$F_{1}(Q_{i})$ with $Q_{i}$ is a residual quadric hypersurface coming from a
pair $(L_{i},P_{4,i})$ in $\mathcal{P}_{5}^{0}$. By Lemma 4.3.24, for each
$i$, the the natural morphism $CH_{1}(F_{1}(Q_{i}\cap L_{i}))_{\mathbb{Q}}\to
CH_{1}(F_{1}(Q_{i}))_{\mathbb{Q}}$ is surjective. Hence, $\tilde{q}_{*}(z)$
lies in the sum of the images of $CH_{1}(F_{1}(L_{i}))_{\mathbb{Q}}\to
CH_{1}(F_{1}(Y))_{\mathbb{Q}}$. By Corollary 4.3.12, the cycle
$\tilde{q}_{*}(z)$ is a multiple of $\Delta^{\vee}$, as desired. ∎

We next prove that all $\Delta_{P}^{\vee}$ has the same Chow class in
$CH_{1}(F_{1}(Y))_{\mathbb{Q}}$.

######  Proposition 4.3.25.

Let $P\subset Y$ be a plane. Then $\Delta_{P}^{\vee}=\Delta^{\vee}$ in
$CH_{1}(F_{1}(Y))_{\mathbb{Q}}$.

###### Proof.

Since the rational equivalence class is a countable union of closed algebraic
subsets [Voi14, Section 1.1.1], we may assume $P\subset Y$ is general. Let
$P^{\prime}\subset Y$ be another plane, general among all planes in $Y$
intersecting $P$ along a line $l$. By Corollary 4.3.15, there is a
$3$-dimensional linear space $L\subset Y$ such that $l$ intersects $L$ at a
point $y$. As $P$ and $P^{\prime}$ are general, we may assume that both $P$
and $P^{\prime}$ intersect $L$ at only one point $y$. The line of lines in $P$
passing through $y$ naturally lifts to a curve
$Z_{1}\subset\widetilde{F_{3,1}}$ contained in the fiber of
$\pi\circ\tilde{f}$ over the point $l_{3}$ of $F_{3}(Y)$ parameterizing
$L_{3}$. We consider the $1$-cycle $z:=[Z_{1}]-[Z_{1}^{\prime}]\in
CH_{1}(\widetilde{F_{3,1}})_{\mathbb{Q}}$. It is clear that
$\tilde{f}_{*}(z)=0$ in $CH_{1}(\mathcal{P}_{5})_{\mathbb{Q}}$ since
$\tilde{f}_{*}([Z_{1}])$ and $\tilde{f}_{*}([Z_{1}^{\prime}])$ is represented
by two lines in the fiber of the $\mathbb{P}^{5}$-bundle $\mathcal{P}_{5}$
over the point $L\in F_{3}(Y)$. By Proposition 4.3.17, we have
$\tilde{q}_{*}(z)=\alpha\Delta^{\vee}$ in $CH_{1}(F_{1}(Y))_{\mathbb{Q}}$. But
it is clear that $\tilde{q}_{*}([Z_{1}])=\Delta_{P}^{\vee}$ and
$\tilde{q}_{*}([Z_{1}^{\prime}])=\Delta_{P^{\prime}}^{\vee}$, and that
$\alpha=0$ by degree reasons. Hence,
$\Delta_{P}^{\vee}=\Delta_{P^{\prime}}^{\vee}$. There exists a
$P_{3}\cong\mathbb{P}^{3}$ containing $P$ and $P^{\prime}$ and the
intersection $P_{3}\cap Y$ is the union of three planes
$P,P^{\prime},P^{\prime\prime}$. The same argument as above shows that
$\Delta_{P}^{\vee}=\Delta_{P^{\prime}}^{\vee}=\Delta_{P^{\prime\prime}}^{\vee}$.
Finally, Proposition 4.3.7 shows that $\Delta_{P}^{\vee}=\Delta^{\vee}$, as
desired. ∎

We now conclude the proof of Theorem 4.3.6 in this section.

###### Proof of Theorem 4.3.6.

It is not hard to show that $H^{2}(F_{1}(Y),\mathbb{Q})=\mathbb{Q}$ since the
restriction map $H^{2}(\mathrm{Gr}(2,10),\mathbb{Q})\to
H^{2}(F_{1}(Y),\mathbb{Q})$ is an isomorphism [DM98]. It suffices to prove
that $CH_{1}(F_{1}(Y))_{\mathbb{Q},hom}=0$. Let $\alpha\in
CH_{1}(F_{1}(Y))_{\mathbb{Q},hom}$ and let $z=\tilde{q}^{*}\alpha\in
CH_{1}(\widetilde{F_{3,1}})_{\mathbb{Q},hom}$. Since
$\tilde{q}_{*}(z)=\deg\tilde{q}\cdot\alpha$, it suffices to prove that
$\tilde{q}_{*}(z)=0$.

Since $\mathcal{P}_{5}$ is a $\mathbb{P}^{5}$-bundle over $F_{3}(Y)$, we have
$CH_{1}(\mathcal{P}_{5})=h^{5}\cdot\pi^{*}CH_{1}(F_{3}(Y))\oplus
h^{4}\cdot\pi^{*}CH_{0}(F_{3}(Y))$. Lemma 4.3.16 shows that
$\tilde{p}_{*}z=0\in CH_{1}(F_{3}(Y))_{\mathbb{Q},hom}$. Hence,
$\tilde{f}_{*}(z)\in h^{4}\cdot\pi^{*}CH_{0}(F_{3}(Y))_{\mathbb{Q},hom}$.
Write $\tilde{f}_{*}(z)=w_{1}+\ldots+w_{s}$ where $w_{i}$ is a $1$-cycle
supported on the fiber of $\pi$ over a point $L_{i}\in F_{3}(Y)$. Let $P_{i}$
be a plane in $Y$ that intersects with $L_{i}$ at only one point $y_{i}$. Let
$P_{5,i}$ be the $5$-dimensional linear subspace spanned by $L_{i}$ and
$P_{i}$. Let $z_{i}$ be the class in $CH_{1}(\widetilde{F_{3,1}})$ represented
by the variety $Z_{i}:=\\{(L_{i},l):y_{i}\in l\subset P_{i}\\}$. Since the
fibers of $\pi:\mathcal{P}_{5}\to F_{3}(Y)$ are projective spaces
$\mathbb{P}^{5}$, the cycle $w_{i}$ is proportional to the class represented
by the variety $\\{(L_{i},P_{4})\in\mathcal{P}_{5}:P_{4}\subset P_{5,i}\\}$,
which is the image of $Z_{i}$ under $\tilde{f}$. Hence, with an appropriate
choice of coefficients $a_{i}\in\mathbb{Q}$, we have

$\tilde{f}_{*}(z)=\sum_{i}w_{i}=\sum_{i}a_{i}\tilde{f}_{*}z_{i}=\tilde{f}_{*}(\sum_{i}a_{i}z_{i}).$

By Proposition 4.3.17, we conclude that

$\tilde{q}_{*}(z)=\tilde{q}_{*}(\sum_{i}a_{i}z_{i})=\sum_{i}a_{i}\Delta_{P_{i}}^{\vee}=a\Delta^{\vee},$

where $a=\sum_{i}a_{i}$. The last equality is due to Proposition 4.3.25. Since
$\tilde{q}_{*}(z)$ is homologue to $0$, the coefficient $a=0$. Hence,
$\tilde{q}_{*}(z)=0$, as desired. This terminates the proof of Theorem 4.3.6.
∎

####  4.3.3 Proof of Theorem B

We prove in this section Theorem B from the introduction.

