a surface and the degree of $\phi$ divides $d$. Furthermore,
if $n\geq 2$, the general fiber of $S^{\prime}\dashrightarrow B$ is of
geometric genus $\leq(\frac{d}{\deg\phi}-1)^{2}$.

###### Proof.

Over the general point $b=(b_{1},\ldots,b_{n})\in B^{\prime\prime}$,
$\psi^{\prime}$ is given by the morphism $\psi^{\prime}_{b}\colon
C\to(\mathbb{P}^{1})^{n}$ induced by the $n$ morphisms $C\to\mathbb{P}^{1}$ of
degree $d$ corresponding to the points $b_{i}\in B^{\prime}$, where $C$ is the
fiber of $f\colon S\dashrightarrow B$ oveer $\pi^{\prime}(b)\in B$. Let
$C^{\prime}$ be the image of $\psi_{b}^{\prime}$. Then the fiber of
$S^{\prime}\dashrightarrow B$ over $\pi^{\prime}(b)\in B$ is $C^{\prime}$ by
construction. Thus $S^{\prime}$ is a surface, and the degree of $\phi$ is the
degree of $C$ over $C^{\prime}$, which divides $d$. This proves the first
statement. To prove the second, we need to prove the geometric genus of
$C^{\prime}$ is $\leq(\frac{d}{\deg\phi}-1)^{2}$. Since $C^{\prime}$ is a
curve of multi-degree $(\frac{d}{\deg\phi},\ldots,\frac{d}{\deg\phi})$ in
$(\mathbb{P}^{1})^{n}$, we can use Lemma 3.3.3 below concerning algebraic
curves in $(\mathbb{P}^{1})^{n}$. ∎

######  Lemma 3.3.3.

Let $n\geq 2$. Let $C$ be an integral curve in $(\mathbb{P}^{1})^{n}$ of
multi-degree $(d,\ldots,d)$. Then the geometric genus of $C$ is less than or
equal to $(d-1)^{2}$.

###### Proof.

We prove it by induction on $n$. When $n=2$, it is the adjunction formula. Now
assume that any curve $C^{\prime}\subset(\mathbb{P}^{1})^{n-1}$ of multi-
degree $(e,\ldots,e)$ has geometric genus $\leq(e-1)^{2}$. Consider the
projection $C\to C^{\prime\prime}\subset(\mathbb{P}^{1})^{n-1}$ to the first
$n-1$ components. The degree of $C\to C^{\prime\prime}$ is of the form $d/e$,
for some $e$. Hence, $C^{\prime\prime}\subset(\mathbb{P}^{1})^{n-1}$ is a
curve of multi-degree $(e,\ldots,e)$, hence it has geometric genus
$\leq(e-1)^{2}$ by induction assumption. Let $\tilde{C}$ and
$\tilde{C}^{\prime\prime}$ be the normalization of $C$ and $C^{\prime\prime}$
respectively. Then $\tilde{C}$ is birational to its image
$C^{\prime\prime\prime}$ in $\tilde{C}^{\prime\prime}\times\mathbb{P}^{1}$,
where the map to the second component is given by the composition map
$\tilde{C}\to C\stackrel{{\scriptstyle i_{n}}}{{\to}}\mathbb{P}^{1}$. Here,
the map $i_{n}\colon C\to\mathbb{P}^{1}$ is the projection map to the $n$-th
component. Note that $C^{\prime\prime\prime}$ is of multi-degree $(d/e,d)$ in
$\tilde{C}^{\prime\prime}\times\mathbb{P}^{1}$ and note that
$NS(C^{\prime\prime}\times\mathbb{P}^{1})=NS(C^{\prime\prime})\oplus
NS(\mathbb{P}^{1})$. Adjunction formula and $g(C^{\prime\prime})\leq(e-1)^{2}$
give us
$p_{a}(C^{\prime\prime\prime})=\frac{d(g(C^{\prime\prime})+d-1)}{e}-d+1\leq
d(e+d/e-2)-d+1\leq(d-1)^{2}$ since $1\leq e\leq d$. Hence,
$p_{g}(C)=g(\tilde{C})=p_{g}(C^{\prime\prime\prime})\leq
p_{a}(C^{\prime\prime\prime})\leq(d-1)^{2}$, as desired. ∎

####  3.3.2 Rational maps between $K3$ surfaces

We treat rational maps between $K3$ surfaces in this part. Let $S$ (resp.
$S^{\prime}$) be a projective $K3$ surface whose Picard group is generated by
an ample line bundle of degree $2D-2$ (resp. $2D^{\prime}-2$). Let $\phi\colon
S\dashrightarrow S^{\prime}$ be a dominant rational map.

######  Proposition 3.3.4.

We have the following inequality

$\frac{1}{(\deg\phi)^{21}}\leq\frac{D-1}{D^{\prime}-1}\leq(\deg\phi)^{21}.$

###### Proof.

Let $\tau\colon\tilde{S}\to S$ and $\tilde{\phi}\colon\tilde{S}\to S^{\prime}$
be a resolution of indeterminacy points of $\phi\colon S\dashrightarrow
S^{\prime}.$ Let $T$ (resp. $T^{\prime}$) be the lattice
$H^{2}(S,\mathbb{Z})_{tr}$ (resp. $H^{2}(S^{\prime},\mathbb{Z})_{tr}$) endowed
with the intersection form. For a positive integer $e$, define $T^{\prime}(e)$
to be the lattice $T^{\prime}$ with the quadratic form multiplied by $e$. For
example, with this notation, the sublattice $eT^{\prime}$ of $T^{\prime}$ is
isometric to $T^{\prime}(e^{2})$ as lattices. The image $E$ of the morphism
$\tilde{\phi}^{*}\colon H^{2}(S^{\prime},\mathbb{Z})_{tr}\to
H^{2}(\tilde{S},\mathbb{Z})_{tr}\cong T$, viewed as a sublattice of $T$, is
isometric to $T^{\prime}(\deg\phi)$. The isomorphism
$H^{2}(\tilde{S},\mathbb{Z})_{tr}\cong T$ is because $\tau^{*}\colon
H^{2}(S,\mathbb{Z})_{tr}\to H^{2}(\tilde{S},\mathbb{Z})_{tr}$ is an
isomorphism. We thus get the following equalities

$[T\colon
E]^{2}=\left|\frac{\mathrm{disc}(E)}{\mathrm{disc}(T)}\right|=(\deg\phi)^{21}\left|\frac{\mathrm{disc}(T^{\prime})}{\mathrm{disc}(T)}\right|=(\deg\phi)^{21}\frac{D^{\prime}-1}{D-1}.$
(3.5)

Here, we have used the fact that the lattice $H^{2}(S,\mathbb{Z})$ is
unimodular since $S$ is a surface, and the fact that the orthogonal complement
of a primitive sublattice in a unimodular lattice and the sublattice itself
have the same discriminant, up to sign ( [Nik80, Proposition 1.6.1]). On the
other hand, we have

######  Lemma 3.3.5.

The morphism of abelian groups $\tilde{\phi}_{*}\colon
H^{2}(\tilde{S},\mathbb{Z})_{tr}\to H^{2}(S^{\prime},\mathbb{Z})_{tr}$ is
injective and sends $E$ onto $(\deg\phi)T^{\prime}$.

###### Proof.

The projection formula shows that
$\tilde{\phi}_{*}\tilde{\phi}^{*}=\deg\phi\cdot Id$. Hence, $\tilde{\phi}_{*}$
sends $E$ onto $(\deg\phi)T^{\prime}$. By Lemma 3.3.7 and the fact that
$\tilde{\phi}_{*}$ is surjective with $\mathbb{Q}$-coefficients, the kernel of
$\tilde{\phi}_{*}$ is of torsion. But $H^{2}(\tilde{S},\mathbb{Z})_{tr}$ is
torsion-free, as $\tilde{S}$ is simply connected. We conclude that the kernel
of $\tilde{\phi}_{*}$ is zero, as desired. ∎

Now Lemma 3.3.5 implies that

$1\leq[T\colon E]\leq[T^{\prime}\colon(\deg\phi)T^{\prime}]=(\deg\phi)^{21}.$
(3.6)

Proposition 3.3.4 follows by combining (3.5) and (3.6). ∎

######  Remark 3.3.6.

One can similarly prove the more general result on hyper-Kähler manifolds,
namely Proposition 3.1.15. The detailed proof is given in Section 3.4. A
sharper lower bound will also be given there.

####  3.3.3 Proof of Theorem 3.1.14;

Let $f\colon S\dashrightarrow B=\mathbb{P}^{1}$ be a fibration into curves
realizing the fibering gonality of $S$. After a resolution of indeterminacies
of $f$, we get a dominant morphism $\tilde{f}\colon\tilde{S}\to B$ whose
general fiber is of gonality $d=\mathrm{Fibgon}(S)$. In Section 3.3.1, we
constructed a surface $S^{\prime}$ that is a fibration over $B$ into curves
and a dominant rational map $\phi\colon\tilde{S}\dashrightarrow S^{\prime}$
over $B$ of degree dividing $d$. The Kodaira dimension $\kappa(S^{\prime})$,
the irregularity $q(S^{\prime})$ and the geometric genus $p_{g}(S^{\prime})$
of $S^{\prime}$ cannot exceed those of $S$ since $S^{\prime}$ is dominated by
$S$. By Enriques-Kodaira classification of algebraic surfaces [Enr49,
Kod64-68], $S^{\prime}$ can only be birational to $\mathbb{P}^{2}$, an
Enriques surface or a K3 surface.

If $S^{\prime}$ is a rational surface, then $\mathrm{Irr}(S)\leq\deg\phi\leq
d=\mathrm{Fibgon}(S)$. Here, the inequality $\deg\phi\leq d$ is because of
Proposition 3.3.2. But clearly $\mathrm{Fibgon}(S)\leq\mathrm{Irr}(S)$. We get
the equality. This is case (a) of the theorem.

If $S^{\prime}$ is birational to an Enriques surface $S^{\prime\prime}$. After
a birational modification of $\tilde{S}$, there is a dominant morphism
$g\colon\tilde{S}\to S^{\prime\prime}$. Since $\tilde{S}$ is simply-connected,
$g$ factors through the universal covering $S^{\prime\prime\prime}$ of
$S^{\prime\prime}$. $S^{\prime\prime\prime}$ is the K3 cover of the Enriques
surface $S^{\prime\prime}$. The Picard number of $S^{\prime\prime\prime}$ is
at least $10$ since the Picard number of the Enriques surface
$S^{\prime\prime}$ is $10$.

######  Lemma 3.3.7.

Let $\phi\colon X\dashrightarrow X^{\prime}$ be a dominant rational map
between projective hyper-Kähler manifolds of the same dimension. Then
$b_{2,\mathrm{tr}}(X)=b_{2,\mathrm{tr}}(X^{\prime})$.

###### Proof.

Let $\tau\colon\tilde{X}\to X$ and $\tilde{\phi}\colon\tilde{X}\to X^{\prime}$
be a resolution of indeterminacy points of $\phi$. Then $\tau^{*}\colon
H^{2}(X,\mathbb{Q})_{tr}\to H^{2}(\tilde{X},\mathbb{Q})_{tr}$ is an
isomorphism and $\tilde{\phi}^{*}\colon H^{2}(X^{\prime},\mathbb{Q})_{tr}\to
H^{2}(\tilde{X},\mathbb{Q})_{tr}$ is injective. They are moreover both
morphisms of Hodge structures. But $H^{2}(X,\mathbb{Q})_{tr}$ and
$H^{2}(X^{\prime},\mathbb{Q})_{tr}$ are simple Hodge structures. This implies
that $\tilde{\phi}^{*}\colon H^{2}(X^{\prime},\mathbb{Q})_{tr}\to
H^{2}(\tilde{X},\mathbb{Q})_{tr}$ is an isomorphism and hence the result. ∎

Lemma 3.3.7 shows that the Picard number of $S^{\prime\prime\prime}$ can only
be $1$ since it is dominated by $S$. This gives us a contradiction. The case
where $S^{\prime}$ is birational to an Enriques surface is thus excluded.

In the rest of this section, we discuss the case when $S^{\prime}$ is
birational to a $K3$ surface. By changing the birational model, we may assume
$S^{\prime}$ _is_ a K3 surface. By Lemma 3.3.7, the Picard number of
$S^{\prime}$ is also $1$. The following proposition shows that in our
situation the Case (b) of Theorem 3.1.14 holds, which concludes the proof of
Theorem 3.1.14.

######  Proposition 3.3.8.

The following inequality holds:

$\mathrm{Fibgen}(S)\leq\mathrm{Fibgon}(S)^{21/2}.$

###### Proof.

Let $D$ and $D^{\prime}$ be the degrees of the $K3$ surfaces $S$ and
$S^{\prime}$, respectively. Let $C^{\prime}$ be the general fiber of
$S^{\prime}\dashrightarrow B$ as in Section 3.3.1. Then we have the following
inequalities

$\displaystyle\left(\frac{\mathrm{Fibgon}(S)}{\deg\phi}-1\right)^{2}$
$\displaystyle\geq p_{g}(C^{\prime})$ by Proposition 3.3.2
$\displaystyle\geq\sqrt{\frac{D^{\prime}}{2}}$ by Ein-Lazarsfeld’s theorem
(Theorem 3.1.11) $\displaystyle\geq\sqrt{\frac{D}{2(\deg\phi)^{21}}}$ by
Proposition 3.3.4
$\displaystyle\geq\frac{\mathrm{Fibgen}(S)}{4(\deg\phi)^{21/2}}$
$\displaystyle\textrm{ by Ein-Lazarsfeld's theorem
(Theorem~{}\ref{EinLazarsfeld})}.$

Note that $\mathrm{Fibgon}(S)\geq 2\deg\phi$. Proposition 3.3.8 follows from
these inequalities. ∎

###  3.4 Some inequalities about Picard lattices of hyper-Kähler manifolds

We prove in this section Propositions 3.1.15 and 3.1.16.

###### Proof of Proposition 3.1.15.

Let $\tau\colon\tilde{X}\to X$ and $\tilde{\phi}\colon\tilde{X}\to X^{\prime}$
be a resolution of indeterminacy points of $\phi\colon X\dashrightarrow
X^{\prime}$. Let $T$ (resp. $T^{\prime}$) be the lattice
$H^{2}(X,\mathbb{Z})_{tr}$ (resp. $H^{2}(X^{\prime},\mathbb{Z})_{tr}$) endowed
with the Beauville-Bogomolov-Fujiki form. As in the proof of Proposition
3.3.4, for a positive integer $e$, define $T^{\prime}(e)$ to be the lattice
$T^{\prime}$ with the quadratic form multiplied by $e$. We claim that the
image $E$ of the morphism $\tilde{\phi}^{*}\colon
H^{2}(X^{\prime},\mathbb{Z})_{tr}\to\tau^{*}H^{2}(X,\mathbb{Z})_{tr}\cong T$,
viewed as a sublattice of $T$, is isometric to
$T^{\prime}((\deg\phi)^{\frac{1}{n}})$. This follows from the equalities
$[q_{X}(\tilde{\phi}^{*}\alpha)]^{n}=c_{X}\cdot(\int_{X}\tilde{\phi}^{*}\alpha^{2n})=(\deg\phi)\cdot
c_{X}\cdot(\int_{X^{\prime}}\alpha^{2n})=(\deg\phi)\cdot[q_{X^{\prime}}(\alpha)]^{n}$,
where $c_{X}=c_{X^{\prime}}$ is the Fujiki constant for the deformation class
of $X$ and we have viewed $\tilde{\phi}^{*}\alpha$ as an element in
$H^{2}(X,\mathbb{Z})$ via the isomorphism
$\tau_{*}:\tau^{*}H^{2}(X,\mathbb{Z})_{tr}\to H^{2}(X,\mathbb{Z})_{tr}$. Now
the claim implies the following equalities

$[T\colon
E]^{2}=\left|\frac{\mathrm{disc}(E)}{\mathrm{disc}(T)}\right|=(\deg\phi)^{\frac{b_{2,\mathrm{tr}}(X)}{n}}\cdot\left|\frac{\mathrm{disc}(T^{\prime})}{\mathrm{disc}(T)}\right|=(\deg\phi)^{\frac{b_{2,\mathrm{tr}}(X)}{n}}\cdot\left|\frac{\mathrm{disc}(\mathrm{Pic}(X^{\prime}))}{\mathrm{disc}(\mathrm{Pic}(X))}\right|.$
(3.7)