######  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 $z\in CH_{0}(X)_{hom}$:

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

Putting together Theorem 4.3.1, Proposition 4.3.3, Proposition 4.3.4 and
Theorem 4.3.6, we conclude that formula (4.11) becomes

$\Psi_{*}z=-8z+\gamma^{\prime}I_{2*}z$ (4.16)

for any cycle $\gamma^{\prime}$ of the form
$ac_{1}^{3}+b^{\prime}c_{1}c_{2}+c^{\prime}c_{3}$ on $X$, where the number $a$
is determined by the class $\gamma$ of (4.11) by
$\gamma=ac_{1}^{3}+bc_{1}c_{2}+cc_{3}$, and $b^{\prime},c^{\prime}$ are
arbitrary. We take for $\gamma^{\prime}$ a multiple of the class of the fixed
locus of $F$ of $\Psi$. Indeed, Proposition 4.3.39 proved in Section 4.3.3
says that the class of $F$ in $\mathrm{CH}^{3}(X)$ has a nonzero coefficient
in $c_{1}^{3}$. Theorem B then follows form

######  Theorem 4.3.26.

The fixed locus $F$ is a constant cycle subvariety in $X$.

Indeed, Theorem 4.3.26 says that the natural morphism $CH_{0}(F)_{hom}\to
CH_{0}(X)_{hom}$ is zero.

######  Remark 4.3.27.

Theorem 4.3.26 had been proved in [Voi08] in the case $r=1$.

##### Proof of Theorem 4.3.26

Let $\mathcal{P}_{2,1}:=\\{(x,l)\in X\times F_{1}(Y):l\subset P_{x}\\}$ be the
incidence variety.

######  Remark 4.3.28.

We have $I_{2}=^{t}\mathcal{P}_{2,1}\circ\mathcal{P}_{2,1}$ as self-
correspondence of $X$.

Let $x\in X=F_{2}(Y)$ be a general point and let $x^{\prime}=\Psi(x)$ where
$\Psi:X\dashrightarrow X$ is the Voisin map. In what follows, we use the
following notation: for a plane $P$, the dual of $P$, defined as the set of
lines in $P$, is denoted by $P^{\vee}$.

######  Proposition 4.3.29.

The cycle $P_{x^{\prime}}^{\vee}-4P_{x}^{\vee}\in
CH_{2}(F_{1}(Y))_{\mathbb{Q}}$ does not depend on the choice of $x\in X$.

Admitting Proposition 4.3.29, we conclude the proof of Theorem 4.3.26.

###### Proof of Theorem 4.3.26.

By Remark 4.3.28, we have

$\mathcal{P}_{2,1}^{*}\circ\mathcal{P}_{2,1*}=I_{2*}:CH_{0}(X)\rightarrow
CH_{3}(X).$

If $x\in F\subset X$, then in the statement of Proposition 4.3.29,
$x^{\prime}=x$ and thus $P_{x}^{\vee}\in CH_{2}(F_{1}(Y))_{\mathbb{Q}}$ is
independent of the choice of $x\in F$. Hence, for any $x_{1},x_{2}\in F$, we
have $\mathcal{P}_{2,1*}(x_{1}-x_{2})=P_{x_{1}}^{\vee}-P_{x_{2}}^{\vee}=0\in
CH_{2}(F_{1}(Y))_{\mathbb{Q}}$. Therefore,
$I_{2*}(x_{1}-x_{2})=\mathcal{P}_{2,1}^{*}\mathcal{P}_{2,1*}(x_{1}-x_{2})=0\in
CH_{0}(X)_{\mathbb{Q}}$. Now, if in Equation (4.11)

$\Psi_{*}z=-8z+\gamma\cdot I_{2*}(z),$

we take $z=x_{1}-x_{2}$, we get $z=-8z\in CH_{0}(X)_{\mathbb{Q}}$. Therefore,
$z=0\in CH_{0}(X)$ as $CH_{0}(X)$ is torsion-free. This implies
$x_{1}=x_{2}\in CH_{0}(X)$. Since $x_{1},x_{2}\in F$ are arbitrarily chosen,
we conclude that $F$ is a constant cycle subvariety. ∎

The rest of this section is devoted to the proof of Proposition 4.3.29. The
proof relies on the geometry of cycles in a cubic fourfold $Y_{4}$ and its
variety of lines $F_{1}(Y_{4})$, which has been studied in [Voi08] and [SV15].
the following relation is established in [Voi08] (see also [SV15]).

######  Theorem 4.3.30 (Voisin [Voi08]).

For a cubic fourfold $Y_{4}\subset\mathbb{P}^{5}$ containing a plane $P$, let
$P^{\vee}\subset F_{1}(Y_{4})$ denote the variety of lines within $P$, and let
$D_{P}\subset F_{1}(Y_{4})$ represent the divisor comprising lines in $Y_{4}$
intersecting $P$. With $l\subset CH^{1}(F_{1}(Y_{4}))$ being the restriction
of the Plücker line bundle class from $\mathrm{Gr}(2,6)$, there exist
constants $\alpha,\beta\in\mathbb{Q}$, and $\gamma\in
CH^{2}(F_{1}(Y_{4}))_{\mathbb{Q}}$, where $\alpha\neq 0$ and $\gamma$ is a
restriction of a class $\delta\in CH^{2}(\mathrm{Gr}(2,6))_{\mathbb{Q}}$ that
is independent of the chose of the plane $P$ and the cubic fourfold $Y_{4}$,
such that:

$P^{\vee}=\alpha D_{P}^{2}+\beta D_{P}\cdot l+\gamma$

within $CH^{2}(F_{1}(Y_{4}))_{\mathbb{Q}}$.

We will partially generalize this relation to the case where $Y_{4}$ has mild
singularities.

######  Corollary 4.3.31.

Consider a cubic hypersurface $Y_{4}\subset\mathbb{P}^{5}$ with at most simple
double points as singularities, containing a plane $P$. Let
$F_{1}(Y_{4})_{sm}$ denote the smooth part of $F_{1}(Y_{4})$. Define
$P^{\vee}\subset F_{1}(Y_{4})_{sm}$, $D_{P}\subset F_{1}(Y_{4})_{sm}$, and
$l\subset CH^{1}(F_{1}(Y_{4})_{sm})$ as in Theorem 4.3.30, but restricted to
the smooth part of $F_{1}(Y_{4})$. There exist constants
$\alpha,\beta\in\mathbb{Q}$, and $\gamma\in
CH^{2}(F_{1}(Y_{4})_{sm})_{\mathbb{Q}}$, with $\alpha\neq 0$ and $\gamma$ as a
restriction of a class $\delta\in CH^{2}(\mathrm{Gr}(2,6))_{\mathbb{Q}}$ that
is independent of the chose of the plane $P$ and the cubic fourfold $Y_{4}$,
such that:

$P^{\vee}=\alpha D_{P}^{2}+\beta D_{P}\cdot l+\gamma$ (4.17)

in $CH^{2}(F_{1}(Y_{4})_{sm})_{\mathbb{Q}}$.

###### Proof.

Let us consider the construction of the family $\mathcal{F}$ over $B$, where
$B$ parametrizes pairs $(P,f)$ with $P$ being a plane in $\mathbb{P}^{5}$ and
$f$ a cubic polynomial such that the hypersurface $Y_{f}$ defined by $f$ has
at most simple double points as singularities, together with the condition
that $f_{|P}=0$. Let

$\mathcal{F}=\\{((P,f),l)\in B\times\mathrm{Gr}(2,6):f|_{l}=0\\}\\}.$

In such a way, we make the Fano variety of lines $F_{1}(Y_{f})$ into family
over $B$.

######  Lemma 4.3.32.

The family $p:\mathcal{F}\to B$ is flat.

###### Proof.

Each fiber of $p$ is a subvariety that is defined as the zero locus of the
vector bundle $\mathrm{Sym}^{3}\mathcal{E}^{*}$ on $\mathrm{Gr}(2,6)$, with
the expected dimension, where $\mathcal{E}$ is the tautological subbundle of
$\mathrm{Gr}(2,6)$. By Koszul’s resolution, each fiber has the same Hilbert
polynomial. This implies that the family is flat. ∎

Within the family $\mathcal{F}$, we have the following subvarieties.
$\mathcal{P}^{\vee}:=\\{((P,f),l)\in B\times\mathrm{Gr}(2,6):l\subset P\\}$
and $\mathcal{D}:=\\{((P,f),l)\in\mathcal{F}:l\cap P\neq\emptyset\\}$,
representing lines within $P$ and lines intersecting $P$, respectively.
Additionally, let $q:\mathcal{F}\to\mathrm{Gr}(2,6)$ be the second projection.
Let $\mathcal{L}=q^{*}\mathcal{O}_{\mathrm{Gr}(2,6)}(1)$ be the pull-back of
the Plücker line bundle. Let $\Gamma=q^{*}\delta\in
CH^{2}(\mathcal{F})_{\mathbb{Q}}$, where $\delta\in
CH^{2}(\mathrm{Gr}(2,6))_{\mathbb{Q}}$ be the constant class as defined in
Theorem 4.3.30. Let $\alpha,\beta\in\mathbb{Q}$ be as in Theorem 4.3.30. We
consider the algebraic cycle
$\mathcal{Z}=\mathcal{P}^{*}-\alpha\mathcal{D}^{2}-\beta\mathcal{D}\cdot\mathcal{L}-\Gamma$.
Theorem 4.3.30 implies that $\mathcal{Z}|_{\mathcal{F}_{t}}=0\in
CH^{2}(\mathcal{F}_{t})_{\mathbb{Q}}$ for $t\in B$ with smooth fibers. By the
specialization of algebraic cycles [Voi15, Proposition 1.4], we conclude that
$\mathcal{Z}|_{\mathcal{F}_{t}}=0\in CH^{2}(\mathcal{F}_{t})$ for all $t\in
B$. For a singular fiber $\mathcal{F}_{t}$, we can restrict further to the
smooth part of $\mathcal{F}_{t}$ and we get the desired result. ∎

######  Remark 4.3.33.

The reason we do not achieve the relation (4.17) for the whole of
$F_{1}(Y_{4})$ is that the divisor $D_{P}$ might encompass the singular locus
of $F_{1}(Y_{4})$, rendering it not a Cartier divisor, hence $D_{P}^{2}$ is
not well-defined. However, upon restriction to the smooth part, all components
are well-defined, and the restriction of $\mathcal{D}^{2}$ equates to
$D_{P}^{2}$.

We will also need the following

######  Lemma 4.3.34.

Let $Y\subset\mathbb{P}^{9}$ be a general cubic eightfold and let $P\subset Y$
be a general plane contained in $Y$. Let $H_{5}$ be a general linear subspace
of $\mathbb{P}^{9}$ containing $P$ such that $H_{5}\cap Y=:Y_{4}$ is a cubic
hypersurface containing the plane $P$. Let $j:F_{1}(Y_{4})\hookrightarrow
F_{1}(Y)$ be the natural inclusion map. Let $D_{P}\subset F_{1}(Y_{4})$ be as
in Theorem 4.3.30. Then the class $j_{*}D_{P}\in CH_{3}(F_{1}(Y))$ is
independent of the choice of the plane $P\subset Y$ and of the linear subspace
$H_{5}\subset\mathbb{P}^{9}$.

###### Proof.

Let $q:\mathcal{P}_{1,0}\to Y$ and $p:\mathcal{P}_{1,0}\to F_{1}(Y)$ be the
universal correspondence of $Y$ and $F_{1}(Y)$. Let $\Sigma_{P}\subset
F_{1}(Y)$ be the variety of lines in $Y$ that intersects the plane $P$. The
class $\Sigma_{P}\in CH^{5}(F_{1}(Y))$ does not depend on the choice of
$P\subset Y$, since $\Sigma_{P}=\mathcal{P}_{1,0}^{*}(P)$ and since
$CH_{2}(Y)=\mathbb{Z}P$ by [Otw99]. Let
$\tilde{\Sigma}_{P}\subset\mathcal{P}_{1,0}$ be the preimage of $P\subset Y$
via $q$. Then similarly, the Chow class of $\tilde{\Sigma}_{P}$ in
$\mathcal{P}_{1,0}$ does not depend on the choice of $P$. Let us define two
vector bundles $\mathcal{E}$ and $\mathcal{H}$ on $\mathcal{P}_{1,0}$ as
follows. $\mathcal{E}$ is the pull-back of the universal subbundle over
$F_{1}(Y)\subset\mathrm{Gr}(2,10)$ via $p$ and $\mathcal{H}$ is the pull-back
of the Hopf bundle over $Y\subset\mathbb{P}^{9}$ via $q$. On
$\tilde{\Sigma}_{P}$, we have a natural inclusion map
$\mathcal{H}|_{\tilde{\Sigma}_{P}}\hookrightarrow\mathcal{E}|_{\tilde{\Sigma}_{P}}$
that induces a sujective morphism of vector bundles
$\phi:\mathcal{E}|_{\tilde{\Sigma}_{P}}^{*}\to\mathcal{H}|_{\tilde{\Sigma}_{P}}^{*}$
that fits into the short exact sequence

$0\to(\det\mathcal{E}^{*}\otimes\mathcal{H})|_{\tilde{\Sigma}_{P}}\to\mathcal{E}|_{\tilde{\Sigma}_{P}}^{*}\to\mathcal{H}|_{\tilde{\Sigma}_{P}}^{*}\to
0.$ (4.18)