On the other hand, with a similar argument to Lemma 3.3.5, we prove that the
morphism of abelian groups $\tilde{\phi}_{*}\colon H^{2}(X,\mathbb{Z})_{tr}\to
H^{2}(X^{\prime},\mathbb{Z})_{tr}$ is injective and sends $E$ to
$(\deg\phi)T^{\prime}$. Hence,

$1\leq[T\colon
E]\leq[T^{\prime}\colon(\deg\phi)T^{\prime}]=(\deg\phi)^{b_{2,\mathrm{tr}}(X)}.$
(3.8)

The proposition follows by combining (3.7) and (3.8). ∎

###### Proof of Proposition 3.1.16.

The only thing that needs proving, in the view of Proposition 3.1.15, is the
following inequality

$\frac{1}{(\deg\phi)^{\rho(X)}}\leq\left|\frac{\mathrm{disc}(\mathrm{Pic}(S))}{\mathrm{disc}(\mathrm{Pic}(S^{\prime}))}\right|.$

Let $\tau\colon\tilde{S}\to S$ and $\tilde{\phi}\colon\tilde{S}\to S^{\prime}$
be a resolution of indeterminacy points of $\phi\colon S\dashrightarrow
S^{\prime}$. Via the morphism
$\tilde{\phi}^{*}\colon\mathrm{Pic}(S^{\prime})\to\mathrm{Pic}(\tilde{S})$, we
can view $\tilde{\phi}^{*}\mathrm{Pic}(S^{\prime})$ as a sublattice of
$\mathrm{Pic}(\tilde{S})$. The sublattice
$\tilde{\phi}^{*}\mathrm{Pic}(S^{\prime})$ is isomorphic to
$\mathrm{Pic}(S^{\prime})(\deg\phi)$, since
$\tilde{\phi}^{*}\alpha\cup\tilde{\phi}^{*}\beta=\tilde{\phi}^{*}(\alpha\cup\beta)=\deg\phi\cdot(\alpha\cup\beta)$
for $\alpha,\beta\in\mathrm{Pic}(S^{\prime})$. Thus

$\mathrm{disc}(\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime})))=(\deg\phi)^{\rho(S)}\mathrm{disc}(\mathrm{Pic}(S^{\prime})).$
(3.9)

######  Lemma 3.4.1.

The sublattice
$\ker(\tilde{\phi}_{*}\colon\mathrm{Pic}(\tilde{S})\to\mathrm{Pic}(S^{\prime}))$
of $\mathrm{Pic}(\tilde{S})$ is the orthogonal complement of
$\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))$ in $\mathrm{Pic}(\tilde{S})$.

###### Proof.

Let $\alpha\in\mathrm{Pic}(\tilde{S})$. Let us show that
$\tilde{\phi}_{*}\alpha=0$ if and only if for any
$\beta\in\mathrm{Pic}(S^{\prime})$, we have
$\alpha\cup\tilde{\phi}^{*}\beta=0$ in $H^{4}(\tilde{S},\mathbb{Z})$. The
projection formula gives
$\tilde{\phi}_{*}(\alpha\cup\tilde{\phi}^{*}\beta)=(\tilde{\phi}_{*}\alpha)\cup\beta$
in $H^{4}(S^{\prime},\mathbb{Z})$. Hence, if $\tilde{\phi}_{*}\alpha=0$, then
$\tilde{\phi}_{*}(\alpha\cup\tilde{\phi}^{*}\beta)=0$. But
$\tilde{\phi}_{*}\colon H^{4}(\tilde{S},\mathbb{Z})\to
H^{4}(S^{\prime},\mathbb{Z})$ is an isomorphism, we must have
$\alpha\cup\tilde{\phi}^{*}\beta=0$. Conversely, if
$\alpha\cup\tilde{\phi}^{*}\beta=0$ for every
$\beta\in\mathrm{Pic}(S^{\prime})$, still by the projection formula, we get
$(\tilde{\phi}_{*}\alpha)\cup\beta=0$, which implies that
$\tilde{\phi}_{*}\alpha=0$ since the intersection product map is nondegenerate
on $\mathrm{Pic}(S^{\prime})$. ∎

Taking into account the fact that the intersection map on
$H^{2}(\tilde{S},\mathbb{Z})$ is nondegenerate on
$\tilde{\phi}^{*}\mathrm{Pic}(S^{\prime})$, Lemma 3.4.1 implies that
$\ker(\tilde{\phi}_{*})\oplus\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))$ is a
direct sum and that
$\ker(\tilde{\phi}_{*})\oplus\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))$ is of
_finite_ index in the abelian group $\mathrm{Pic}(\tilde{S})$. Hence,

$\left|\frac{\mathrm{disc}(\ker(\tilde{\phi}_{*}))\cdot\mathrm{disc}(\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))}{\mathrm{disc(\mathrm{Pic}(\tilde{S}))}}\right|=[\mathrm{Pic}(\tilde{S})\colon\ker(\tilde{\phi}_{*})\oplus\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))]^{2}.$
(3.10)

Since
$\tilde{\phi}_{*}\colon\mathrm{Pic}(\tilde{S})\to\mathrm{Pic}(S^{\prime}))$
sends $\ker(\tilde{\phi}_{*})\oplus\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))$
onto $(\deg\phi)\mathrm{Pic}(S^{\prime})$, and since the induced morphism

$\overline{\tilde{\phi}_{*}}\colon\mathrm{Pic}(\tilde{S})/\ker\tilde{\phi}_{*}\to\mathrm{Pic}(S^{\prime})$

is injective, we have

$\begin{split}[\mathrm{Pic}(\tilde{S}):(\ker(\tilde{\phi}_{*})\oplus\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime})))]&=[(\mathrm{Pic}(\tilde{S})/\ker\tilde{\phi}_{*}):\overline{\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))}]\\\
&\leq[\mathrm{Pic}(S^{\prime}):(\deg\phi)\mathrm{Pic}(S^{\prime})]\\\
&=(\deg\phi)^{\rho(S)},\end{split}$ (3.11)

where $\overline{\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))}$ is the image of
$\tilde{\phi}^{*}(\mathrm{Pic}(S^{\prime}))$ in
$\mathrm{Pic}(\tilde{S})/\ker\tilde{\phi}_{*}$. Notice the following Lemma.

######  Lemma 3.4.2.

$|\mathrm{disc}(\mathrm{Pic}(\tilde{S}))|=|\mathrm{disc}(\mathrm{Pic}(S))|$.

###### Proof.

$\tilde{S}$ is obtained by a sequence of blowing-ups of points from $S$.
Therefore,

$\mathrm{Pic}(\tilde{S})=\tau^{*}\mathrm{Pic}(S)\oplus\bigoplus_{i}\mathbb{Z}E_{i},$

where $E_{i}$ is the _total transform_ in $\tilde{S}$ of the exceptional
divisor of the $i$-th blowing-up. We have the following formula for the
intersection numbers of $E_{i}$:

$E_{i}.E_{j}=\left\\{\begin{array}[]{cc}-1&\textrm{ if }i=j\\\ 0&\textrm{
otherwise. }\end{array}\right.$

Hence,
$\mathrm{disc}(\mathrm{Pic}(\tilde{S}))=\mathrm{disc}(\mathrm{Pic}(S))\cdot\mathrm{disc}(\oplus_{i}\mathbb{Z}E_{i})=\pm\mathrm{disc}(\mathrm{Pic}(S))$,
as desired. ∎

The proposition now follows from Lemma 3.4.2 and from inequalities (3.9),
(3.10) and (3.11), noticing that $\mathrm{disc}(\ker(\tilde{\phi}_{*}))\geq
1$. ∎

###  3.5 Related results

The study of rational maps between K3 surfaces is an intriguing question. One
of the circulated expectations in the area is the following question.

######  Problem 3.5.1.

Let $S$ be a very general projective K3 surface. Let $\phi:S\dashrightarrow
S^{\prime}$ be a dominant rational map to another K3 surface $S^{\prime}$. Is
that true that $\phi$ can only be an isomorphism (i.e. the degree of $\phi$ is
$1$)?

As is clear from the proof of Theorem 3.1.14, if Problem 3.5.1 has a positive
answer, then the Case (b) in Theorem 3.1.14 can be improved to be
$\mathrm{Fibgon}(S)^{2}\geq\mathrm{Fibgen}(S)$. We do not expect a purely
lattice theoretic answer.

We can also study self-rational maps of hyper-Kähler manifolds, the higher
dimensional generalization of K3 surfaces. As we can see from the following
theorem, self-rational maps of hyper-Kähler manifolds has very restrictive
numeric properties.

######  Theorem 3.5.2.

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$ such that
$b_{2,tr}(X)$ is odd. Let $\sigma_{X}$ be its holomorphic symplectic form. Let
$\phi:X\dashrightarrow X$ be a dominant self-rational map. Then the degree of
$\phi$ is of the form $k^{2n}$ where $k$ is an integer. In this case,
$\phi^{*}\sigma_{X}=\pm k\sigma_{X}$.

###### Proof.

Let $\tau:\tilde{X}\to X$ and $\tilde{\phi}:\tilde{X}\to X$ be a resolution of
indeterminacies of $\phi:X\dashrightarrow X$. Let $T$ be the lattice
$H^{2}(X,\mathbb{Z})_{tr}$ endowed with the Beauville–Bogomolov–Fujiki form
$q_{X}$. For a positive integer $e$, define $T(e)$ to be the lattice $T$ with
the quadratic form multiplied by $e$. We claim that the image $E$ of the
morphism
$\tilde{\phi}^{*}:H^{2}(X,\mathbb{Z})_{tr}\to\tau^{*}H^{2}(X,\mathbb{Z})_{tr}\cong
T$, viewed as a sublattice of $T$, is isometric to
$T((\deg\phi)^{\frac{1}{n}})$. This follows from the equalities
$[q_{X}(\tilde{\phi}^{*}\alpha)]^{n}=c_{X}\cdot(\int_{X}\tilde{\phi}^{*}\alpha^{2n})=(\deg\phi)\cdot
c_{X}\cdot(\int_{X}\alpha^{2n})=(\deg\phi)\cdot[q_{X}(\alpha)]^{n}$, where
$c_{X}$ is the Fujiki constant for the deformation class of $X$ and we have
viewed $\tilde{\phi}^{*}\alpha$ as an element in $H^{2}(X,\mathbb{Z})$ via the
isomorphism $\tau_{*}:\tau^{*}H^{2}(X,\mathbb{Z})_{tr}\to
H^{2}(X,\mathbb{Z})_{tr}$.. The claim already implies that $\deg\phi$ is of
the form $m^{n}$ with $m$ an integer. Let us show that $m$ is a perfect
square. Note that the claim also implies the following equalities

$[T:E]^{2}=\left|\frac{\mathrm{disc}(E)}{\mathrm{disc}(T)}\right|=(\deg\phi)^{\frac{b_{2,tr}(X)}{n}}\cdot\left|\frac{\mathrm{disc}(T)}{\mathrm{disc}(T)}\right|=(\deg\phi)^{\frac{b_{2,tr}(X)}{n}}=m^{b_{2,tr}(X)}.$
(3.12)

Since $b_{2,tr}(X)$ is assumed to be odd, comparing the two sides of the
equality shows that $m$ is a perfect square. Hence, the degree of $\phi$ is of
the form $k^{2n}$.

Therefore, for each $\alpha\in H^{2}(X,\mathbb{Z})_{tr}$, we have
$q_{X}(\tilde{\phi}^{*}\alpha)=k^{2}q_{X}(\alpha)$. Therefore, the morphism
$\psi:=\frac{1}{k}\phi^{*}:H^{2}(X,\mathbb{Q})_{tr}\to
H^{2}(X,\mathbb{Q})_{tr}$ is an Hodge isometry. Since $b_{2,tr}(X)$ is assumed
to be odd, $\psi$ has an eigenvector $v$ with $\pm 1$ as its eigenvalue. But
now the fact that $H^{2}(X,\mathbb{Q})_{tr}$ is a _simple_ Hodge structure
implies that $\psi=\pm Id$. Theorem 3.5.2 then follows. ∎

######  Example 3.5.3.

(i) Assume that $X$ admits a Lagrangian fibration $f:X\to B$ with a (rational)
zero section. Let $x\in X$ be a general point of $X$ and $b=f(x)\in B$. The
fact that the general fiber of the fibration is Lagrangian implies that

$\sigma_{X,x}=\alpha\wedge f^{*}\beta,$

where $\alpha\in\Omega_{X_{b},x}$ and $\beta\in\Omega_{B,b}$. Since the
general fiber $X_{b}$ is an abelian variety of dimension $n$, the self map
$m_{k,b}:X_{b}\to X_{b}$ defined as the multiplication by a natural number $k$
is of degree $k^{2n}$, and $m_{k,b}^{*}\alpha=k\alpha$. The existence the
global zero section of this Lagrangian fibration implies that
$m_{k,b}:X_{b}\to X_{b}$ can be defined globally as a self-rational map
$m_{k}:X\dashrightarrow X$. The degree of $m_{k}$ is $k^{2n}$ and
$m_{k}^{*}\sigma_{X}=k\sigma_{X}$.
(ii) Let $X$ be the Fano variety of lines of a cubic fourfold $Y$. The
following self-rational map $\phi:X\dashrightarrow X$, called the Voisin map,
is constructed in [Voi04] as follows. Let $\ell\in X$ be a general line in
$Y$. Then there is a unique plane in $\mathbb{P}^{5}$ that passes $\ell$
twice. The intersection of $Y$ and this plane is of the form
$2\ell+\ell^{\prime}$ where $\ell^{\prime}$ is another line in $Y$. Then
$\phi(\ell)$ is defined to be $\ell^{\prime}$. As is shown in [Voi04],
$\phi:X\dashrightarrow X$ is of degree $16=2^{4}$ and
$\phi^{*}\sigma_{X}=-2\sigma_{X}$.

## Chapter 4 On the Geometry of the Higer Dimension Voisin Maps

Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as
varieties of $r$-planes in cubic hypersurfaces of adequate dimension. This map
has been thoroughly studied in the case $r=1$, which is the Beauville-Donagi
case. In this chapter, we compute the action of $\Psi$ on holomorphic forms
for any $r$. For $r=2$, we compute the action of $\Psi$ on the Chow group of
$0$-cycles, and confirm that it is as expected from the generalized Bloch
conjecture.

###  4.1 Introduction

Let $X$ be a smooth projective variety of dimension $n$ defined over the field
of complex numbers $\mathbb{C}$. The Chow group of $k$-cycles of $X$, denoted
as $CH_{k}(X)$, is the quotient by the rational equivalence of the free
abelian group generated by closed irreducible subvarieties of dimension $k$.
In this work, we will ignore the complexities introduced by the torsion part
of the Chow groups, focusing instead on $CH_{k}(X)_{\mathbb{Q}}$, which is the
Chow group after tensoring with $\mathbb{Q}$.

Hodge theory emerges as a pivotal framework for investigating complex smooth
projective varieties. The interaction between Hodge structures and the Chow
groups is both enlightening and enigmatic. The cycle class map
$cl:CH_{k}(X)_{\mathbb{Q}}\to H^{2n-2k}(X,\mathbb{Q})$ plays a central role in
this interplay. A plethora of conjectures have been proposed, relating the
complexity of Chow groups and that of Hodge structures. Among these, the Hodge
conjecture stands out predicting that the image of $cl$ coincides with the set
of Hodge classes in $H^{2n-2k}(X,\mathbb{Q})$.

This chapter is primarily motivated by a conjecture put forth by Voisin, which
we will explore in detail:

######  Conjecture 4.1.1 (Voisin).

Let $X$ be a strict Calabi-Yau manifold of dimension $n$. Let $C\subset
CH_{0}(X)_{\mathbb{Q}}$ be the subgroup generated by the intersections of
divisors and of Chern classes of $X$. Then the cycle class map

$cl:CH_{0}(X)_{\mathbb{Q}}\to H^{2n}(X,\mathbb{Q})$

is injective on $C$.

Subsequent sections will delve into the terminology, underlying motivations,
and the conceptual backdrop of Conjecture 4.1.1, setting the stage for a
comprehensive exploration of its implications and relevance to the broader
field of algebraic geometry.