Over $\tilde{\Sigma}_{P}$, the defining equations of
$H_{5}\subset\mathbb{P}^{9}$ induces a section $s$ of
$\mathcal{E}^{*}|_{\tilde{\Sigma}_{P}}^{\oplus 4}$ that is zero when projected
to $H^{0}(\tilde{\Sigma}_{P},\mathcal{H}^{*}|_{\tilde{\Sigma}_{P}}^{\oplus
4})$. Hence, we can view $s$ as a section $\sigma_{s}$ of
$(\det\mathcal{E}^{*}\otimes\mathcal{H})|_{\tilde{\Sigma}_{P}}^{\oplus 4}$.
Let $\tilde{D}_{P}$ be the zero locus of $\sigma_{s}$. Then $\tilde{D}_{P}$
parametrizes the pairs $(l,y)\in F_{1}(Y)\times Y$ such that $l\subset H_{5}$
and $l\cap P=y$. By the projection formula, we find that the Chow class of
$\tilde{D}_{P}$ in $\mathcal{P}_{1,0}$ is $\tilde{\Sigma}_{P}\cdot
c_{1}((\det\mathcal{E}^{*}\otimes\mathcal{H}))^{4}$, which is independent of
the choice of $P$ and $H_{5}$. Since $D_{P}=p_{*}\tilde{D}_{P}$, the Chow
class of $D_{P}$ in $F_{1}(Y)$ is independent of the choice of $P$ and $H_{5}$
as well. ∎

For a plane $P\subset Y$, let $P^{\vee}\subset F_{1}(Y)$ be the subvariety of
lines contained in $P$. Let $l\in CH^{1}(F_{1}(Y))$ be the restriction of the
Plücker line bundle class of $\mathrm{Gr}(2,10)$. Let $Y_{4}\subset Y$ be a
linear section of $Y$ that has at most simple double points as singularities.
Let $\Sigma\subset F_{1}(Y_{4})$ be the singular locus of $F_{1}(Y_{4})$ and
let $j:F_{1}(Y_{4})_{sm}\hookrightarrow F_{1}(Y)-\Sigma=:F_{1}(Y)^{0}$ be the
inclusion map. Let $D_{P}$ be defined as in Corollary 4.3.31. Then Corollary
4.3.31 and Lemma 4.3.34 imply the following

######  Corollary 4.3.35.

In $CH_{2}(F_{1}(Y)^{0})_{\mathbb{Q}}$, we have the following relation

$P^{\vee}=\alpha j_{*}(D_{P}^{2})+c|_{F_{1}(Y)^{0}},$

where $\alpha\neq 0$ is a rational number and $c\in
CH_{2}(F_{1}(Y))_{\mathbb{Q}}$ is a Chow class that is independent of the
choice of the plane $P$.

###### Proof of Proposition 4.3.29.

By the definition of the Voisin map, there is a unique linear subspace
$H\subset\mathbb{P}^{9}$ of dimension $3$, such that $H\cap
Y=2P_{x}+P_{x^{\prime}}$. Let $H_{5}\subset\mathbb{P}^{9}$ be a linear
subspace containing $H$ (thus containing both $P_{x}$ and $P_{x^{\prime}}$),
and let $Y_{4}=Y\cap H_{5}$.

######  Lemma 4.3.36.

For a general choice of $H_{5}$, the cubic hypersurface $Y_{4}$ has $4$ simple
double points as the only singularities.

###### Proof.

The base locus of the linear system $L=\\{H_{5}^{\prime}\cap Y:H\subset
H_{5}^{\prime}\subset\mathbb{P}^{9}\\}$ is $P_{x}\cup P_{x^{\prime}}$. Hence,
by Bertini’s theorem, for a general $H_{5}$, the cubic hypersurface
$Y_{4}:=Y\cap H_{5}$ is smooth outside $P_{x}\cup P_{x^{\prime}}$. Let us
write $H_{5}=\\{(x_{0}:x_{1}:\ldots:x_{5})\\}$ and we can assume that
$H=\\{(x_{0}:x_{1}:x_{2}:x_{3}:0:0)\\}$,
$P_{x}=\\{(x_{0}:x_{1}:x_{2}:0:0:0)\\}$ and
$P_{x^{\prime}}=\\{(0:x_{1}:x_{2}:x_{3}:0:0)\\}$. The fact that $H\cap
Y_{4}=2P_{x}+P_{x^{\prime}}$ means that the defining equation of $Y_{4}\subset
H_{5}$ can be written as

$f(x_{0},\ldots,x_{5})=x_{3}^{2}x_{0}+x_{4}Q_{1}(x_{0},\ldots,x_{5})+x_{5}Q_{2}(x_{0},\ldots,x_{5}),$

where $Q_{1}$ and $Q_{2}$ are quadratic polynomials. Let $y\in Y_{4}$ be a
singular point. Since $y\in H$, one must have $x_{4}(y)=x_{5}(y)=0$. Writing
$f_{x_{i}}=\frac{\partial f}{\partial x_{i}}$, we find by direct calculations
that $f_{x_{0}}(y)=x_{3}(y)^{2}$, $f_{x_{1}}(y)=f_{x_{2}}(y)=0$,
$f_{x_{3}}(y)=2x_{0}(y)x_{3}(y)$, $f_{x_{4}}(y)=Q_{1}(y)$ and
$f_{x_{5}}(y)=Q_{2}(y)$. The fact $y\in Y_{4}$ is singular implies that
$f_{x_{i}}(y)=0$ for any $x_{i}$. Taken together, we find that $y\in H_{5}$
satisfies the following system of equations

$\left\\{\begin{array}[]{cc}x_{4}&=0\\\ x_{5}&=0\\\ x_{3}&=0\\\
Q_{1}(x_{0},\ldots,x_{5})&=0\\\
Q_{2}(x_{0},\ldots,x_{5})&=0\end{array}\right.$

By the generality of $Y\subset\mathbb{P}^{9}$ and $H_{5}$, the solutions of
this system of equations are four points in $H_{5}$. To show that the four
singular points are simple double points. We do a local check. Let $y$ be one
of the singular points. Up to a change of coordinates of $P_{x}$, we may
assume without loss of generality that $x_{0}(y)\neq 0$ and
$x_{1}(y)=x_{2}(y)=0$, namely, $y=(1:0:0:0:0:0)\in H_{5}$. On the open affine
subset $U_{0}$ of $H_{5}$ defined by $x_{0}=1$, the cubic hypersurface
$Y_{4}\cap U_{0}$ is defined by the equation
$x_{3}^{2}+x_{4}Q_{1}(1,x_{1},\ldots,x_{5})+x_{5}Q_{2}(1,x_{1},\ldots,x_{5})$.
The Taylor expansion of this polynomial around the point
$y=(0,0,0,0,0)\in\mathbb{A}^{5}\cong U_{0}$ is the polynomial itself. The fact
that this polynomial does not have degree $1$ term is simply because $y$ is a
singular point. To show that $y$ is a double point, we only need to make sure
that the degree $2$ term of the expression
$x_{3}^{2}+x_{4}Q_{1}(1,x_{1},\ldots,x_{5})+x_{5}Q_{2}(1,x_{1},\ldots,x_{5})$
is non-degenerate, and this condition is clearly satisfied for general $Q_{1}$
and $Q_{2}$. ∎

Let $y_{1},y_{2},y_{3},y_{4}$ be the four singular points of $Y_{4}$. Let
$\Sigma\subset F_{1}(Y_{4})$ be the singular locus of $F_{1}(Y_{4})$. Then
$\Sigma$ is the union of four surfaces $\Sigma_{1}$, $\Sigma_{2}$,
$\Sigma_{3}$ and $\Sigma_{4}$, parametrizing the lines in $Y_{4}$ that pass
through the point $y_{1},y_{2},y_{3}$ and $y_{4}$, respectively. Let
$F_{1}(Y)^{0}=F_{1}(Y)-\Sigma$ Let $j:F_{1}(Y_{4})_{sm}\hookrightarrow
F_{1}(Y)^{0}$ be the natural inclusion. By Corollary 4.3.35, we have

$P_{x}^{*}=\alpha j_{*}(D_{P_{x}}^{2})+c^{\prime}|_{F_{1}(Y)^{0}}\in
CH_{2}(F_{1}(Y))_{\mathbb{Q}}$ (4.19)

and

$P_{x^{\prime}}^{*}=\alpha
j_{*}(D_{P_{x^{\prime}}}^{2})+c^{\prime}|_{F_{1}(Y)^{0}}\in
CH_{2}(F_{1}(Y)^{0})_{\mathbb{Q}}$ (4.20)

for some constant $c^{\prime}\in CH_{2}(F_{1}(Y))_{\mathbb{Q}}$. Now let
$\mathcal{P}=\\{(l,y)\in F_{1}(Y_{4})_{sm}\times Y_{4}:y\in l\\}$ be the
incidence correspondence, then for any plane $P\subset Y_{4}$, we have
$D_{P}=\mathcal{P}^{*}(P)$. It is clear that $2P_{x}+P_{x^{\prime}}=h^{2}$ in
$CH_{2}(Y_{4})$, so $2D_{P_{x}}+D_{P_{x^{\prime}}}=\mathcal{P}^{*}(h^{2})=l\in
CH_{3}(F_{1}(Y_{4})_{sm})$. Taken into account of the equations (4.19) and
(4.20), we find that

$P_{x^{\prime}}^{*}=4P_{x}^{*}-4\alpha j_{*}D_{P_{x}}\cdot l+\alpha
l^{2}-3c^{\prime}\in CH_{2}(F_{1}(Y)^{0})_{\mathbb{Q}}.$

Taking into account of Lemma 4.3.34,
$P_{x^{\prime}}^{*}-4P_{x}^{*}=c|_{F_{1}(Y)^{0}}\in
CH_{2}(F_{1}(Y)^{0})_{\mathbb{Q}}$, where $c\in CH_{2}(F_{1}(Y))_{\mathbb{Q}}$
is a constant $2$-cycle on $F_{1}(Y)$. Hence, in $F_{1}(Y)$, the cycle
$P_{x^{\prime}}^{*}-4P_{x}^{*}-c$ is supported on $\Sigma$, the singular locus
of $F_{1}(Y_{4})$. Thus, we need to understand the geometry and Chow classes
of $\Sigma$.

Following Lemma 4.3.36, let $y_{1},y_{2},y_{3},y_{4}$ be the four singular
points of $Y_{4}$. Then $\Sigma$ is the union of four surfaces $\Sigma_{1}$,
$\Sigma_{2}$, $\Sigma_{3}$ and $\Sigma_{4}$, parametrizing the lines in
$Y_{4}$ that pass through the point $y_{1},y_{2},y_{3}$ and $y_{4}$,
respectively. We have the following two lemmas about the geometry of the
surfaces $\Sigma_{i}$ that we will prove later.

######  Lemma 4.3.37.

The surfaces $\Sigma_{1}$, $\Sigma_{2}$, $\Sigma_{3}$ and $\Sigma_{4}$ are
irreducible. In particular,
$CH_{2}(\Sigma)=\oplus_{i=1}^{4}CH_{2}(\Sigma_{i})$.

######  Lemma 4.3.38.

Let $H_{5}\subset\mathbb{P}^{9}$ be a general linear subspace of dimension $5$
such that the linear section $Y_{4}:=Y\cap H_{5}$ has only simple double
singularities and let $y\in Y_{4}$ be a singular point. Let $S$ be the surface
of lines in $Y_{4}$ that pass through the point $y$. Then the Chow class of
$S$ in $F_{1}(Y)$ does not depend on the choice of $H_{5}$ and $y$.

By Lemma 4.3.37 and the fact that $P_{x^{\prime}}-4P_{x}-c$ is supported on
$\Sigma$, we conclude that
$P_{x^{\prime}}-4P_{x}-c=\sum_{i=1}^{4}a_{i}\Sigma_{i}$, with
$a_{i}\in\mathbb{Q}$. The cohomological class of $P_{x^{\prime}}-4P_{x}-c$ is
clearly a constant, thus the cohomological class of
$\sum_{i=1}^{4}a_{i}\Sigma_{i}$ is constant. Lemma 4.3.38 then implies that
the Chow class of $\sum_{i=1}^{4}a_{i}\Sigma_{i}$ is also constant. Therefore,
$P_{x^{\prime}}-P_{x}=c+\sum_{i=1}^{4}a_{i}\Sigma_{i}$ is a constant in
$CH_{2}(F_{1}(Y))_{\mathbb{Q}}$, as desired. This concludes the proof of
Proposition 4.3.29. ∎

###### Proof of Lemma 4.3.37.

To check the irreducibility, we return to the proof of Lemma 4.3.36. Using the
notation there, let

$g(x_{1},\ldots,x_{5})=f(0,x_{1},\ldots,x_{5})=x_{4}Q_{1}(0,x_{1},\ldots,x_{5})+x_{5}Q_{2}(0,x_{1},\ldots,x_{5})$

and let $q(x_{1},\ldots,x_{5})$ be the degree $2$ part of the polynomial
$x_{3}^{2}+x_{4}Q_{1}(1,x_{1},\ldots,x_{5})+x_{5}Q_{2}(1,x_{1},\ldots,x_{5})$.
Then the surface $\Sigma_{1}$ of lines passing through the singular point
$y=(1:0:0:0:0:0)$ is the subvariety in
$\mathbb{P}^{4}=\\{(x_{1}:\ldots:x_{5})\\}$ cut by the equations
$g(x_{1},\ldots,x_{5})=0$ and $q(x_{1},\ldots,x_{5})=0$. From the expression,
we see that $g(x_{1},\ldots,x_{5})$ depends on the coefficients of the terms
$x_{i}x_{j}$ with $1\leq i,j\leq 5$ in $Q(x_{0},x_{1},\ldots,x_{5})$ whereas
$q(x_{1},\ldots,x_{5})$ depends on the coefficients of the terms $x_{0}x_{i}$
for $1\leq i\leq 5$. So the choice of coefficients of $g(x_{1},\ldots,x_{5})$
and $q(x_{1},\ldots,x_{5})$ does not have influence on each other. Now we fix
one choice of smooth $Q=\\{q=0\\}$. Then varying $C=\\{g=0\\}$, the base
points of $C\cap Q$ is the line $(s:t:0:0:0)$ corresponding to the lines on
the plane $P_{x}$ passing through the point $y$. Hence, for a general choice
of $q$ and $g$, the surface $\Sigma_{1}=C\cap Q$ is smooth outside the line
$(s:t:0:0:0)$. Writing
$Q_{1}(x_{0},\ldots,x_{5})=\sum_{i,j=0}^{5}a_{ij}x_{i}x_{j}$ and
$Q_{2}(x_{0},\ldots,x_{5})=\sum_{i,j=0}^{5}b_{ij}x_{i}x_{j}$, the Jacobian
matrix of the polynomials $q(x_{1},\ldots,x_{5})$ and $g(x_{1},\ldots,x_{5})$
at the point $(s:t:0:0:0)$ is