####  4.1.1 Strict Calabi-Yau manifolds

######  Definition 4.1.2.

A _strict Calabi-Yau manifold_ is a simply connected compact Kähler manifold
$X$ of dimension at least $3$, with trivial canonical bundle and such that
$H^{0}(X,\Omega_{X}^{k})=0$ for $0<k<{\rm dim}\,X$.

######  Remark 4.1.3.

It follows from Kodaira’s embedding theorem that such $X$ is automatically
projective.

Strict Calabi-Yau manifolds are pivotal in the study of algebraic varieties
with trivial canonical bundles, known as $K$-trivial varieties. This class
also includes hyper-Kähler manifolds, which are simply connected manifolds and
possess a non-degenerate holomorphic $2$-form, for which Conjecture 4.1.1 is
also expected to be true when they are projective (this is the Beauville-
Voisin conjecture [Bea07, Voi08]), and complex tori, which are defined as
quotients of $\mathbb{C}^{n}$ by a lattice $\Gamma$, for which Conjecture
4.1.1 is definitely wrong, when they are projective.

##### Voisin’s examples of Calabi-Yau manifolds

We focus on specific families of $K$-trivial varieties as constructed in
[Voi04], generalizing the Beauville-Donagi construction [BD85]. Let
$Y\subset\mathbb{P}^{n}$ be a smooth cubic hypersurface of dimension $n-1$,
and let $r\geq 0$ denote a nonnegative integer. Define $X=F_{r}(Y)$ as the
Hilbert scheme that parametrizes the $r$-dimensional linear subspaces in $Y$.
As proved in [Voi04, (4.41)], for $n+1=\binom{r+3}{2}$ and a general $Y$, the
variety $X$ is a $K$-trivial variety of dimension
$N=(r+1)(n-r)-\binom{r+3}{3}$. Specifically, when $r=0$, $X$ is an elliptic
curve; and as established in [BD85], $X$ is a hyper-Kähler manifold for $r=1$.
We have the following

######  Lemma 4.1.4.

For $r\geq 2$ and $n+1=\binom{r+3}{2}$, the constructed variety $X=F_{r}(Y)$
is a strict Calabi-Yau manifold.

###### Proof.

Reference [DM98, Proposition 3.1(a)] indicates that, generally, if
$n\geq\frac{2}{r+1}\binom{r+3}{r}+r+1$, then $F_{r}(Y)$ is simply connected.
This inequality is consistently satisfied for $r\geq 1$ when
$n+1=\binom{r+3}{2}$. Consequently, $X$ is simply connected in our context.
The Beauville-Bogomolov decomposition theorem [Bea83, Théorème 1] gives a
decomposition of $X\cong T\times W\times CY$, where $T$ is a complex torus,
$W$ a product of hyper-Kähler manifolds, and $CY$ a product of strict Calabi-
Yau manifolds. As shown in [DM98, Théorème 3.4], the restriction morphism
$H^{i}(\mathrm{Gr}(r+1,n+1),\mathbb{Q})\to H^{i}(F_{r}(Y),\mathbb{Q})$ is an
isomorphism for $i<\min\\{\dim F_{r}(Y),n-2r-1\\}$. Under the condition
$n+1=\binom{r+3}{2}$, $2<\min\\{\dim F_{r}(Y),n-2r-1\\}$ for any $r\geq 2$.
Thus, $H^{2,0}(X)=H^{2,0}(\mathrm{Gr}(r+1,n+1))=0$, and the Picard number of
$X$ is $1$ for $r\geq 2$. Hence, in the Beauville-Bogomolov decomposition of
$X$, only the strict Calabi-Yau manifold component remains, and it is
irreducible since the Picard number is $1$. ∎

####  4.1.2 Conjectures on Chow groups

We begin by examining some foundational conjectures related to the Chow groups
of smooth projective varieties. This discussion is then extended to include
conjectures specifically concerning the Chow groups of projective hyper-Kähler
and strict Calabi-Yau manifolds. These sections provide the necessary
background and motivation for Conjecture 4.1.1, with our analysis closely
following the insights and frameworks presented in [Voi04II], [Voi14], and
[Voi16].

##### Bloch-Beilinson filtration and the generalized Bloch conjecture

Bloch and Beilinson have conjectured the existence of a descending filtration,
denoted as $F^{i}CH_{k}(X)_{\mathbb{Q}}$, on the Chow groups with rational
coefficients for any smooth complex projective variety $X$. This filtration is
conjectured to satisfy a set of axioms.

######  Conjecture 4.1.5 (Bloch-Beilinson Conjecture [Voi04II]).

For every smooth projective variety $X$, there exists a descending filtration
$F^{\bullet}$ on $CH^{i}(X)_{\mathbb{Q}}$ characterized by:

  * (i)

(Non-Triviality) $CH^{i}(X)_{\mathbb{Q}}=F^{0}CH^{i}(X)_{\mathbb{Q}}$, and
$F^{1}CH^{i}(X)\mathbb{Q}=CH^{i}(X)_{\mathbb{Q},hom}$.

  * (ii)

(Functoriality) For a cycle $Z$ in $CH^{k}(X\times Y)_{\mathbb{Q}}$, the
pushforward of $F^{i}CH^{l}(X)_{\mathbb{Q}}$ by $Z_{*}$ is included within
$F^{i}CH^{l+k-n}(Y)_{\mathbb{Q}}$, where $n$ represents the dimension of $X$.

  * (iii)

(Graded Component) If $[Z]=0$ in $H^{2k}(X\times Y,\mathbb{Q})$, the induced
map $Z_{*}:Gr_{F}^{i}CH^{l}(X)_{\mathbb{Q}}\to
Gr_{F}^{i}CH^{l+k-n}(Y)_{\mathbb{Q}}$ vanishes for any $i$.

  * (iv)

(Finiteness) The filtration terminates with $F^{k+1}CH^{k}(X)_{\mathbb{Q}}=0$
for all varieties $X$ and integers $k$.

The conjecture further suggests that the filtration of $CH_{0}(X)$ is
intricately linked to the Hodge structures modulo Hodge substructures of
coniveau $\geq 1$, that is, to holomorphic forms.

######  Conjecture 4.1.6 (Generalized Bloch Conjecture for $0$-Cycles).

Given a correspondence $Z\in CH^{n}(X\times Y)_{\mathbb{Q}}$ between smooth
projective varieties $X$ and $Y$, both of dimension $n$, if the map
$[Z]^{*}:H^{i,0}(Y)\to H^{i,0}(X)$ vanishes for some $i\leq n$, then the
pushforward $Z_{*}:Gr_{F}^{i}CH_{0}(X)_{\mathbb{Q}}\to
Gr_{F}^{i}CH_{m-k}(Y)_{\mathbb{Q}}$ also vanishes for that $i$. Here,
$F^{\bullet}$ represents the Bloch-Beilinson filtration and $Gr_{F}^{\bullet}$
its graded part.

This conjecture is named the generalized Bloch conjecture as it extends the
classical Bloch conjecture, stated as follows in [Blo80].

######  Conjecture 4.1.7 (Bloch [Blo80]).

For a correspondence $Z\in CH^{2}(S\times T)_{\mathbb{Q}}$ between surfaces
that induces a null map $[Z]^{*}:H^{2,0}(T)\to H^{2,0}(S)$, the induced
morphism $Z_{*}:F^{2}CH_{0}(S)\to F^{2}CH_{0}(T)$ is identically zero. Here,
$F^{2}CH_{0}(S)$ is defined as the kernel of the Albanese map from
$CH_{0}(S)_{hom}$ to $Alb(S)$, and similarly for $F^{2}CH_{0}(T)$.

##### Conjectures of Beauville and Voisin

Let $X$ be a projective hyper-Kähler manifold. Beauville’s splitting
conjecture, as introduced in [Bea07], suggests that the Bloch-Beilinson
filtration on the Chow ring of $X$ undergoes a natural multiplicative
splitting. One weak version is now often referred to as the "weak splitting
conjecture," detailed in the introduction of [Voi16].

######  Conjecture 4.1.8 (Beauville’s Weak Splitting Conjecture [Bea07]).

For a projective hyper-Kähler manifold $X$, the cycle class map is injective
on the subalgebra of $CH^{*}(X)$ that is generated by divisors.

This conjecture was further expanded by Voisin in [Voi08] to include not only
divisors but also Chern classes into the generating elements of the
subalgebra.

######  Conjecture 4.1.9 (Voisin [Voi08]).

In the case of a projective hyper-Kähler manifold $X$, consider $C^{*}$ to be
the subalgebra of $CH^{*}(X)$ generated by divisors and Chern classes. The
cycle class map is injective on $C^{*}$.

However, generalizing these conjectures to strict Calabi-Yau manifolds often
leads to contradictions. Following [Bea07, Examples 1.7], denote by $Y$ the
blow-up of $\mathbb{P}^{3}$ along a smooth curve of genus 2 and degree 5. Take
a smooth divisor $D$ in $|-2K_{Y}|$ and let $X$ be the double covering of $Y$
ramified along $D$, $X$ is shown to be a strict Calabi-Yau threefold. However,
the cycle class map $cl:CH_{1}(X)_{\mathbb{Q}}\to H^{4}(X,\mathbb{Q})$ fails
to be injective on the subgroup generated by intersections of divisors,
invaliding the “strict Calabi-Yau version" of Conjectures 4.1.8 and 4.1.9.

Despite these counterexamples concerning the “strict Calabi-Yau version" of
Conjecture 4.1.9 for $1$-cycles, it remains anticipated that the conjecture
holds true for $0$-cycles. This expectation takes the form of Conjecture
4.1.1, with compelling evidence provided by the constructions in [Baz17,
Theorem 1.2].

######  Theorem 4.1.10 (Bazhov [Baz17]).

Let $Y$ be a projective homogeneous variety of dimension $n+1\geq 4$, and let
$X$ be a general element of the anti-canonical system $|-K_{Y}|$. Then $X$ is
a strict Calabi-Yau manifold satisfying Conjecture 4.1.1.

####  4.1.3 Voisin maps

The distinct feature of the manifolds $X=F_{r}(Y)$ of Section 4.1.1 among all
$K$-trivial manifolds revolves around the presence of a self-rational map,
$\Psi:X\dashrightarrow X$, referred to as the Voisin map. This map was
introduced in [Voi04] through the following construction: Consider a general
point $x\in X$, representing an $r$-dimensional linear space $P_{x}$ within
$Y$. As demonstrated in [Voi04, Lemma 8], there exists a unique
$(r+1)$-dimensional linear subspace $H_{x}$ in $\mathbb{P}^{n}$ tangent to $Y$
along $P_{x}$. The intersection $H_{x}\cap Y$ is a cubic hypersurface
containing $P_{x}$ doubly, leaving a residual $r$-dimensional linear subspace
in $Y$ represented by a point $x^{\prime}\in X$. This process defines the
Voisin map as $\Psi(x)=x^{\prime}$.

The result from [AC08, Corollaire 2.2] indicates that for $r\geq 2$, the
Voisin map does not preserve any non-trivial fibrations. Given this, exploring
the dynamics of the Voisin map becomes a compelling avenue of study, although
it falls outside the scope of this work. Notably, investigating the dynamics
of the Voisin map in the hyper-Kähler case ($r=1$) has yielded significant
insights into the geometry of $X$, as reported in [Voi04, AV08, ABR11].

Our first result is a computation of two basic invariants of the map $\Psi$.

######  Theorem A.

Let $X=F_{r}(Y),\,r\geq 0$ be as in 4.1.1, and let $\Psi:X\dashrightarrow X$
be the Voisin map. Then

  * (i)

For any $\omega\in H^{0}(X,K_{X})$, we have

$\Psi^{*}\omega=(-2)^{r+1}\omega.$

  * (ii)

The map $\Psi:X\dashrightarrow X$ is of degree $4^{r+1}$.

######  Remark 4.1.11.

Theorem A is trivial for $r=0$. For $r=1$, the results have been previously
established in [Voi04].

Having Theorem A, Conjecture 4.1.6 leads us to the following

######  Conjecture 4.1.12.

Let $X=F_{r}(Y)$ with $r\geq 2$. Then for any $z\in CH_{0}(X)_{hom}$,

$\Psi_{*}z=(-2)^{r+1}z.$

Indeed, let us explain how Conjecture 4.1.6 implies Conjecture 4.1.12. Let
$N=\dim X$. As $X$ is a strict Calabi-Yau manifold, we have $H^{k,0}(X)=0$ for
all $0<k<N$. Consider $\Delta_{X}\in CH^{n}(X\times X)$ as the diagonal of
$X\times X$. Then, for $0<k<N$, $[\Delta_{X}]^{*}|_{H^{k,0}(X)}=0$. Conjecture
4.1.6 implies that $\Delta_{X*}|_{Gr^{k}_{F}CH_{0}(X)}=0$ for $0<k<N$, leading
to $Gr^{k}_{F}CH_{0}(X)=0$ for these values of $k$. Thus, the Bloch-Beilinson
filtration on $CH_{0}(X)$ simplifies to:

$0=F^{N+1}CH_{0}(X)\subseteq
F^{N}CH_{0}(X)=\ldots=F^{1}CH_{0}(X)=CH_{0}(X)_{hom}\subseteq
F^{0}CH_{0}(X)=CH_{0}(X).$

Now, define $Z=\Gamma_{\Psi}-(-2)^{r+1}\Delta_{X}$. Given Theorem A,
$[Z]^{*}|_{H^{N,0}(X)}=0$. Applying Conjecture 4.1.6 once more, we find
$Z_{*}|_{Gr^{N}_{F}CH_{0}(X)}=0$, confirming that $Z_{*}$ is null on
$Gr^{N}_{F}CH_{0}(X)=F^{N}CH_{0}(X)=F^{1}CH_{0}(X)=CH_{0}(X)_{hom}$, yielding
the desired conclusion.

Our second main result is the proof of Conjecture 4.1.12 when $r=2$

######  Theorem B.

Let $Y\subset\mathbb{P}^{9}$ be a general cubic $8$-fold, and let $X=F_{2}(Y)$
be the Fano variety of planes in $Y$. Let $\Psi:X\dashrightarrow X$ be the
Voisin map. Then for any $z\in CH_{0}(X)_{hom}$:

$\Psi_{*}z=-8z.$

In the proof of Theorem B, the notion of constant-cycle subvarieties, as
described in [Huy14, Voi16], plays a pivotal role. Let us revisit the
definition for clarity.

######  Definition 4.1.13 ([Huy14, Voi16]).

Let $X$ be a smooth algebraic variety. A closed subvariety $j:Z\hookrightarrow
X$ is a _constant-cycle subvariety_ if every two points $z_{1},z_{2}\in Z$ are
rationally equivalent in $X$. Equivalently, the image of the morphism,
$j_{*}:CH_{0}(Z)\to CH_{0}(X)$, is $\mathbb{Z}$.

Let $F$ denote the closure of the fixed locus under the Voisin map
$\Psi:X\dashrightarrow X$. We next observe that Conjecture 4.1.12 immediately
leads us to the following

######  Conjecture 4.1.14.

The variety $F\subset X$ is a constant-cycle subvariety for $r\geq 2$.

Let us explain how Conjecture 4.1.14 is implied by Conjecture 4.1.12. Consider
two points $x,y\in F$. Since the rational equivalence class is a countable
union of closed algebraic subsets [Voi14, Section 1.1.1], we may assume that
$x,y$ are general. Assuming $x,y$ are within the fixed locus of $\Psi$, we
have $\Psi(x)=x$ and $\Psi(y)=y$, leading to $\Psi_{*}(x-y)=x-y$. However,
Proposition 4.1.12 indicates $\Psi_{*}(x-y)=(-2)^{r}(x-y)$ under Conjecture
4.1.6. Consequently, $x-y=0\in CH_{0}(X)_{\mathbb{Q}}$. Roitman’s theorem
[Roi80] implies $CH_{0}(X)_{hom}$ is torsion-free due to the triviality of
$Alb(X)$, so it follows that $x=y\in CH_{0}(X)$. Thus, $F\subset X$ forms a
constant-cycle subvariety.

In the present paper, we will prove directly Conjecture 4.1.14 for $r=2$, and
this will be one step in our proof of Theorem B, that is, Conjecture 4.1.12
for $r=2$.

######  Remark 4.1.15.

Interestingly, even in the scenario of $r=1$—not addressed in Conjecture
4.1.14—it has been proved [Voi08] that $F\subset X$ is a constant-cycle
subvariety.

Now, let us examine the indeterminacy locus of the Voisin map
$\Psi:X\dashrightarrow X$. This locus comprises two components, described as
follows. The first component, $\mathrm{Ind}_{0}$, contains points $x\in X$
that represent $P_{x}\subset Y$, such that there is more than one linear
subspace of dimension $r+1$ tangent to $Y$ along $P_{x}$. The second
component, $\mathrm{Ind}_{1}$, includes points $x\in X$ representing
$P_{x}\subset Y$, where an $(r+1)$-dimensional linear space exists that
contains $P_{x}$ and is contained within $Y$. It is not hard to prove
(Proposition 4.4.4 and Proposition 4.4.5) that $\mathrm{Ind}_{0}$ has
codimension $2$, while $\mathrm{Ind}_{1}$ has codimension $r+2$ for $r\geq 2$
(and $\mathrm{Ind}_{1}$ is empty for $r=1$). We prove as a consequence of
Theorem B the following result

######  Theorem C.

Assume $r=2$. Then any $0$-cycle of $X$ which is a polynomial in the Chern
classes of $X$ and divisor classes is rationally equivalent to a cycle
supported on Ind.

######  Corollary 4.1.16.

If $\mathrm{Ind}\subset X$ is a constant cycle subvariety, Conjecture 4.1.1
holds true for $X=F_{2}(Y)$.

This leaves us with an open

######  Question 4.1.17.

Is $\mathrm{Ind}\subset X$ a constant cycle subvariety for $r\geq 2$?

######  Remark 4.1.18.

In the hyper-Kähler case ($r=1$), the indeterminacy locus $\mathrm{Ind}_{0}$
is _not_ a constant-cycle subvariety, nor is it Lagrangian, as indicated by
[Ame09, Lemma 2] and [Voi04]. However, our understanding of why
$\mathrm{Ind}_{0}$ is not a constant-cycle subvariety in this scenario does
not contradict the evidence - that we will present in Section 4.4.3 \- for the
positive answer to Question 4.1.17.

####  4.1.4 Notations and supplementary results

For the continuity and coherence of this chapter, we establish some notations
that will be used throughout, unless specified otherwise:

  * $Y$

denotes a cubic hypersurface, which is typically smooth.

  * $X$

refers to a strict Calabi-Yau or hyper-Kähler manifold. Concrete examples of
strict Calabi-Yau and hyper-Kähler manifolds in this chapter are constructed
as the Fano variety of $r$-linear spaces in a smooth cubic hypersurface $Y$.
The dimension of $X$ is $(r+1)(n-r)-\binom{r+3}{3}$ and is denoted as $N$.

  * $P_{x}$

denotes the $r$-dimensional linear subspace in $Y$ (resp. $\mathbb{P}^{n}$)
for $x\in X$ (resp. $x\in\mathrm{Gr}(r+1,n+1)$).

  * $\Psi$

: $X\dashrightarrow X$ denotes the Voisin map.

  * $F$

represents the closure of the fixed locus of the Voisin map
$\Psi:X\dashrightarrow X$.

Throughout the chapter, we establish several auxiliary results that not only
support the proof of the main theorems but also hold intrinsic interest. Here
is a summary of these findings:

  * •

(Referencing Theorem 4.3.26 and Theorem 4.3.39): For $X=F_{2}(Y)$, the Fano
variety of planes in a general cubic eightfold, it is established that
$F\subset X$ is a constant cycle subvariety with a Chow class of
$-404c_{1}^{3}+110c_{1}c_{2}+49c_{3}$.

  * •

(Referencing Theorem 4.3.6): For a general cubic eightfold $Y$, it is
demonstrated that $CH_{1}(F_{1}(Y))_{hom,\mathbb{Q}}=0$.

  * •

(Referencing Theorem 4.4.10): For the strict Calabi-Yau manifolds
$X=F_{r}(Y)$, with $r\geq 2$, as presented in Section 4.1.1, the map
$F_{r}:\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))/PSL_{n+1}(\mathbb{C})\dashrightarrow\mathrm{Def}(X)$
is a local isomorphism.

###  4.2 Action of the Voisin map on top degree holomorphic forms

In this section, we compute the action $\Psi^{*}:H^{0}(X,K_{X})\rightarrow
H^{0}(X,K_{X})$ of the Voisin map and its degree, proving Theorem A.

######  Remark 4.2.1.

For $r=0$, where $Y$ is a plane cubic curve and $X=F_{0}(Y)=Y$ denotes an
elliptic curve, $\Psi:X\to X$ acts by mapping $x$ to $-2x$, in accordance with
the addition law of the elliptic curve. Consequently, the degree of $\Psi:X\to
X$ is $4$. In the scenario where $r=1$ and $X$ is a hyper-Kähler fourfold, the
result $\deg\Psi=16$ is first discovered by Voisin [Voi04], and can be derived
using either Chow-theoretic techniques [Voi04], [AV08, Corollary 1.7] or
vector bundle methods [Huy24, Lemma 4.12 and Proposition 4.17].

Let us first note the following

######  Lemma 4.2.2.

The two assertions in Theorem A are equivalent, up to sign.

###### Proof.

(i) leads to (ii) as follows: Given that $\sigma\wedge\bar{\sigma}$
constitutes a volume form on $X$, it follows that

$\deg\Psi\int_{X}\sigma\wedge\bar{\sigma}=\int_{X}\Psi^{*}\sigma\wedge\Psi^{*}\bar{\sigma}=\int_{X}(-2)^{r+1}\sigma\wedge(-2)^{r+1}\bar{\sigma}=4^{r+1}\int_{X}\sigma\wedge\bar{\sigma}.$

Conversely, let us demonstrate that (ii) implies $\Psi^{*}\sigma=\pm
2^{r+1}\sigma$, aligning closely with (i). The rational map can be
consistently defined across the family of cubic hypersurfaces, implying the
degree of $\Psi$ remains invariant across different choices of the generic
cubic hypersurface $Y$. Here, “generic” means that the cubic hypersurface is
chosen outside a proper Zariski closed subset. Therefore, we may presume $Y$
is defined over $\mathbb{Q}$. Consequently, the rational map
$\Psi:X\dashrightarrow X$ is also defined over $\mathbb{Q}$. Thus,
$\Psi^{*}:H^{0}(X,K_{X/\mathbb{Q}})\to H^{0}(X,K_{X/\mathbb{Q}})$ operates by
multiplication by a rational number, given that $H^{0}(X,K_{X/\mathbb{Q}})$ is
a one-dimensional $\mathbb{Q}$-vector space. If $\Psi^{*}\sigma=\lambda\sigma$
with $\lambda\in\mathbb{Q}$, then $\lambda^{2}=4^{r+1}$, leading to
$\lambda=\pm 2^{r+1}$. ∎

####  4.2.1 Proof of Theorem A (i)

Let $\mathrm{Fix}(\Psi):=\\{x\in X:\Psi\textrm{ is defined at }x\textrm{ and
}\Psi(x)=x\\}$ be the fixed locus of $\Psi:X\dashrightarrow X$.

######  Proposition 4.2.3.

For $Y$ general, the fixed locus $\mathrm{Fix}(\Psi)$ is not empty and is of
codimension $r+1$ in $X$.

###### Proof.

We will suppose that $r\geq 2$ since the case $r=1$ is shown in [ABR11,
Proposition 3.1] already (see also [GK20, Corollary 3.13]). Let
$B=\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}(3))$ be the parametrizing space
of cubic hypersurfaces. Let
$\mathrm{Fl}=\\{(t,s)\in\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+2,n+1):P_{t}\subset\Pi_{s}\\}$
be the flag variety of pairs of linear subspaces in $\mathbb{P}^{n}$. Let

$\mathcal{I}=\\{(f,t,s)\in B\times\mathrm{Fl}:\Pi_{s}\textrm{ is the only
}\mathbb{P}^{r+1}\textrm{ such that }Y_{f}\cap\Pi_{s}=3P_{t}\\},$
$\tilde{\mathcal{I}}=\\{(f,t,s)\in B\times\mathrm{Fl}:Y_{f}\cap\Pi_{s}\supset
3P_{t}\\}.$

Let $p:\mathcal{I}\to B$ (resp. $\tilde{p}:\tilde{\mathcal{I}}\to B$) and
$q:\mathcal{I}\to\mathrm{Fl}$ (resp.
$\tilde{q}:\tilde{\mathcal{I}}\to\mathrm{Fl}$) be the canonical projection
maps. By definition of $\Psi$, for any $f\in B$, the fiber
$p^{-1}(f)\subset\mathcal{I}$ coincides with the fixed locus of $\Psi$ for the
cubic hypersurface $Y_{f}$. Hence, it suffices to show that the map
$p:\mathcal{I}\to B$ is dominant and that the dimension of the general fibers
is $\dim X-r-1$.

######  Lemma 4.2.4.

The variety $\mathcal{I}$ is open dense in $\tilde{\mathcal{I}}$. The
dimension of $\mathcal{I}$ is $\dim X+\dim
H^{0}(\mathbb{P}^{n},\mathcal{O}(3))-r-2$.

###### Proof.

Let us consider the fibers of the map
$\tilde{q}:\tilde{\mathcal{I}}\to\mathrm{Fl}$. For any element
$(t,s)\in\mathrm{Fl}$ representing linear subspaces $P$ and $\Pi$
respectively, we may assume without loss of generality that

$P=\\{(x_{0},x_{1},\ldots,x_{r},0,\ldots,0)\\},$
$\Pi=\\{(x_{0},x_{1},\ldots,x_{r},x_{r+1},0,\ldots,0)\\}.$

The fiber $\tilde{q}^{-1}((t,s))$ parametrizes the cubic hypersurfaces $Y$
such that $\Pi\cap Y\supset 3P$. The last condition implies that the defining
equation $f$ of $Y$ in $\mathbb{P}^{n}$ is given by

$f(Y_{0},\ldots,Y_{n})=\alpha Y_{r+1}^{3}+Y_{r+2}Q_{r+2}+\ldots+Y_{n}Q_{n},$

where $\alpha\in\mathbb{C}$ is a constant and $Q_{r+1},\ldots,Q_{n}$ are
quadratic polynomials. To write the above fact more formally, the fiber
$\tilde{q}^{-1}((t,s))$ is identified with the image of the following map

$\begin{array}[]{cccc}\Phi:&(\mathbb{C}\oplus
H^{0}(\mathbb{P}^{n},\mathcal{O}(2))^{\oplus n-r-1})-\\{0\\}&\to&B\\\
&(\alpha,Q_{r+1},\ldots,Q_{n})&\mapsto&\alpha
Y_{r+1}^{3}+Y_{r+2}Q_{r+2}+\ldots+Y_{n}Q_{n}.\end{array}$

It is not hard to see that $\mathcal{I}\cap\tilde{q}^{-1}((t,s))$ parametrizes
the cubic hypersurfaces $Y$ whose defining equation $f$ is given by

$f(Y_{0},\ldots,Y_{n})=\alpha Y_{r+1}^{3}+Y_{r+2}Q_{r+2}+\ldots+Y_{n}Q_{n},$

subject to $\alpha\neq 0$ and
$Q_{r+1}(Y_{0},\ldots,Y_{r},0,\ldots,0),\ldots,Q_{n}(Y_{0},\ldots,Y_{r},0,\ldots,0)$
are linearly independent in $H^{0}(\mathbb{P}^{r},\mathcal{O}(2))$. Notice
that $n-r-1=\dim H^{0}(\mathbb{P}^{r},\mathcal{O}(2))$, the above conditions
give an open dense subset in $(\mathbb{C}\oplus
H^{0}(\mathbb{P}^{n},\mathcal{O}(2))^{\oplus n-r-1})-\\{0\\}$, which implies
that $\mathcal{I}\cap\tilde{q}^{-1}((t,s))$ is open dense in
$\tilde{q}^{-1}((t,s))$ for any $(t,s)\in\mathrm{Fl}$. Therefore,
$\mathcal{I}$ is open dense in $\tilde{\mathcal{I}}$.

For the next step, let us calculate the dimension of $\mathcal{I}$. To this
end, we use now the projection $\mathcal{I}\rightarrow\mathrm{Fl}$ and compute
the dimensions of its fibers. Since $\mathrm{Fl}$ is a
$\mathbb{P}^{r+1}$-bundle over $\mathrm{Gr}(r+2,n+1)$, it is clear that
$\dim\mathrm{Fl}=(r+2)(n-r-1)+(r+1)$. By the above description of the fiber
$q^{-1}((t,s))$, the dimension of the fiber of $q:\mathcal{I}\to\mathrm{Fl}$
is the dimension of the space of cubic polynomials in $Y_{0},\ldots,Y_{n}$
such that each monomial of it contains either one of the variables
$Y_{r+2},\ldots,Y_{n}$, and the latter is $\dim
H^{0}(\mathbb{P}^{n},\mathcal{O}(3))-\dim
H^{0}(\mathbb{P}^{r+1},\mathcal{O}(3))$. Taken together, we have

$\dim{\mathcal{I}}=(r+2)(n-r-1)+(r+1)+\dim
H^{0}(\mathbb{P}^{n},\mathcal{O}(3))-\dim
H^{0}(\mathbb{P}^{r+1},\mathcal{O}(3)).$

Taking into account of the relation $n+1=\dim
H^{0}(\mathbb{P}^{r+1},\mathcal{O}(2))$ and the fact that $\dim
X=(n-r)(r+1)-\dim H^{0}(\mathbb{P}^{r},\mathcal{O}(3))$, we find that
$\dim{\mathcal{I}}=\dim X+\dim H^{0}(\mathbb{P}^{n},\mathcal{O}(3))-r-2$, as
desired. ∎

######  Lemma 4.2.5.

The map $\tilde{p}:\tilde{\mathcal{I}}\to B$ is surjective.

###### Proof.

It is equivalent to showing that for any cubic hypersurface $Y$, there are
linear subspaces $P,\Pi\subset\mathbb{P}^{n}$ of dimension $r$, $r+1$
respectively such that $\Pi\cap Y\supset 3P$. To construct such examples, it
is worthwhile to notice that for $r\geq 2$, the variety $F_{r+1}(Y)$ is non-
empty by a dimension counting argument. For any $\Pi\in F_{r+1}(Y)$ and any
$P\subset\Pi$, we have $\Pi\cap Y\supset 3P$. That concludes the proof. ∎

By the two lemmas above, we conclude that the map $p:\mathcal{I}\to B$ is
dominant. Since the map $p:\mathcal{I}\to B$ is dominant, the dimension of the
general fiber is $\dim\mathcal{I}-\dim B$, which is equal to $\dim X-r-1$. ∎

######  Proposition 4.2.6.

Let $x\in X$ be a generic fixed point of $\Psi:X\dashrightarrow X$
representing an $r$-dimensional linear subspace $P\subset Y$. Then the linear
map

$\Psi_{*,x}:T_{X,x}\to T_{X,x}$

has $N-r-1$ eigenvalues equal to $1$, corresponding to the tangent space to
the fixed locus $F$ of $\Psi$ at $x$, and $r+1$ eigenvalues equal to $-2$,
corresponding to the action of $\Psi_{*}$ in the normal direction to $F$ at
$x$.