$\begin{pmatrix}0&0&0&a_{01}s+a_{02}t&b_{01}s+b_{02}t\\\
0&0&0&2a_{12}st&2b_{12}st\end{pmatrix}.$

This matrix does not have full rank only if $s=0$ or $t=0$ or
$(a_{01}-b_{01})s+(a_{02}-b_{02})t=0$, corresponding to the three lines
passing through the other three singular points. Therefore, the singular locus
of $\Sigma_{1}$ is of codimension $2$. But a reducible complete intersection
of dimension $\geq 1$ in the projective space always have codimension $1$
singular locus by the Fulton–Hansen connectedness theorem [Ful79]. Hence,
$\Sigma_{1}$ is irreducible, as desired. ∎

###### Proof of Lemma 4.3.38.

Let $H$ be the projective tangent space of $Y$ at point $y$. It is a linear
subspace of dimension $8$ in $\mathbb{P}^{9}$. The linear section $Y_{4}$
being singular at $y$ is equivalent to the relation $H_{5}\subset H$. Let
$F_{1,y}(Y)$ be the variety of lines in $Y$ that passes through $y$. Let
$\mathcal{P}=\\{(l,y)\in F_{1}(Y)\times Y:y\in l\\}$ be the incidence variety.
Then $F_{1,y}(Y)$ can be identified with a subvariety of $\mathcal{P}$ given
by the preimage of the point $y$ under the projection map $q:\mathcal{P}\to
Y$. Since every line in $Y$ passing through $y$ is contained in $H$, the
subvariety $S$ of $F_{1,y}(Y)$ parametrizing the lines that is furthermore
included in $H_{5}$ is given by the zero locus of a section $\sigma$ of the
vector bundle $(\mathcal{E}^{*})^{\oplus 3}$. However, as the point $y$
already lies in $H_{5}$, if we restrict $\sigma$ on $y$ via the following
morphism $(\mathcal{E}^{*})^{\oplus 3}\to\mathcal{O}^{\oplus 3}$, we get zero.
Thus, $S$ can be viewed as the zero locus of a section of
$(\det(\mathcal{E}^{*}))^{3}$, with expected dimension. Therefore, the Chow
class of $S$ in $F_{1}(Y)$ is given by $c_{1}(\mathcal{E}^{*})^{3}\cdot
F_{1,y}(Y)$. Note that $F_{1,y}(Y)=\mathcal{P}^{*}(y)\in CH(F_{1}(Y))$ with
$\mathcal{P}$ the canonical correspondence between $F_{1}(Y)$ and $Y$. Since
$CH_{0}(Y)$ is trivial, the class $F_{1,y}(Y)\in CH(F_{1}(Y))$ is independent
of $y$. In conclusion, the constant Chow class of $S$ in $F_{1}(Y)$ is
independent of the choice of $H_{5}$ and $y$. ∎

##### Chow Class of $F$ in $X$

In this section, we determine the Chow class of $F$ within $X$. Denote by
$\mathcal{F}$ the tautological subbundle on $\mathrm{Gr}(4,10)$ (resp.
$\mathrm{Gr}(r+2,n+1)$ in the general case), and $\mathcal{E}$ as the
tautological subbundle of $X=F_{2}(Y)$ (resp. $X=F_{r}(Y)$ in the general
case). Let $c_{i}$ represent $c_{i}(\mathcal{E}^{*})$ over $X$. In the case
$r=2$, we have

######  Proposition 4.3.39.

The Chow class of $F$ within $X$ is expressed as
$-20c_{1}^{3}+110c_{1}c_{2}+49c_{3}$ in $CH^{3}(X)$.

###### Proof.

Let us start by giving the general method for any $r$ and then specialize to
$r=2$ for explicit calculations. Consider the following stratification of
$X\times\mathrm{Gr}(r+2,n+1)$:

${X\times\mathrm{Gr}(r+2,n+1)}$${\tilde{\tilde{X}}:=\\{(x,P_{r+1})\in
X\times\mathrm{Gr}(r+2,n+1)|P_{r+1}\cap Y\supset
P_{x}\\}}$${\tilde{X}:=\\{(x,P_{r+1})\in
X\times\mathrm{Gr}(r+2,n+1)|P_{r+1}\cap Y\supset
2P_{x}\\}}$${\tilde{F}:=\\{(x,P_{r+1})\in
X\times\mathrm{Gr}(r+2,n+1)|P_{r+1}\cap Y\supset
3P_{x}\\}}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$

Let $pr_{1}:X\times\mathrm{Gr}(r+2,n+1)\to X$ and
$pr_{2}:X\times\mathrm{Gr}(r+2,n+1)\to\mathrm{Gr}(r+2,n+1)$ be the projection
maps. It is not hard to see that
$F=(pr_{1}\circ\gamma\circ\beta\circ\alpha)(\tilde{F})$. We will see shortly
that the Chow class of $\tilde{F}$ in $X\times\mathrm{Gr}(r+2,n+1)$ lies in a
subring generated by the Chern classes of $pr_{1}^{*}\mathcal{E}^{*}$ and
$pr_{2}^{*}\mathcal{F}^{*}$, say

$[\tilde{F}]=\sum_{i}pr_{1}^{*}P_{i}(c_{k}(\mathcal{E}^{*}))\cdot
pr_{2}^{*}Q_{i}(c_{l}(\mathcal{F}^{*}))$ (4.21)

in $CH^{r+1+(n-r-1)(r+2)}(X\times\mathrm{Gr}(r+2,n+1))$, where $P_{i}$ and
$Q_{i}$ are polynomials with adequate degrees. Then the class of $F$ in $X$ is
$pr_{1*}([\tilde{F}])=\sum_{i}P_{i}(c_{k}(\mathcal{E}^{*}))\cdot
pr_{1*}pr_{2}^{*}Q_{i}(c_{l}(\mathcal{F}^{*}))$. In this expression,
$pr_{1*}pr_{2}^{*}Q_{i}(c_{l}(\mathcal{F}^{*}))\neq 0$ only if the weighted
degree of $Q_{i}(c_{l}(\mathcal{F}^{*}))$ is $(n-r-1)(r+2)$, namely the
dimension of $\mathrm{Gr}(r+2,n+1)$ (or equivalently, the weignted degree of
$P_{i}(c_{k}(\mathcal{E}^{*}))$ is $r+1$). In this case, in
$CH_{0}(\mathrm{Gr}(r+2,n+1))$, the class $Q_{i}(c_{l}(\mathcal{F}^{*}))$ is a
multiple of a point $o\in\mathrm{Gr}(r+2,n+1)$. The coefficient, denoted by
$q_{i}$, is calculable using Schubert calculus, once we know the expression of
$Q_{i}(c_{l}(\mathcal{F}^{*}))$. Taken together, the class of $F$ in
$CH^{r+1}(X)$ is given by