###### Proof.

Since $x\in X$ is a generic fixed point of $\Psi$ representing a plane
$P\subset Y$, there is a unique $(r+1)$-plane $\Pi\subset\mathbb{P}^{n}$ such
that $\Pi\cap Y=3P$ as algebraic cycles. Without loss of generality, we may
assume that

$P=\\{(x_{0},x_{1},\ldots,x_{r},0,\ldots,0)\\},$
$\Pi=\\{(x_{0},x_{1},\ldots,x_{r},x_{r+1},0,\ldots,0)\\}.$

The fact $\Pi\cap Y=3P$ implies that the defininig equation $f$ of $Y$ in
$\mathbb{P}^{n}$ is given by

$f(Y_{0},\ldots,Y_{n})=Y_{r+1}^{3}+Y_{r+2}Q_{r+2}+\ldots+Y_{n}Q_{n},$ (4.1)

where $Q_{r+1},\ldots,Q_{n}$ are quadratic polynomials. Let $v_{1}\in T_{X,x}$
be a nonzero tangent vector given by a path $\\{P_{1,t}\\}_{t\in\Delta}$ with
$P_{1,0}=P$, where $\Delta$ is the unit disc in the complex plane. To
understand $\Psi_{*,x}(v_{1})\in T_{X,x}$, we have to determine
$P_{2,t}:=\Psi(P_{1,t})$ for each small $t\in\Delta$. By the construction, for
each small $t\in\Delta$, there is a unique $(r+1)$-plane
$\Pi_{t}\subset\mathbb{P}^{n}$ such that $\Pi_{t}\cap Y=2P_{1,t}+P_{2,t}$.
Since the inclusions $P_{1,t}\subset\Pi_{t}\subset\mathbb{P}^{n}$ vary
holomorphically with respect to $t\in\Delta$, we may assume that $\Pi_{t}$ is
given by the image of a linear map $G_{t}:\mathbb{P}^{r+1}\to\mathbb{P}^{n}$
varying holomorphically with $t\in\Delta$ and that

$P_{1,t}=\\{G_{t}(x_{0},\ldots,x_{r},0)\\}.$ (4.2)

Then the relation $\Pi_{t}\cap Y=2P_{1,t}+P_{2,t}$ can be translated as

$f\circ G_{t}(x_{0},\ldots,x_{r+1})=x_{r+1}^{2}L_{t}(x_{0},\ldots,x_{r+1})$
(4.3)

for some linear function $L_{t}$ and that

$P_{2,t}=\\{G_{t}(x_{0},\ldots,x_{r+1}):L_{t}(x_{0},\ldots,x_{r+1})=0\\}.$
(4.4)

Let us simplify the notation and write $\mathbf{x}=(x_{0},\ldots,x_{r+1})$.
Let us write $G_{t}(\mathbf{x})$ as

$G_{t}(\mathbf{x})=(x_{0}+tY_{0}(\mathbf{x}),x_{1}+tY_{1}(\mathbf{x}),\ldots,x_{r+1}+tY_{r+1}(\mathbf{x}),\\\
tY_{r+2}(\mathbf{x}),\ldots,tY_{n}(\mathbf{x}))+O(t^{2}).$ (4.5)

Then by (4.1), we have

$f\circ
G_{t}(\mathbf{x})=x_{r+1}^{3}+3tx_{r+1}^{2}Y_{r+1}(\mathbf{x})+t\sum_{i=r+2}^{n}Y_{i}(\mathbf{x})Q_{i}(\mathbf{x},\vec{0})+O(t^{2}).$
(4.6)

Comparing (4.3) and (4.6), we get that $x_{r+1}^{2}$ divides
$\sum_{i=r+2}^{n}Y_{i}(\mathbf{x})Q_{i}(\mathbf{x},\vec{0})$, and that

$L_{t}(\mathbf{x})=x_{r+1}+t\left\\{3Y_{r+1}(\mathbf{x})+\frac{Y_{i}(\mathbf{x})Q_{i}(\mathbf{x},\vec{0})}{x_{r+1}^{2}}\right\\}+O(t^{2}).$
(4.7)

Hence, the equation $L_{t}(\mathbf{x})=0$ has an explicit expression as

$x_{r+1}=-t\left\\{3Y_{r+1}(\mathbf{x}^{\prime},0)+\frac{\sum_{i=r+2}^{n}Y_{i}Q_{i}}{X_{r+1}^{2}}(\mathbf{x}^{\prime},0)\right\\}+O(t^{2}),$
(4.8)

where $\mathbf{x}^{\prime}=(x_{0},\ldots,x_{r})$. Comparing (4.4)(4.5)(4.8),
we get

$\displaystyle P_{2,t}=$
$\displaystyle\\{(x_{0}+tY_{0}(\mathbf{x}^{\prime},0),\ldots,x_{r}+tY_{r}(\mathbf{x}^{\prime},0),-2tY_{r+1}(\mathbf{x}^{\prime},0)-t\frac{\sum
Y_{i}Q_{i}}{X_{r+1}^{2}}(\mathbf{x},0),$ $\displaystyle
tY_{r+2}(\mathbf{x}^{\prime},0),\ldots,tY_{n}(\mathbf{x}^{\prime},0))+O(t^{2})\\}.$

We view now $v_{1}\in T_{X,x}$ as a $(r+1)\times(n-r)$-matrix via the
following natural inclusion and identification $T_{X,x}\subset
T_{\mathrm{Gr}(r+1,n+1),x}=Hom(\mathbb{C}^{r+1},\mathbb{C}^{n+1}/\mathbb{C}^{r+1})$.
Then $v_{1}$ is represented by the matrix

$\begin{pmatrix}Y_{r+1}\\\ Y_{r+2}\\\ \vdots\\\ Y_{n}\end{pmatrix},$

where we view each $Y_{i}$ as a row vector $(a_{0},a_{1},\ldots,a_{r})$ if
$Y_{i}(x_{0},\ldots,x_{r},0)=a_{0}x_{0}+\ldots+a_{r}x_{r}$. By the above
calculations, we see that $\Psi_{*,x}(v_{1})$ is represented by the matrix

$\begin{pmatrix}-2Y_{r+1}-\frac{\sum_{i=r+2}^{n}Y_{i}Q_{i}}{X_{r+2}^{2}}\\\
Y_{r+2}\\\ \vdots\\\ Y_{n}\end{pmatrix}$

with the same explanation about the notations. Hence, $\Psi_{*,P}$ sends
$\begin{pmatrix}Y_{r+1}\\\ Y_{r+2}\\\ \vdots\\\ Y_{n}\end{pmatrix}$ to
$\begin{pmatrix}-2Y_{r+1}-\frac{\sum_{i=r+2}^{n}Y_{i}Q_{i}}{X_{r+2}^{2}}\\\
Y_{r+2}\\\ \vdots\\\ Y_{n}\end{pmatrix}.$ The representing matrix of
$\Psi_{*,x}$ is uppper triangular with the diagonal $r+1$ copies of $-2$ and
$N-r-1$ copies of $1$. Therefore, the eigenpolynomial of
$\Psi_{*,x}:T_{X,x}\to T_{X,x}$ is $(t+2)^{r+1}(t-1)^{N-r-1}$, as desired. ∎

###### Proof of Theorem A (i).

By Proposition 4.2.6, the eigenpolynomial of
$\Psi^{*}_{P}:\Omega_{X,P}\to\Omega_{X,P}$ is $(t+2)^{r+1}(t-1)^{N-r-1}$ for a
generic fixed point $P$ of $\Psi$. Hence, the map $\Psi^{*}_{P}:K_{X,P}\to
K_{X,P}$ is given as the multiplication by $(-2)^{r+1}$. Let $\omega\in
H^{0}(X,K_{X})$ be a nowhere zero top degree holomorphic differential form on
$X$. Then, as $H^{0}(X,K_{X})$ has dimension $1$,

$\Psi^{*}\omega=\lambda\omega$ (4.9)

for some $\lambda\in\mathbb{C}$. By evaluating the equation (4.9) at the point
$P\in X$, we find $\lambda=(-2)^{r+1}$, and we finish the proof of Theorem A
(i). ∎

####  4.2.2 A direct proof of Theorem A (ii)

While Theorem A (ii) has been established as a consequence of Theorem A (i)
using Lemma 4.2.2, we present a direct proof, using an enumerative geometry
viewpoint.

###### Proof of Theorem A (ii).

The beginning of the proof is similar to that of [Huy24, Lemma 4.12]. Let
$P^{\prime}\in X$ be a generic $r$-plane in $Y$. The preimage of $P^{\prime}$
via $\Psi$ is the set of $r$-planes $P$ in $Y$ such that there is an
$(r+1)$-plane $\Pi$ in $\mathbb{P}^{n}$ such that $\Pi\cap Y=2P+P^{\prime}$ as
algebraic cycles. In general, let $\Pi\subset\mathbb{P}^{n}$ be an
$(r+1)$-plane containing $P^{\prime}$. Then $\Pi\cap Y=Q\cup P^{\prime}$ where
$Q\subset\Pi$ is a quadratic hypersurface that corresponds to a quadratic form
$q_{\Pi}$. The preimage of $P^{\prime}$ via $\Psi$ corresponds to the
$(r+1)$-planes $\Pi$ such that the quadratic form $q_{\Pi}$ is of rank $1$.
Now let $\pi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{n-r-1}$ be the
projection map induced by the $r$-plane $P^{\prime}$. One can resolve the
indeterminacies of $\pi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{n-r-1}$ by
blowing up $\mathbb{P}^{n}$ along $P^{\prime}$. Let
$p:\mathcal{P}\to\mathbb{P}^{n}$ be the blowup map. Then the induced map
$q:\mathcal{P}\to\mathbb{P}^{n-r-1}$ is a projective bundle induced by a
vector bundle $\mathcal{E}$ over $\mathbb{P}^{n-r-1}$. We now prove

######  Lemma 4.2.7.

(i) We have
$\mathcal{E}\cong\mathcal{O}_{\mathbb{P}^{n-r-1}}^{r+1}\oplus\mathcal{O}_{\mathbb{P}^{n-r-1}}(-1)$.
(ii) There is a section $q\in
H^{0}(\mathbb{P}^{n-r-1},Sym^{2}\mathcal{E}^{*}\otimes\mathcal{O}_{\mathbb{P}^{n-r-1}}(1))$
such that on each point $\Pi\in\mathbb{P}^{n-r-1}$, $q$ coincides with
$q_{\Pi}$.

###### Proof.

See [Huy24, Section 1.5.1-1.5.2]. ∎

Hence, to calculate the degree of $\Psi:X\dashrightarrow X$, we need to
calculate the number of points in $\mathbb{P}^{n-r-1}$ over which $q$ is of
rank $1$. Since $P^{\prime}$ is generic, by a dimension counting argument we
find that there is no plane $\Pi$ containing $P^{\prime}$ that is contained in
$Y$. Therefore, we are reduced to calculate the degree of the locus where a
generic section of $Sym^{2}\mathcal{E}^{*}\otimes\mathcal{O}(1)$ has rank
$\leq 1$. Let $f:\mathbb{P}^{n-r-1}\to\mathbb{P}^{n-r-1}$ be a morphism of
degree $2^{n-r-1}$ (e.g.,
$f:(x_{0},\ldots,x_{n-r-1})\mapsto(x_{0}^{2},\ldots,x_{n-r-1}^{2})$). Let
$Z_{q}\subset\mathbb{P}^{n-r-1}$ be the locus where $q\in
H^{0}(\mathbb{P}^{n-r-1},Sym^{2}\mathcal{E}^{*}\otimes\mathcal{O}(1))$ is of
rank $\leq 1$. Then $f^{*}Z_{q}$ is the locus where $f^{*}q\in
H^{0}(Sym^{2}\mathcal{F})$ is of rank $\leq 1$. Here,
$\mathcal{F}=\mathcal{O}(1)^{r+1}\oplus\mathcal{O}(3)$. Then

$\deg f^{*}Z_{q}=\deg f\cdot\deg Z_{q}=2^{n-r-1}\deg Z_{q}.$ (4.10)

Let $p:\mathbb{P}(\mathcal{F})\to\mathbb{P}^{n-r-1}$ and
$\pi:\mathbb{P}(\mathrm{Sym}^{2}\mathcal{F})\to\mathbb{P}^{n-r-1}$ be the
projective bundles over $\mathbb{P}^{n-r-1}$ corresponding to the vector
bundles $\mathcal{F}$ and $\mathrm{Sym}^{2}\mathcal{F}$, respectively.
Consider the Veronese map of degree $2$ over the projective space
$\mathbb{P}^{n-r-1}$.

$\begin{array}[]{cccc}v_{2}:&\mathbb{P}(\mathcal{F})&\to&\mathbb{P}(\mathrm{Sym}^{2}\mathcal{F})\\\
&\alpha&\mapsto&\alpha^{2}.\end{array}$

The section $f^{*}q$ induces a section $\sigma_{q}$ of the projective bundle
$\pi:\mathbb{P}(Sym^{2}\mathcal{F})\to\mathbb{P}^{n-r-1}$ such that
$f^{*}Z_{q}$ coincides with
$\pi_{*}(\mathrm{Im}\sigma_{q}\cap\mathrm{Im}(v_{2}))=p_{*}(v_{2}^{*}(\mathrm{Im}(\sigma_{q}))$.
Let $\mathcal{S}$ (resp. $\mathcal{S}^{\prime}$) be the tautological subbundle
of $\mathbb{P}(\mathcal{F})$ (resp. $\mathbb{P}(Sym^{2}\mathcal{F})$). Let
$\mathcal{Q}^{\prime}$ be the tautological quotient bundle of
$\mathbb{P}(Sym^{2}\mathcal{F})$. Then the pull-back of $f^{*}q\in
H^{0}(\mathbb{P}^{n-r-1},Sym^{2}\mathcal{F})$ gives a section of
$H^{0}(\mathbb{P}(Sym^{2}\mathcal{F}),\mathcal{Q}^{\prime})$ whose zero locus
coincides with $\mathrm{Im}(\sigma_{q})$. Writing $c(\mathcal{Q}^{\prime})$ as
$\pi^{*}c(\mathrm{Sym}^{2}\mathcal{F})\cdot c(\mathcal{S}^{\prime})^{-1}$ and
noticing that $v_{2}^{*}\mathcal{S}^{\prime}=\mathcal{S}^{\otimes 2}$, we see
that $f^{*}Z_{q}$ is the part of degree $n$ of $p_{*}(c(S^{\otimes
2})^{-1}\cdot p^{*}c(\mathrm{Sym}^{2}\mathcal{F}))$. Let $s(\mathcal{F})$ be
the formal series of the Segre classes of $\mathcal{F}$ [Ful84, Chapter 3] and
let $h=c_{1}(\mathcal{O}_{\mathbb{P}^{n}}(1))$. By a direct calculation, we
find that

$\displaystyle f^{*}Z_{q}$
$\displaystyle=2^{r+1}\sum_{i}2^{i}s_{i}(\mathcal{F})\cdot
c_{n-r-1-i}(\mathrm{Sym}^{2}\mathcal{F})$ $\displaystyle=\textrm{The degree
$n-r-1$ part of
}2^{r+1}\frac{(1+2h)^{\frac{(r+1)(r+2)}{2}}\cdot(1+4h)^{r+1}\cdot(1+6h)}{(1+2h)^{r+1}\cdot(1+6h)}$
$\displaystyle=2^{r+1}\cdot 2^{\frac{(r+1)r}{2}}\cdot 4^{r+1}h^{n-r-1}.$

Taking into account of the relation (4.10), we find that $\deg Z_{q}=4^{r+1}$,
as desired. ∎

###  4.3 Action of the Voisin map on $CH_{0}(X)_{hom}$

The main purpose of this section is to prove Theorem B. Part of the argument
will work for any $r$ and will be given in Section 4.3.1.

####  4.3.1 Decomposition of the action of $\Psi_{*}$

Let $X=F_{r}(Y)$ be as presented in Section 4.1.1. Let
$I_{r}=\\{(x,x^{\prime})\in X\times X:\dim(P_{x}\cap P_{x^{\prime}})\geq
r-1\\}$.

######  Theorem 4.3.1.

There is an $(r+1)$-cocycle $\gamma\in CH^{r+1}(X)$ lying in the image of the
restriction map $CH^{r+1}(\mathrm{Gr}(r+1,n+1))\to CH^{r+1}(X)$ such that for
any $z\in CH_{0}(X)$, we have in $CH_{0}(X)_{\mathbb{Q}}$

$\Psi_{*}z=(-2)^{r+1}z+\gamma\cdot I_{r*}(z).$ (4.11)

###### Proof.

The argument is quite similar to that in the proof of [Voi14, Theorem 4.16].
Let $B=\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}(3))$ be the projective space
parametrizing all cubic hypersurfaces in $\mathbb{P}^{n}$. For $f\in B$, we
denote $Y_{f}\subset\mathbb{P}^{9}$ the hypersurface defined by $f$. There is
a universal Fano variety of $r$-spaces defined as follows.

$\mathcal{X}:=\\{(f,x)\in B\times\mathrm{Gr}(r+1,n+1):P_{x}\subset Y_{f}\\}.$

The fiber of the projection map $\pi:\mathcal{X}\to B$ over a point $f\in B$
is the Fano variety of $r$-spaces of $Y_{f}$. The Voisin map
$\Psi:X\dashrightarrow X$ can also be defined universally. In fact, we define
the universal graph $\Gamma_{\Psi_{\mathrm{univ}}}$ as the closure of the set
of pairs $(x,x^{\prime})\in\mathcal{X}\times_{B}\mathcal{X}$ such that there
exists a $(r+1)$-dimensional linear subspace $H_{r+1}\subset\mathbb{P}^{n}$
such that $H_{r+1}\cap Y_{\pi(x)}=2P_{x}+P_{x^{\prime}}$. Denote

${\Gamma_{\Psi_{\mathrm{univ}}}}$${\mathcal{X}\times_{B}\mathcal{X}}$${\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1)}$${B}$$\scriptstyle{i}$$\scriptstyle{q}$$\scriptstyle{p}$

the inclusion map and the natural projection maps. Then over a point $f\in B$
corresponding to a general smooth cubic hypersurface $Y_{f}$, the fiber
$(p\circ i)^{-1}(f)$ is the graph of the Voisin map on $F_{r}(Y_{f})$. There
is a stratification of $\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1)$ given
as follows.

${\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1)}$${I^{G}_{1}:=\\{(x,x^{\prime}):P_{x}\cap
P_{x^{\prime}}\neq\emptyset\\}}$${I^{G}_{2}:=\\{(x,x^{\prime}):P_{x}\cap
P_{x^{\prime}}\textrm{ contains a line
}\\}}$${\vdots}$${I_{r}^{G}:=\\{(x,x^{\prime}):P_{x}\cap
P_{x^{\prime}}\textrm{ contains an
}(r-1)\textrm{-space}\\}}$${I_{r+1}^{G}=\Delta_{G}:=\\{(x,x)\\}}$

In other words, the subvariety $I_{k}^{G}$ is defined as

$I_{k}^{G}=\\{(x,x^{\prime})\in\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1):\dim(P_{x}\cap
P_{x^{\prime}})\geq k-1\\}.$

The map
$q:\mathcal{X}\times_{B}\mathcal{X}\to\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1)$
has a projective bundle structure over each stratum. Precisely, let $d=\dim
B$. Let $I_{k}^{\mathcal{X}}$ be the preimage of $I_{k}^{G}$ under the map
$q:\mathcal{X}\times_{B}\mathcal{X}\to\mathrm{Gr}(r+1,n+1)\times\mathrm{Gr}(r+1,n+1)$.
Over the open subset $I_{k}^{G}-I_{k+1}^{G}$, the map
$q_{|I_{k}^{\mathcal{X}}}$ is a $\mathbb{P}^{d-\delta_{k}}$-bundle where
$\delta_{k}=2h^{0}(\mathbb{P}^{r},\mathcal{O}(3))-h^{0}(\mathbb{P}^{k-1},\mathcal{O}(3))$.
Let
$i_{k}^{\mathcal{X}}:I_{k}^{\mathcal{X}}\hookrightarrow\mathcal{X}\times_{B}\mathcal{X}$
be the inclusion maps. The Chow ring of $\mathcal{X}\times_{B}\mathcal{X}$ is
thus equal to

$\bigoplus_{i}h^{i}\cdot\left(\bigoplus_{k}\left(q_{|I_{k}^{\mathcal{X}}}\right)^{*}CH^{*}(I_{k}^{G})\right),$

where $h=p^{*}\mathcal{O}_{B}(1)$. Now let us consider the Chow class of
$\Gamma_{\Psi_{\mathrm{univ}}}$ in $CH^{N}(\mathcal{X}\times_{B}\mathcal{X})$,
where $N=(r+1)(n-r)-\binom{r+3}{3}$ is the dimension of $X$. By the
construction, we have $\Gamma_{\Psi_{\mathrm{univ}}}\subset
I_{r}^{\mathcal{X}}$, thus

$\Gamma_{\Psi_{\mathrm{univ}}}\in\bigoplus_{i}h^{i}\cdot\left(i_{r+1,*}^{\mathcal{X}}(q_{|\Delta_{\mathcal{X}}})^{*}CH^{-i}(\Delta_{G})+i_{r*}^{\mathcal{X}}\left(q_{|I_{k}^{\mathcal{X}}}\right)^{*}CH^{m-i}(I_{r}^{G})\right),$

where $m$ is the relative dimension of $pr_{1}:I_{r}\to X$ that can be
calculated as follows. Over a point $x\in X$, the fiber of $pr_{1}:I_{r}\to X$
is the set of points $x^{\prime}\in X$ such that the intersection $P_{x}\cap
P_{x^{\prime}}$ contains a $\mathbb{P}^{r-1}$ in $P_{x}$. The set of
$\mathbb{P}^{r-1}$ contained in $P_{x}$ is a $\mathbb{P}^{r}$. Furthermore,
for each given $\mathbb{P}^{r-1}\subset Y$, the set of $r$-spaces
$P_{x^{\prime}}\subset\mathbb{P}^{n}$ that contain the given
$\mathbb{P}^{r-1}$ is a $\mathbb{P}^{n-r}$, and the condition that
$P_{x^{\prime}}\subset Y$ is equivalent to saying that the defining equation
of the residual quadric is identically zero, which gives
$\frac{(r+1)(r+2)}{2}$ independent conditions. Taking everything into account,
we find that $m=n-\frac{(r+1)(r+2)}{2}=r+1$.

Now since $\Gamma_{\Psi}$ is a fiber of $p\circ
i:\Gamma_{\Psi_{\mathrm{univ}}}\to B$, we conclude that

$\Gamma_{\Psi}=\Gamma_{\Psi_{\mathrm{univ}}|{X\times
X}}\in(i_{r+1,*}^{\mathcal{X}}q^{*}CH^{0}(\Delta_{G})+i_{r*}^{\mathcal{X}}q^{*}CH^{r+1}(I_{r}^{G}))|_{X\times
X}.$

Therefore, we can write

$\Gamma_{\Psi}=\alpha\Delta_{X}+\delta,$ (4.12)

where $\alpha\in\mathbb{Z}$ is a coefficient and
$\delta\in(i_{{r}*}^{\mathcal{X}}q^{*}CH^{r+1}(I_{r}^{G}))|_{X\times X}$.
Write $\delta=i_{r*}^{\mathcal{X}}(q^{*}\delta_{G})$ for some $\delta_{G}\in
CH^{r+1}(I_{r}^{G})$. Notice that $I_{r}^{G}$ is a fiber bundle over
$\mathrm{Gr}(r+1,n+1)$ whose fiber is a closed Schubert subvariety $\Sigma$ of
$\mathrm{Gr}(r+1,n+1)$, that is defined as the closure of the set of
$r$-spaces in $\mathbb{P}^{n}$ that intersects with a given $r$-space along a
$(r-1)$-space. This fiber bundle has a universal cellular decomposition into
affine bundles in the sense of [Ful84, Example 1.9.1]. By [Ful84, Example
19.1.11] and [Voi03, Théorème 7.33], we have

$CH^{*}(I_{r}^{G})=CH^{*}(\mathrm{Gr}(r+1,n+1))\otimes CH^{*}(\Sigma).$

Therefore, we can write

$\delta_{G}=\sum_{i=0}^{r+1}(p_{1}^{*}\alpha_{i}).\beta_{i},$

where $\alpha_{i}\in CH^{i}(\mathrm{Gr}(r+1,n+1)$ and $\beta_{i}\in
CH^{r+1-i}(\Sigma)$. One easily checks that the morphism
$pr_{2}^{*}:CH^{*}(Gr(r+1,n+1))\to CH^{*}(\Sigma)$ is surjective. Hence, there
exists $\gamma_{i}\in CH^{r+1-i}(\mathrm{Gr}(r+1,n+1))$ such that
$\beta_{i}=pr_{2}^{*}\gamma_{i}$. Therefore,

$\delta_{G}=\sum_{i=0}^{r+1}p_{1}^{*}\alpha_{i}.p_{2}^{*}\gamma_{i}.$ (4.13)

Now we prove

######  Lemma 4.3.2.

The coefficient $\alpha$ in the decomposition (4.12) equals $(-2)^{r+1}$ .

###### Proof.

For $\omega\in H^{N,0}(X)$, we have
$\delta^{*}\omega=p_{2*}(\sum_{i}p_{1}^{*}([\alpha_{i}]\cup\omega)\cup\gamma_{i|X}\cup[I_{r}])$.
Since $\omega$ is a top degree form, $[\alpha_{i}]\cup\omega=0$ unless $i=0$
and thus, $\delta^{*}\omega=c[\gamma_{0|X}]\cup I_{r}^{*}\omega$, where $c$ is
some constant number. Now let us prove that $I_{r}^{*}\omega=0$. Let
$\mathcal{P}_{r,r-1}=\\{(x,\lambda)\in F_{r}(Y)\times
F_{r-1}(Y):P_{\lambda}\subset P_{x}\\}$ be the flag variety. Since
$I_{r}=\mathcal{P}_{r,r-1}^{t}\circ\mathcal{P}_{r,r-1}$ as correspondences, we
have $I_{r}^{*}\omega=\mathcal{P}_{r,r-1}^{*}(\mathcal{P}_{r,r-1*}(\omega))=0$
since $\mathcal{P}_{r,r-1*}\omega\in H^{N-1,-1}(F_{1}(Y))$. Therefore,
$\Psi^{*}\omega=\alpha\omega+\delta^{*}\omega=\alpha\omega$. By Theorem A, we
get $\alpha=(-2)^{r+1}$. ∎

Since $z\in CH_{0}(X)_{\mathbb{Q}}$ is a $0$-cycle,
$\delta_{*}z=p_{2*}(\sum_{i}p_{1}^{*}(\alpha_{i}\cdot z)\cdot
p_{2}^{*}\gamma_{i|X}\cdot I_{r})=p_{2*}(p_{1}^{*}(\alpha_{0}\cdot z)\cdot
p_{2}^{*}\gamma_{0|X}\cdot I_{r})=\gamma\cdot I_{r*}z$ where $\gamma$ is some
multiple of $\gamma_{0|X}$, which is an element in the image of the
restriction map $CH^{r+1}(\mathrm{Gr}(r+1,n+1))\to CH^{r+1}(X)$. Taking Lemma
4.3.2, formula (4.12) and formula (4.13) into account, we prove the formula

$\Psi_{*}z=(-2)^{r+1}z+\gamma\cdot I_{r*}z,$ (4.14)

as desired. ∎

The cycle gamma appearing in (4.14) is a polynomial in the Chern classes
$c_{i}$ of the tautological bundle of the Grassmannian, restricted to $X$. Let
us now prove

######  Proposition 4.3.3.

We have $c_{r+1}\cdot I_{r*}z$=0 for $z\in CH_{0}(X)_{hom}$.

###### Proof.

Let $H_{n-1}\subset\mathbb{P}^{n}$ be a general hyperplane. Let us define the
Schubert variety $\Sigma_{r+1}^{H_{n-1}}$ as

$\Sigma_{r+1}^{H_{n-1}}:=\\{y\in\mathrm{Gr}(r+1,n+1):P_{y}\subset H_{n-1}\\}.$

Then $c_{r+1}$ is represented by the class of $\Sigma_{r+1}^{H_{n-1}}$. Let
$x\in X$ be a general point and let

$\Theta_{x}:=\\{y\in X:\dim(P_{y}\cap P_{x})\geq r-1\\}.$

Then by the definition of $I_{r}$, $\Theta_{x}$ represents $I_{r*}x$. By
definition, the intersection $\Sigma_{r+1}^{H_{n-1}}\cap\Theta_{x}$ is the set

$\\{y\in X:P_{y}\subset H_{n-1}\cap Y,\,{\rm dim}\,P_{x}\cap P_{y}\geq
r-1\\},$

which we can also rewrite, if $P_{x}$ is not contained in $H_{n-1}$, as

$\Sigma_{r+1}^{H_{n-1}}\cap\Theta_{x}=\\{y\in X:\,Y\cap H_{n-1}\supset
P_{y}\supset P_{x}\cap H_{n-1}\\}.$

Let $\Delta_{x}=P_{x}\cap H_{n-1}$. Then $\Delta_{x}$ provides a point
$\delta_{x}$ of the variety $F_{r-1}(Y\cap H_{n-1})$. Let $i:F_{r}(Y\cap
H_{n-1})\hookrightarrow X$ be the natural embedding map. Let
$\mathcal{P}_{r,r-1}^{H_{n-1}}:=\\{(y,\lambda)\in F_{r}(Y\cap H_{n-1})\times
F_{r-1}(Y\cap H_{n-1}):P_{\lambda}\subset P_{y}\\}$ be the incidence variety.
By the above description of $\Sigma_{r+1}^{H_{n-1}}\cap\Theta_{x}$, one finds
that

$c_{r+1}\cdot I_{r*}x=i_{*}(\mathcal{P}_{r,r-1}^{H_{n-1}*}({\delta_{x}})),$

where $\Delta_{x}$ is viewed as an element in $CH_{0}(F_{r-1}(Y\cap
H_{n-1}))$. One can verify that $F_{r-1}(Y\cap H_{n-1})$ is a Fano manifold
and thus, $CH_{0}(F_{r-1}(Y\cap H_{n-1}))=\mathbb{Z}$. Therefore, the Chow
class of $c_{r+1}\cdot I_{r*}x$ does not depend on $x$. That is, for any $z\in
CH_{0}(X)_{hom}$, we get $c_{r+1}\cdot I_{r*}z=0$ in $CH_{0}(X)$, as desired.
∎

######  Proposition 4.3.4.

If $CH_{1}(F_{r-1}(Y))_{\mathbb{Q}}$ is trivial, then $c_{r}\cdot I_{r*}(z)=0$
in $CH_{1}(X)$ for any $z\in CH_{0}(X)_{hom}$.

###### Proof.

We only write down the case $r=2$, for which the assumption of Proposition
4.3.4 will be proved in the next section. The general case is similar. Let
$H_{7}$ be a $7$-dimensional linear subspace in $\mathbb{P}^{9}$. In the
Grassmannian $\mathrm{Gr}(3,10)$, the class $c_{2}$ is represented by the
subvariety $\Sigma_{2}^{H_{7}}$ defined as

$\Sigma_{2}^{H_{7}}:=\\{y\in\mathrm{Gr}(3,10):P_{y}\cap H_{7}\textrm{ contains
a line }{\Delta_{y,H}}\\}.$

For general $x\in X$, the plane $P_{x}$ intersects $H_{7}$ at a single point
$z_{x}$, and the class $I_{2,*}x$ is represented by the following subvariety
$\Theta_{x}$ in $X$ defined as

$\Theta_{x}:=\\{y\in X:P_{y}\cap P_{x}\textrm{ contains a line
}{\Delta_{xy}}\\}.$

Let $H_{6}\subset H_{7}$ be a linear subspace of dimension $6$ not containing
the point ${z_{x}}$.

######  Lemma 4.3.5.

For general $x\in X$, and any $y\in\Theta_{x}$, we have $z_{x}\in\Delta_{xy}$.

###### Proof.

Let $P_{y}$ be a plane in $Y$ that intersects $P_{x}$ along the line
$\Delta_{xy}$ and intersects $H_{7}$ along the line $\Delta_{y,H}$. Then
$\Delta_{xy}$ and $\Delta_{y,H}$ must intersect, since they are two lines in a
projective plane; and the intersection point of $\Delta_{xy}$ and
$\Delta_{y,H}$ must be ${z_{x}}$ since ${z_{x}}$ is the only intersection
point of $P_{x}$ and $H_{7}$. Therefore, $z_{x}\in\Delta_{xy}$, as desired. ∎

Let $\Sigma^{H_{6}}_{1}:=\\{y\in\mathrm{Gr}(3,10):P_{y}\cap
H_{6}\neq\emptyset\\}.$ Let $\Xi_{x}$ be the variety of points $y\in X$ such
that $P_{y}\cap P_{x}$ contains a line $\Delta_{xy}$ containing ${z_{x}}$. By
Lemma 4.3.5, we now conclude that

$\Sigma_{2}^{H_{7}}\cap\Theta_{x}=\Sigma_{1}^{H_{6}}\cap\Xi_{x}.$ (4.15)

Indeed, by Lemma 4.3.5, $\Sigma_{2}^{H_{7}}\cap\Theta_{x}$ contains all the
points $y\in X$ such that $P_{y}\cap P_{x}$ contains a line $\Delta_{xy}$
containing ${z_{x}}$ and such that $P_{y}\cap H_{7}$ contains a line
$\Delta_{y,H}$. This variety coincides with $\Sigma_{1}^{H_{6}}\cap\Xi_{x}$
since knowing that ${z_{x}}\in P_{y}$ and that $z_{x}\not\in H_{6}$,
$P_{y}\cap H_{7}$ contains a line if and only if $P_{y}\cap H_{6}$ is
nonempty.