$[F]=\sum_{i:\deg P_{i}=r+1}q_{i}P_{i}(c_{k}(\mathcal{E}^{*})).$

Now let us prove (4.21), i.e., $[\tilde{F}]$ is generated by
$pr_{1}^{*}\mathcal{E}^{*}$ and $pr_{2}^{*}\mathcal{F}^{*}$. The subvariety
$\tilde{\tilde{X}}$ of $X\times\mathrm{Gr}(r+2,n+1)$ is defined by the
condition $P_{x}\subset P_{r+1}$. Hence, we can view $\tilde{\tilde{X}}$ as
the locus where the composite map of vector bundles $pr_{1}^{*}\mathcal{E}\to
V_{n+1}\to V_{n+1}/pr_{2}^{*}\mathcal{F}$ is zero, where $V_{n+1}$ is the
trivial bundle of rank $n+1$ over $X\times\mathrm{Gr}(r+2,n+1)$. Hence,
$\tilde{\tilde{X}}$ is the zero locus of a section of
$pr_{1}^{*}\mathcal{E}^{*}\otimes V_{n+1}/pr_{2}^{*}\mathcal{F}$. Now over
$\tilde{\tilde{X}}$, we have a natural injection
$pr_{1}^{*}\mathcal{E}|_{\tilde{\tilde{X}}}\to
pr_{2}^{*}\mathcal{F}|_{\tilde{\tilde{X}}}$. The defining polynomial $f$ of
$Y$ induces a section of
$pr_{2}^{*}\mathrm{Sym}^{3}\mathcal{F}^{*}|_{\tilde{\tilde{X}}}$. Since
$P_{x}\subset Y$ for any $x\in X$, this section vanishes when passing to
$pr_{1}^{*}\mathrm{Sym}^{3}\mathcal{E}^{*}|_{\tilde{\tilde{X}}}$. Hence, the
section is actually a section $\tilde{\tilde{f}}$ of
$pr_{2}^{*}\mathrm{Sym}^{2}\mathcal{F}^{*}|_{\tilde{\tilde{X}}}\otimes(pr_{2}^{*}\mathcal{F}|_{\tilde{\tilde{X}}}/pr_{1}^{*}\mathcal{E}|_{\tilde{\tilde{X}}})^{*}$.
The subvariety $\tilde{X}$ is the locus of $\tilde{\tilde{X}}$ where, when
passing to
$pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*}|_{\tilde{\tilde{X}}}\otimes(pr_{2}^{*}\mathcal{F}|_{\tilde{\tilde{X}}}/pr_{1}^{*}\mathcal{E}|_{\tilde{\tilde{X}}})^{*}$,
the section $\bar{\tilde{\tilde{f}}}$ is zero. Hence, $\tilde{X}$ is the zero
locus of a section of the vector bundle
$pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*}|_{\tilde{\tilde{X}}}\otimes(pr_{2}^{*}\mathcal{F}|_{\tilde{\tilde{X}}}/pr_{1}^{*}\mathcal{E}|_{\tilde{\tilde{X}}})^{*}$.
Similar argument shows that $\tilde{F}$ is the zero locus of a section of the
vector bundle
$pr_{1}^{*}\mathcal{E}^{*}|_{\tilde{X}}\otimes(pr_{2}^{*}\mathcal{F}|_{\tilde{X}}/pr_{1}^{*}\mathcal{E}|_{\tilde{X}})^{*\otimes
2}$. Taken together, the class of $\tilde{F}$ in
$CH(X\times\mathrm{Gr}(r+2,n+1))$ is given by the following class

$e(pr_{1}^{*}\mathcal{E}^{*}\otimes(V_{n+1}-pr_{2}^{*}\mathcal{F}))\cdot
e((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})\otimes
pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*})\cdot
e((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})^{\otimes 2}\otimes
pr_{1}^{*}\mathcal{E}^{*})$

where $e(...)$ is the Euler class and we have written the vector bundles as
their class in K groups to avoid any confusions. This gives an expression of
the class $\tilde{F}$ as generated by the Chern classes of
$pr_{1}^{*}\mathcal{E}^{*}$ and $pr_{2}^{*}\mathcal{F}^{*}$.

In what follows, we do the explicit calculations in the case $r=2$, with the
help of Mathematica and SageMath.

##### Calculations of
$e(pr_{1}^{*}\mathcal{E}^{*}\otimes(V_{10}-pr_{2}^{*}\mathcal{F}))$

We may assume that $pr_{1}^{*}\mathcal{E}^{*}$ is formally a direct sum of
three line bundles whose classes are $l_{1},l_{2},l_{3}$ and that
$V_{10}/pr_{2}^{*}\mathcal{F}$ is formally a direct sum of
$m_{1},\ldots,m_{6}$. Then the Euler class of
$pr_{1}^{*}\mathcal{E}^{*}\otimes V_{10}/pr_{2}^{*}\mathcal{F}$ is
$\prod_{i,j}(l_{i}+m_{j})$. The Euler class of
$pr_{1}^{*}\mathcal{E}^{*}\otimes V_{10}/pr_{2}^{*}\mathcal{F}$ expressed by
$c_{i}$, the Chern classes of $pr_{1}^{*}\mathcal{E}^{*}$, and by $d_{j}$, the
Chern classes of $V_{10}/pr_{2}^{*}\mathcal{F}$ is calculated, using
Mathematica, in Appendix A.1.1. The result is tremendously long. Fortunately,
we do not really need the full form of the result. As our general method
indicates, we only need to expand the Chern classes of
$pr_{1}^{*}\mathcal{E}^{*}$ up to degree $3$. With this in mind, the Euler
class of $pr_{1}^{*}\mathcal{E}^{*}\otimes(V_{10}-pr_{2}^{*}\mathcal{F})$ is

        d6^3 + c1 d5 d6^2 + c1^2 d4 d6^2 + c2 (d5^2 d6 - 2 d4 d6^2) +
        c1 c2 (d4 d5 d6 - 3 d3 d6^2) + c1^3 d3 d6^2 +
        c3 (d5^3 - 6 d3 d6^2 + 3 (-d4 d5 d6 + 3 d3 d6^2))+ higher terms in ci

##### Calculations of
$e((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})\otimes
pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*})$ and
$e(((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})^{\otimes 2}\otimes
pr_{1}^{*}\mathcal{E}^{*})$

Let $C_{i}=c_{i}(pr_{2}^{*}\mathcal{F}^{*})$ be the Chern classes of
$pr_{2}^{*}\mathcal{F}^{*}$ and let
$S_{i}=c_{i}(pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*})$. Then $S_{i}$ can be
represented by the $c_{i}$, the Chern classes of $pr_{1}^{*}\mathcal{E}^{*}$
as follows.

        S1[c1_, c2_, c3_] := 4 c1;
        S2[c1_, c2_, c3_] := 5 c1^2 + 5 c2;
        S3[c1_, c2_, c3_] := 2 c1^3 + 11 c1 c2 + 7 c3;

It is a general fact [Ful84, Example 3.2.2] that if $L$ is a line bundle and
$V$ is a vector bundle of rank $s$, then

$e(L\otimes V)=\sum_{i=0}^{s}c_{1}(L)^{i}c_{s-i}(V).$

Since
$c_{1}(pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})=c_{1}(pr_{2}^{*}\mathcal{F}^{*})-c_{1}(pr_{1}^{*}\mathcal{E}^{*})$,
the class $e((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})\otimes
pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*})$ is given by the expression

        (C1 - c1)^6 + (C1 - c1)^5 S1[c1, c2, c3] + (C1 - c1)^4 S2[c1, c2, c3]
        + (C1 - c1)^3 S3[c1, c2, c3]+ higher terms in ci

and the class
$e(((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})^{\otimes 2}\otimes
pr_{1}^{*}\mathcal{E}^{*})$ is given by

        8 (C1 - c1)^3 + 4 (C1 - c1)^2 c1 + 2 (C1 - c1) c2 + c3.