Let $\mathcal{P}_{2,1}:=\\{(x^{\prime},[l])\in F_{2}(Y)\times
F_{1}(Y):l\subset P_{x^{\prime}}\\}$ be the flag variety. Let
$\Delta_{P_{x},{z_{x}}}^{\vee}=\\{[l]\in F_{1}(Y):{z_{x}}\in l\subset
P_{x}\\}$. Then $\Delta_{P_{x},z_{x}}^{\vee}$ provides a class
$\delta_{P_{x},z_{x}}^{\vee}\in CH_{1}(F_{1}(Y))$. The equation (4.15) shows
that $c_{2}\cdot
I_{2,*}x=c_{1}\cdot\mathcal{P}_{2,1}^{*}(\delta_{P_{x},{z_{x}}}^{\vee})$.
Therefore, for $z\in CH_{0}(X)_{\mathbb{Q},hom}$, we have $c_{2}\cdot
I_{2,*}z=c_{1}\cdot\mathcal{P}_{2,1}^{*}(Z)$ for some $Z\in
CH_{1}(F_{1}(Y))_{\mathbb{Q},hom}$. But we have
$CH_{1}(F_{1}(Y))_{\mathbb{Q},hom}=0$ by Theorem 4.3.6 below (or by assumption
for $r>2$). Hence, $c_{2}\cdot I_{2,*}z=0\in CH_{1}(X)_{\mathbb{Q}}$, as
desired. ∎

####  4.3.2 Triviality of $CH_{1}(F_{1}(Y))_{\mathbb{Q}}$

Let $Y\subset\mathbb{P}^{9}$ be a cubic eightfold. It has been established
that $H^{p,q}(F_{1}(Y))=0$ for $p\leq 1$ and $p\neq q$ [DM98], indicating that
the coniveau of $F_{1}(Y)$ is at least $2$. According to the generalized Bloch
conjecture, this suggests that $CH_{i}(F_{1}(Y))_{\mathbb{Q},hom}=0$ for
$i\leq 1$. This section is devoted to proving this statement, namely

######  Theorem 4.3.6.

The Chow group $CH_{1}(F_{1}(Y))_{\mathbb{Q}}$ is isomorphic to $\mathbb{Q}$.

##### Lines of lines

Let $P$ be a plane contained in $Y$ and let $x\in P$ be a point. Let
$\Delta_{P,x}^{\vee}$ be the variety of lines in $P$ passing through $x$.
Viewed as a $1$-cycle of $F_{1}(Y)$, it is clear that the Chow class of
$\Delta_{P,x}^{\vee}$ does not depend on the choice of $x\in P$ and we let
$\Delta_{P}^{\vee}\in CH_{1}(F_{1}(Y))$ denote the Chow class of
$\Delta_{P,x}^{\vee}$ for some (and thus any) $x\in P$.

######  Proposition 4.3.7.

Let $L\cong\mathbb{P}^{3}\subset\mathbb{P}^{9}$ be a $3$-dimensional linear
subspace whose intersection with $Y$ contains three planes $P_{1},P_{2}$ and
$P_{3}$ (not necessarily distinct). Then the Chow class

$\Delta_{P_{1}}^{\vee}+\Delta_{P_{2}}^{\vee}+\Delta_{P_{3}}^{\vee}\in
CH_{1}(F_{1}(Y))$

does not depend on the choice of $L$.

######  Remark 4.3.8.

The linear spaces $L\cong\mathbb{P}^{3}$ whose intersection with $Y$ is the
union of three planes form a projective variety of general type, which is not
rationally connected. To prove the proposition, we consider a larger space,
namely the variety of cubic surfaces in $Y$ which are cones and prove that
this space is rationally connected.

###### Proof of Proposition 4.3.7.

Let $\mathcal{M}$ be the space of cubic surfaces in $Y$ which are cones. An
element of $\mathcal{M}$ is of the form $(S,y_{0})$ where $S$ is a cubic
surface that is a cone from a vertex $y_{0}$. If $P_{1},P_{2},P_{3}$ are
planes as in the Proposition, and $y_{0}$ is a point in the intersection of
the three planes, then $P_{1}\cup P_{2}\cup P_{3}$ is a cone with vertex
$y_{0}$, hence $(P_{1}\cup P_{2}\cup P_{3},y_{0})$ is an element of
$\mathcal{M}$. Consider the incidence variety

$\mathcal{E}=\\{((S,y_{0}),l)\in\mathcal{M}\times F_{1}(Y):l\textrm{ is a line
in }S\textrm{ passing through }y_{0}\\}.$

Then $\Delta_{P_{1}}^{\vee}+\Delta_{P_{2}}^{\vee}+\Delta_{P_{3}}^{\vee}$ is
the $1$-cycle given by $\mathcal{E}_{*}(P_{1}\cup P_{2}\cup P_{3},y_{0})$.
Therefore, to prove the Proposition, it suffices to show that $\mathcal{M}$ is
rationally connected.

Let $\pi:\mathcal{M}\to Y$ be the rational map that sends a cubic surface
$(S,y_{0})$ that is a cone onto its vertex $y_{0}$. The fiber of $\pi$ over
$y_{0}$ parametrizes the cubic surfaces in $Y$ that are cones with vertex
$y_{0}$. Then $S$ is contained in the singular hyperplane section
$Y_{y_{0}}:=Y\cap\bar{T}_{Y,y_{0}}$ where $\bar{T}_{Y,y_{0}}$ is the
projective tangent space of $Y$ at the point $y_{0}$. We may take $y_{0}$ a
general point so that the cubic hypersurface $Y_{y_{0}}$ of dimension $7$ has
an ordinary double point at $y_{0}$, and does not contain any
$\mathbb{P}^{3}$, and so that no $3$-dimensional linear subspaces in
$\mathbb{P}^{9}$ passing through $y_{0}$ is contained in $Y$. The last
condition is satisfied since the $3$-dimensional linear subspaces contained in
$Y$ form a divisor in $Y$, as can be shown by a simple dimension counting
argument.

######  Lemma 4.3.9.

Under the assumptions above on $y_{0}$, the fiber of $\pi$ over $y_{0}$ is in
bijection with the set of planes contained in the Hessian quadric $Q_{y_{0}}$
of $Y_{y_{0}}$ at the point $y_{0}$.

###### Proof.

A cubic surface $S=\mathbb{P}^{3}\cap Y_{y_{0}}$ in $Y_{y_{0}}$ passing
through $y_{0}$ is a cone with vertex $y_{0}$ if and only if the equation
$f_{y_{0}}$ defining $Y_{y_{0}}$ has vanishing Hessian, which means that the
Hessian quadric vanishes on the tangent space $T_{\mathbb{P}^{3},y_{0}}$ . ∎

One knows that the variety of planes in a quadric sixfold is a connected Fano
manifold. Thus, by Lemma 4.3.9, the map $\pi$ is a fibration whose base and
general fiber are rationally connected. Therefore, the total space
$\mathcal{M}$ is rationally connected [GHS03], as desired. ∎

######  Definition 4.3.10.

Let $\Delta^{\vee}$ be an element of $CH_{1}(F_{1}(Y))_{\mathbb{Q}}$ defined
by
$\frac{1}{3}(\Delta_{P_{1}}^{\vee}+\Delta_{P_{2}}^{\vee}+\Delta_{P_{3}}^{\vee})$
where $P_{1},P_{2},P_{3}$ are the planes as in Proposition 4.3.7.

######  Corollary 4.3.11.

Let $L\subset Y$ be a $3$-dimensional linear space in $Y$. For any plane
$P\subset L$, we have $\Delta^{\vee}_{P}=\Delta^{\vee}$ in
$CH_{1}(F_{1}(Y))_{\mathbb{Q}}$.

###### Proof.

This is because the triple $(P,P,P)$ satisfies Proposition 4.3.7 and thus
$3\Delta^{\vee}_{P}=3\Delta^{\vee}$, from which we conclude. ∎

######  Corollary 4.3.12.

Let $L\subset Y$ be a $3$-dimensional linear space in $Y$. Then the image of
the natural morphism

$CH_{1}(F_{1}(L))_{\mathbb{Q}}\to CH_{1}(F_{1}(Y))_{\mathbb{Q}}$

is of dimension $1$ and is generated by $\Delta^{\vee}$. In particular, the
image does not depend on the choice of $L\in F_{3}(Y)$.

######  Remark 4.3.13.

We will see, in Proposition 4.3.25, that for any $P\subset Y$, we have
$\Delta^{\vee}_{P}=\Delta^{\vee}$.

##### The geometry of $F_{1}(Y)$ and its $1$-cycles

The following Lemma is proved by an easy dimension count.

######  Lemma 4.3.14.

Assume $Y$ is general. The linear subspaces $\mathbb{P}^{3}\subset Y$ cover a
divisor $D$ in $Y$.

######  Corollary 4.3.15.

A general line $\Delta\subset Y$ meets finitely many $\mathbb{P}^{3}\subset
Y$.

Indeed, if $\Delta\not\subset D$, the $\mathbb{P}^{3}\subset Y$ intersecting
$\Delta$ are in bijection with the intersection points of $\Delta$ and $D$.
Let $F_{3,1}\subset F_{3}(Y)\times F_{1}(Y)$ be the set of pairs $(x,s)$ such
that $P_{x}\cap\Delta_{s}\not=\emptyset$. Let $p:F_{3,1}\rightarrow F_{3}(Y)$
(resp. $q:F_{3,1}\rightarrow F_{1}(Y)$) be the first (resp. second)
projection. The second projection $q:F_{3,1}\rightarrow F_{1}(Y)$ is dominant
by Lemma 4.3.14 and generically finite by Corollary 4.3.15.

######  Lemma 4.3.16.

(i) The morphism $q^{*}:CH_{1}(F_{1}(Y))\rightarrow CH_{1}(F_{3,1})$ is
injective.
(ii) The morphism $p_{*}\circ q^{*}:CH_{1}(F_{1}(Y))_{hom}\rightarrow
CH_{1}(F_{3}(Y))_{hom}$ is zero.

###### Proof.

(i) follows from the fact that $q$ is dominant. For (ii), we observe that
there is a natural birational map

$\pi:\mathcal{P}_{3}\times_{Y}\mathcal{P}_{1}\to F_{3,1},$

where

$\mathcal{P}_{3}\subset F_{3}(X)\times Y,\,\,\mathcal{P}_{1}\subset
F_{1}(X)\times Y$

are the incidence correspondences. The morphism $\pi$ is commutative with the
projections to $F_{3}(Y)$ and to $F_{1}(Y)$. By direct calculations,

$\displaystyle\mathcal{P}_{3}^{*}\circ\mathcal{P}_{1*}$
$\displaystyle=p_{*}\circ\pi_{*}\circ\pi^{*}\circ q^{*}$
$\displaystyle=p_{*}\circ q^{*}$

Thus the morphism $p_{*}\circ q^{*}$ factors through the morphism
$(\mathcal{P}_{1})_{*}:CH_{1}(F_{1}(Y))_{hom}\rightarrow CH_{2}(Y)_{hom}$,
which is zero because $CH_{2}(Y)_{hom}=0$ by [Otw99]. ∎

The variety $F_{3,1}$ admits a rational map $f$ to the projective bundle
$\mathcal{P}_{5}$ over $F_{3}(Y)$ whose fiber over $x\in F_{3}(Y)$ is the set
of $\mathbb{P}^{4}$ containing $P_{x}$. The variety $\mathcal{P}_{5}$ is of
dimension $9$ since it is a $\mathbb{P}^{5}$-bundle over a $4$-fold
$F_{3}(Y)$. To a pair $(P_{x},\Delta_{t})$ of a $\mathbb{P}^{3}$ and a line
which intersect, this map associates $\langle P_{x},\Delta_{t}\rangle$. We
introduce now a desingularization $\tau:\widetilde{F_{3,1}}\rightarrow
F_{3,1}$ on which $f$ desingularizes to a morphism
$\tilde{f}:\widetilde{F_{3,1}}\rightarrow\mathcal{P}_{5}$. We observe now the
following: for each $x\in F_{3}(Y)$ and $4$-dimensional space $P^{\prime}_{x}$
containing $P_{x}$, the intersection $P^{\prime}_{x}\cap Y$ is the union
$P_{x}\cup Q_{x}$ where $Q_{x}$ is a $3$-dimensional quadric intersecting
$P_{x}$ along a $2$-dimensional quadric. The general fiber of $\tilde{f}$ over
$(x,P^{\prime}_{x})$ is birational to the family of lines in the
$3$-dimensional quadric $Q_{x}$. We are now going to prove

######  Proposition 4.3.17.

Let $z\in CH_{1}(\widetilde{F_{3,1}})_{\mathbb{Q}}$ be a $1$-cycle such that
$\tilde{f}_{*}(z)=0\in CH_{1}(\mathcal{P}_{5})$. Then
$\tilde{q}_{*}(z)=\alpha\Delta^{\vee}\in CH_{1}(F_{1}(Y))_{\mathbb{Q}}$ for
some $\alpha\in\mathbb{Q}$.

We will use for this the following general result of Bloch-Srinivas type
[Voi14]:

######  Lemma 4.3.18.

Let $f:Z\to B$ be a surjective projective morphism between algebraic
varieties. Let $B^{0}$ be an open dense subset of $B$ such that
(a) $Z-f^{-1}(B^{0})$ is of codimension at least $2$, and that
(b) Every fiber of $f$ over $B^{0}$ has trivial $CH_{0}$, i.e.,
$CH_{0,hom}=0$.
Let $z\in CH_{1}(Z)_{\mathbb{Q}}$ be a $1$-cycle in $Z$ such that
$f_{*}(z)=0\in CH_{1}(B)_{\mathbb{Q}}$. Then $z$ is supported on the fibers of
$f$ over $B^{0}$. More precisely, there are points $b_{1},\ldots,b_{r}\in
B^{0}$, such that $z$ is $\mathbb{Q}$-rationally equivalent to a $1$-cycle
supported on $f^{-1}(b_{1})\cup\ldots\cup f^{-1}(b_{r})$.

###### Proof.

The case when $\dim B=1$ is rather trivial, so let us assume $\dim B\geq 2$.
Let $W\subset Z$ be a multisection of degree $N$ of $f:Z\to B$. Let
$f_{W}:W\to B$ be the restriction of $f$ to $W$. Since $Z-f^{-1}(B^{0})$ is of
codimension at least $2$, by Chow moving lemma, we may assume that $z$ is
supported on $f^{-1}(B^{0})$. We may also assume that none of the components
of $z$ lies entirely in the branched locus of $f_{W}$. Write
$z=z_{1}+z_{2}+\ldots+z_{s}-z_{s+1}-\ldots-z_{t}$ with $z_{i}$ irreducible
curves in $f^{-1}(B^{0})$. Consider $B_{z_{i}}=f(z_{i})\subset B^{0}$. We may
assume $B_{z_{i}}$ is dimension $1$ since otherwise the component $z_{i}$ is
supported on the fibers of $f$. Let $f_{i}:f^{-1}(B_{z_{i}})\to B_{z_{i}}$ be
the restriction of $f$. Since $\dim B\geq 2$, we may further assume, by Chow
moving lemma, that the map $f_{i}:f^{-1}(B_{z_{i}})\to B_{z_{i}}$ restricted
to $z_{i}$ is a birational map. Since every fiber of $f$ over $B^{0}$ has
trivial $CH_{0}$, the $1$-cycle $z_{i}-\frac{1}{N}f^{-1}_{W}(B_{z_{i}})$ in
$CH_{1}(f^{-1}(B_{z_{i}}))$ restricts to $0$ for the general fiber of
$f_{i}:f^{-1}(B_{z_{i}})\to B_{z_{i}}$, so by the Bloch-Srinivas construction
[Voi03], the cycle $z_{i}-\frac{1}{N}f^{-1}_{W}(B_{z_{i}})$ is supported on
the fibers of $f_{i}$. Summing up the components, we find that the $1$-cycle
$z-\frac{1}{N}f^{*}_{W}f_{*}(z)\in CH_{1}(Z)_{\mathbb{Q}}$ is supported on the
fibers of $f$ over $\bigcup_{i}B_{z_{i}}\subset B^{0}$. However, $f_{*}(z)=0$
by assumption. Hence, $z\in CH_{1}(Z)_{\mathbb{Q}}$ is supported on the fibers
of $f$ over $B^{0}$. ∎

We are going to apply Lemma 4.3.18 to
$\tilde{f}:\widetilde{F_{3,1}}\to\mathcal{P}_{5}$ and to the following open
set $\mathcal{P}_{5}^{0}$ defined as the set parametrizing the pairs
$(L_{3},P_{4})$ such that
(a) either $Q$ is smooth,
(b) or $Q$ is singular at only one point $y$ and $L$ does not contain $y$.
It is clear that the fibers of $\tilde{f}$ over $\mathcal{P}^{0}_{5}$ are
$CH_{0}$ trivial. Our first step is thus to check assumption (a) in Lemma
4.3.18. We prove by a case by case analysis the following

######  Lemma 4.3.19.

$\tilde{f}^{-1}(\mathcal{P}_{5}^{0})\subset\widetilde{F_{3,1}}$ has complement
of codimension $\geq 2$.

###### Proof.

The complement $R$ of $\mathcal{P}^{0}_{5}$ in $\mathcal{P}_{5}$ is stratified
by the following subsets $R_{4},R_{3},R_{2},R_{1}$, where $R_{i}$ parametrizes
$(L,P_{4})\in R$ such that the residual $Q$ is of rank $i$.

Analysis of $R_{4}$. The stratum $R_{4}$ parametrizes
$(L,P_{4})\in\mathcal{P}_{5}$ such that $Q$ is singular at one point
(equivalently, the rank of $Q$ is $4$) and that $L$ contains the singular
point of $Q$. In this case, $Q$ is a cone from its singular point over a
smooth quadric surface.

######  Sublemma 4.3.20.

For any $3$-dimensional linear subspace $L$ contained in $Y$, the set of
$4$-dimensional subspaces $P_{4}\subset\mathbb{P}^{9}$ containing $L$ such
that $Q$ has a single singular point lying on $L$ has codimension at least $2$
in $\mathbb{P}^{5}=${$4$-dimensional subspaces $P_{4}\subset\mathbb{P}^{9}$
containing $L$}.

###### Proof.

Without loss of generality, let us assume
$L=\\{(x_{0}:\ldots:x_{3}:0:\ldots:0)\\}\subset\mathbb{P}^{9}$. The fact that
$L\subset Y$ implies that the defining equation of $Y$ is of the form

$f(x_{0},\ldots,x_{9})=\sum_{i=4}^{9}x_{i}Q_{i}(x_{0},\ldots,x_{9}),$

where $Q_{i}$ is a quadratic polynomial in the variables $x_{0},\ldots,x_{9}$,
for each $i\in\\{4,\ldots,9\\}$. Let
$\mathbf{a}:=(a_{0}:a_{1}:\ldots:a_{5})\in\mathbb{P}^{5}$. Then $\mathbf{a}$
determines a dimension $4$ linear subspace $P_{4}$ containing $L$ as

$P_{4}=\\{(x_{0}:x_{1}:x_{2}:x_{3}:ta_{0},ta_{1}:\ldots:ta_{5}):(x_{0}:x_{1}:x_{2}:x_{3}:t)\in\mathbb{P}^{4}\\}.$

The corresponded residual quadric hypersuface $Q_{\mathbf{a}}$ is thus defined
by

$Q_{\mathbf{a}}(x_{0},x_{1},x_{2},x_{3},t)=\sum_{i=4}^{9}a_{i-4}Q_{i}(x_{0},x_{1},x_{2},x_{3},ta_{0},ta_{1},\ldots,ta_{5}).$

We can identify the quadratic form $Q_{\mathbf{a}}(x_{0},x_{1},x_{2},x_{3},t)$
with a $5\times 5$ symmetric matrix that we still denote as $Q_{\mathbf{a}}$.
Viewed as a function of $\mathbf{a}\in\mathbb{P}^{5}$, the matrix
$Q_{\mathbf{a}}$ is a section of
$\mathrm{Sym}^{2}(\mathcal{O}(1)_{\mathbb{P}^{5}}\oplus\mathcal{O}_{\mathbb{P}^{5}}^{4})\otimes\mathcal{O}_{\mathbb{P}^{5}}(1)$,
and thus the locus of $\mathbf{a}\in\mathbb{P}^{5}$ where $Q_{\mathbf{a}}$
degenerates is a degree $7$ hypersurface in $\mathbb{P}^{5}$. On the other
hand, the quadratic polynomial of $Q_{\mathbf{a}}\cap L$ is
$Q_{\mathbf{a}}(x_{0},x_{1},x_{2},x_{3},t=0)$, which, viewed as a $4\times 4$
symmetric matrix varing with $\mathbf{a}$, is a section of
$\mathrm{Sym}^{2}(\mathcal{O}_{\mathbb{P}^{5}}^{4})\otimes\mathcal{O}_{\mathbb{P}^{5}}(1)$,
so that the degenerate locus of $Q_{\mathbf{a}}\cap L$ is of degree $4$.
Therefore, there exists a point $\mathbf{a}\in\mathbb{P}^{5}$ such that
$Q_{\mathbf{a}}$ is singular whereas $Q_{\mathbf{a}}\cap L$ is smooth.
Therefore, the locus of $\mathbf{a}\in\mathbb{P}^{5}$ such that
$Q_{\mathbf{a}}$ is singular at exactly one point and that this point lies in
$L$ is strictly contained, as a closed subset, in the locus where
$Q_{\mathbf{a}}$ is singular at only one point. The latter locus is of
codimension $1$ in $\mathbb{P}^{5}$ since it is defined by the vanishing of
the determinant of the matrix $Q_{\mathbf{a}}$. Hence, the set of
$4$-dimensional subspaces $P_{4}\subset\mathbb{P}^{9}$ containing $L$ such
that $Q$ has a single singular point lying on $L$ has codimension at least $2$
in $\mathbb{P}^{5}$, as desired. ∎

######  Sublemma 4.3.21.

The fiber of $\tilde{f}:\widetilde{F_{3,1}}\to\mathcal{P}_{5}$ over an element
$(L,P_{4})$ such that the corresponding residual quadric $Q$ has only one
singularity is isomorphic to the union

$\mathcal{P}_{2,1}^{\vee}\cup\mathcal{P}_{2,2}^{\vee},$

where $\mathcal{P}_{2,i}^{\vee}$ ($i=1,2$) is a $\mathbb{P}^{2}$-bundle over
$\mathbb{P}^{1}$, and $\mathcal{P}_{2,1}^{\vee}\cap\mathcal{P}_{2,2}^{\vee}$
is a smooth quadric surface.

###### Proof.

The singular quadric hypersurface $Q$ is a cone with vertex at its only
singular point $y$ over a smooth quadric surface $Q^{\prime}$. Each line in
$Q^{\prime}$, together with $x$ determines a plane, and every line in $Q$ lies
in some of these planes. There are two pencils of lines in $Q^{\prime}$, each
of which induces a pencil of planes in $Q^{\prime}$. Each plane contains a
$\mathbb{P}^{2}$ of lines. Hence, the set of lines in $Q$ is the union of two
$\mathbb{P}^{2}$-bundles over $\mathbb{P}^{1}$. The intersection of these
$\mathbb{P}^{2}$-bundles is the set of lines in $Q$ that pass through the
singular point $y$, which is in turn isomorphic to $Q^{\prime}$. ∎

Sublemma 4.3.20 implies that $R_{4}\subset\mathcal{P}_{5}$ is of codimension
at least $2$. Sublemma 4.3.21 shows that the dimension of the fiber of
$\tilde{f}$ over $R_{4}$ is $3$, the same with that of the general fiber of
$\tilde{f}:\widetilde{F_{3,1}}\to\mathcal{P}_{5}$. Hence, the codimension of
$\tilde{f}^{-1}(R_{4})$ is at least $2$. This complete the analysis of
$R_{4}$.

Analysis of $R_{3}$. The stratum $R_{3}$ parametrizes
$(L,P_{4})\in\mathcal{P}_{5}$ such that $Q$ is of rank $3$. In this case, $Q$
is singular along a line and is a cone with its vertex one of its singular
point over a quadric cone surface.

######  Sublemma 4.3.22.

The stratum $R_{3},R_{2},R_{1}$ is of codimension $3,6,10$, respectively, in
$\mathcal{P}_{5}$. In particular, $R_{1}=\emptyset$.

###### Proof.

Let $\mathcal{E}_{4}$ and $\mathcal{Q}_{6}$ be the tautological subbundle,
respectively, tautological quotient bundle over $F_{3}(Y)$. Then
$\mathcal{P}_{5}$ is nothing but the projectivization of $\mathcal{Q}_{6}$.
Let $\mathcal{H}$ be the Hopf bundle of $\mathbb{P}(\mathcal{Q}_{6})$, which
is a subbundle of $\pi^{*}\mathcal{Q}_{6}$. Let $\mathcal{F}_{5}$ be the
kernel of the composite of the canonical maps
$V_{10}\to\pi_{*}\mathcal{Q}_{6}\to\pi_{*}\mathcal{Q}_{6}/\mathcal{H}$, where
the first map is the universal quotient map of bundles of
$\mathbb{P}(\mathcal{Q}_{6})$. Then there is a natural exact sequence of
vector bundles on $\mathbb{P}(\mathcal{Q}_{6})$:

$0\to\pi^{*}\mathcal{E}_{4}\to\mathcal{F}_{5}\to\mathcal{H}\to 0.$

The defining equation of $Y\subset\mathbb{P}^{9}$ gives a section of the
bundle $\mathrm{Sym}^{3}\mathcal{F}_{5}^{*}$ whose image in
$\pi^{*}\mathrm{Sym}^{3}\mathcal{E}_{4}^{*}$ vanishes since the fibers of
$\mathbb{P}(\mathcal{E}_{4})$ are by definition $3$-dimensional linear
subspaces in $Y$. Hence, the universal residual quadric hypersurface is
defined by a section of the bundle
$\mathrm{Sym}^{2}\mathcal{F}_{5}^{*}\otimes\mathcal{H}^{*}$. Notice that
$\mathrm{Sym}^{2}\mathcal{F}_{5}^{*}\otimes\mathcal{H}^{*}$ is generated by
global sections. Since $Y$ is general, the locus where such a quadratic form
is of rank $\leq r$ is $\binom{6-r}{2}$, for $r\in\\{0,\ldots,4\\}$, as
desired. ∎

######  Sublemma 4.3.23.

The fiber of $\tilde{f}:\widetilde{F_{3,1}}\to\mathcal{P}_{5}$ over a point
$(L,P_{4})$ in $R_{3}$ is of dimension $3$.

###### Proof.

The residual quadric hypersurface $Q$ is singular along a line and is a cone
from one of its singular point over a quadric cone surface. By the same
argument as in Sublemma 4.3.21, the variety of lines in $Q$ is isomorphic to a
$\mathbb{P}^{2}$-bundle over a smooth conic, thus has dimension $3$. ∎

By Sublemma 4.3.22 and Sublemma 4.3.23, the codimension of
$\tilde{f}^{-1}(R_{3})$ is $3$. This completes the analysis of $R_{3}$.

Analysis of $R_{2}$. The stratum $R_{2}$ parametrizes
$(L,P_{4})\in\mathcal{P}_{5}$ such that $Q$ is of rank $2$. In this case, $Q$
is a union of two $3$-dimensional linear subspaces intersecting along a plane.
It is clear that in this case, the variety of lines in $Q$ is isomorphic to
the union of two $\mathrm{Gr}(2,4)$ intersecting along a $\mathbb{P}^{2}$.
Hence, the dimension of fibers of $\tilde{f}$ over $R_{2}$ is $4$. By Sublemma
4.3.22, the codimension of $\tilde{f}^{-1}(R_{2})$ is $5$. This completes the
analysis of $R_{2}$.

Analysis of $R_{1}$. The stratum $R_{1}$ parametrizes
$(L,P_{4})\in\mathcal{P}_{5}$ such that $Q$ is of rank $1$. By Sublemma
4.3.22, this case does not happen.

The case where $Q$ is of rank $0$ does not happen, since in this case, $P_{4}$
is contained in $Y$, which cannot happen if $Y$ is a general cubic eightfold.
∎

######  Lemma 4.3.24.

For any $(L,P_{4})$ in $\mathcal{P}_{5}^{0}$, with associated $3$-dimensional
quadric $Q$, the natural morphism

$CH_{1}(F_{1}(L\cap Q))_{\mathbb{Q}}\to CH_{1}(F_{1}(Q))_{\mathbb{Q}}$

is surjective.

###### Proof.

Recall that after the proof of Lemma 4.3.18, we have divided the points in
$\mathcal{P}^{0}_{5}$ into two cases.

Case (a). In this case, $CH_{1}(F_{1}(Q))_{\mathbb{Q}}=\mathbb{Q}$ since
$F_{1}(Q)$ is a connected Fano manifold. Therefore, $CH_{1}(F_{1}(L\cap
Q))_{\mathbb{Q}}\to CH_{1}(F_{1}(Q))_{\mathbb{Q}}$ has to be surjective.

Case (b). In this case, $Q$ is a cone from its singular point $y$ over a
smooth quadric surface $Q^{\prime}$. By Lemma 4.3.21, the Fano variety of
lines $F_{1}(Q)$ is isomorphic to
$\mathcal{P}_{2,1}^{\vee}\cup\mathcal{P}_{2,2}^{\vee}$ where
$\mathcal{P}_{2,i}^{\vee}$ is a $\mathbb{P}^{2}$-bundle over $\mathbb{P}^{1}$
and the $\mathcal{P}_{2,1}^{\vee}\cap\mathcal{P}_{2,2}^{\vee}$ is a smooth
quadric surface $Q^{\prime}$. The variety $F_{1}(L\cap Q)$ is a disjoint union
of two lines $\ell_{1}$ and $\ell_{2}$, each representing a pencil of lines on
the smooth quadric surface $L\cap Q$. Hence, $CH_{1}(F_{1}(L\cap Q))$ is
generated freely by two cycles $z_{1}$ and $z_{2}$ where $z_{i}$ represents
$\ell_{i}$ for $i=1,2$. Let us analyze $CH_{1}(F_{1}(Q))$. Let us consider
$\mathcal{P}_{2,1}^{\vee}$ which is a $\mathbb{P}^{2}$-bundle over $\ell_{1}$.
Let $p_{1}:\mathcal{P}_{2,1}^{\vee}\to\ell_{1}$ be the corresponding map, and
let $h$ be the auti-tautological class of this projective bundle. The Chow
group $CH_{1}(\mathcal{P}_{2,1}^{\vee})$ is then generated by $h^{2}\cdot
p_{1}^{*}(\ell_{1})$ and $h\cdot p_{1}^{*}(pt)$. The image of the class
$h^{2}\cdot p_{1}^{*}(\ell_{1})$ in $CH_{1}(F_{1}(Q))$ is the same as the
class of $z_{1}$ in $CH_{1}(F_{1}(Q))$ and the image of the class $h\cdot
p_{1}^{*}(pt)$ is the as the class of $z_{2}$ in $CH_{1}(F_{1}(Q))$. Hence,
the image of $CH_{1}(F_{1}(L\cap Q))\to CH_{1}(F_{1}(Q))$ contains the image
of $CH_{1}(\mathcal{P}_{2,1}^{\vee})\to CH_{1}(F_{1}(Q))$. Similarly, the
image of $CH_{1}(F_{1}(L\cap Q))\to CH_{1}(F_{1}(Q))$ contains the image of
$CH_{1}(\mathcal{P}_{2,2}^{\vee})\to CH_{1}(F_{1}(Q))$. Therefore,
$CH_{1}(F_{1}(L\cap Q))\to CH_{1}(F_{1}(Q))$ is surjective. ∎

We finally conclude the proof of Proposition 4.3.17.

###### Proof of Proposition 4.3.17.

Let $z\in CH_{1}(\widetilde{F_{3,1}})_{\mathbb{Q}}$ be a $1$-cycle such that
$\tilde{f}_{*}(z)=0\in CH_{1}(\mathcal{P}_{5})_{\mathbb{Q}}$. Lemma 4.3.19 and
the discussion above shows that Lemma 4.3.18 can be applied to the map
$\tilde{f}:\widetilde{F_{3,1}}\to\mathcal{P}_{5}$ with $\mathcal{P}_{5}^{0}$
the open dense subset, so that we conclude that $z$ is supported on fibers of
$\tilde{f}$ over $\mathcal{P}_{5}^{0}$. Write $z=z_{1}+\ldots+z_{r}\in
CH_{1}(\widetilde{F_{3,1}})_{\mathbb{Q}}$ where $z_{i}$ is supported on