##### Calculation of $[F]$

Now we can calculate
$e(pr_{1}^{*}\mathcal{E}^{*}\otimes(V_{n+1}-pr_{2}^{*}\mathcal{F}))\cdot
e((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})\otimes
pr_{1}^{*}\mathrm{Sym}^{2}\mathcal{E}^{*})\cdot
e(((pr_{2}^{*}\mathcal{F}^{*}-pr_{1}^{*}\mathcal{E}^{*})^{\otimes 2}\otimes
pr_{1}^{*}\mathcal{E}^{*})$ (up to degree $3$ in $c_{i}$). The detailed
Mathematica calculations and results are presented in Appendix A.1.2. We only
need the expression up to degree $3$ in $c_{i}$. The expression of
$[\tilde{F}]$ is

        c1^3 (8 C1^9 d3 d6^2 - 36 C1^8 d4 d6^2 + 56 C1^7 d5 d6^2
                - 20 C1^6 d6^3)+
        c3 (8 C1^9 d5^3 - 24 C1^9 d4 d5 d6 + 24 C1^9 d3 d6^2 + 57 C1^6 d6^3)+
        c1 c2 (8 C1^9 d4 d5 d6 - 36 C1^8 d5^2 d6 - 24 C1^9 d3 d6^2 +
                72 C1^8 d4 d6^2 + 42 C1^7 d5 d6^2 - 178 C1^6 d6^3)+
        + higher terms in ci.

Now we need to calculate the coefficients in the above expression. To this
ends, we do the Schubert calculus with the help of SageMath. We replace
$C_{i}$ by $s[1,...,1]$ with $i$ copies of $1$ and replace $d_{i}$ by
$(-1)^{i}s[i]$, where $s[1,...,1]$ and $s[i]$ are the Schur’s polynomials with
the corresponding weights. We do the Schubert calculus using Schur’s
polynomials and find the coefficients of $s[6,6,6,6]$. The output result of
the SageMath is exactly $-20c_{1}^{3}+110c_{1}c_{2}+49c_{3}$, as desired. ∎

######  Remark 4.3.40.

Using the same argument (but with much simpler calculations), we find that for
the Fano variety of lines of cubic fourfold, the fixed locus of the Voisin map
is $21c_{2}$—a result that coincides with the result of [GK20, Theorem A].

###  4.4 Indeterminacy Locus of the Voisin Map

In this section, we explore the indeterminacy locus of the Voisin map
$\Psi:X\dashrightarrow X$. As previously discussed in Section 4.1.3, the
indeterminacy locus comprises two components, described as follows. The first
component, $\mathrm{Ind}_{0}$, parametrizes points $x\in X$ that represent
$P_{x}\subset Y$, where there are more than one linear subspaces of dimension
$r+1$ tangent to $Y$ at $P_{x}$. The second component, $\mathrm{Ind}_{1}$,
includes points $x\in X$ representing $P_{x}\subset Y$, where an
$(r+1)$-dimensional linear space both contains $P_{x}$ and is contained within
$Y$. The main purpose of this section is to prove Theorem C.

####  4.4.1 Proof of Theorem C

Let us analyse the action of the Voisin map $\Psi$ on the Chern classes of $X$
and on the divisor classes of $X$ respectively. Note that for $r\geq 2$,
$CH^{1}(X)$ is generated by only one divisor $h$, which is the restriction of
Plücker line bundle on $\mathrm{Gr}(r+1,n+1)$.

######  Lemma 4.4.1.

For any $r\geq 1$, we have $\Psi^{*}h=(3r+4)h$.

###### Proof.

Let $\mathcal{E}$ be the tautological subbundle of
$X\subset\mathrm{Gr}(r+1,n+1)$. This is a rank $r+1$ subbundle of the trivial
vector bundle $V_{n+1}\otimes\mathcal{O}_{X}$. Let $\mathcal{F}$ be the kernel
of the following morphism of $\mathcal{O}_{X}$-modules

$\begin{array}[]{cccc}\mathcal{G}:&V_{n+1}\otimes\mathcal{O}_{X}&\to&\mathcal{S}ym^{2}\mathcal{E}^{*}\\\
&(a_{0},\ldots,a_{n})&\mapsto&(\sum_{i}a_{i}\frac{\partial f}{\partial
x_{i}}|_{P_{x}})_{x\in X}.\end{array}$

The codimension of $\mathrm{Ind}_{0}\subset X$ is $2$. Hence, the result will
not change if we replace $X$ by $X^{\prime}:=X-\mathrm{Ind}_{0}$. We use the
same notations for the $\mathcal{O}_{X}$-modules restricted to $X^{\prime}$.
Then $\mathcal{F}$ becomes a vector bundle of rank $r+2$ on $X^{\prime}$ which
fits in the following short exact sequence

$0\to\mathcal{F}\to
V_{n+1}\otimes\mathcal{O}_{X}\to\mathrm{Sym}^{2}\mathcal{E}^{*}\to 0.$ (4.22)

From this short exact sequence, we deduce that

$c_{1}(\mathcal{F})=-(r+2)h.$ (4.23)

The homogeneous polynomial $f$ induces a section $\sigma_{f}$ of the vector
bundle $\mathrm{Sym}^{3}\mathcal{F}^{*}$, and the fact that
$\mathbb{P}(\mathcal{F}|_{x})$ is the $(r+1)$-space that is tangent to $Y$
along $P_{x}$ shows that $\sigma_{f}$ can be viewed as an element in
$H^{0}(X^{\prime},(\mathcal{F}/\mathcal{E})^{*}\otimes(\mathcal{F}/\mathcal{E})^{*}\otimes\mathcal{F}^{*})$
via the following short exact sequences

$0\to(\mathcal{F}/\mathcal{E})^{*}\otimes\mathrm{Sym}^{2}\mathcal{F}^{*}\to\mathrm{Sym}^{3}\mathcal{F}^{*}\to\mathrm{Sym}^{3}\mathcal{E}^{*}\to
0,$
$0\to(\mathcal{F}/\mathcal{E})^{*}\otimes(\mathcal{F}/\mathcal{E})^{*}\otimes\mathcal{F}^{*}\to(\mathcal{F}/\mathcal{E})^{*}\otimes\mathrm{Sym}^{2}\mathcal{F}^{*}\to(\mathcal{F}/\mathcal{E})^{*}\otimes\mathrm{Sym}^{2}\mathcal{E}^{*}\to
0.$

Therefore, $\sigma_{f}$ induces a morphism
$\phi:\mathcal{F}\to(\mathcal{F}/\mathcal{E})^{*}\otimes(\mathcal{F}/\mathcal{E})^{*}$.
Then the locus of $X$ where $\phi$ is not surjective is $\mathrm{Ind}_{1}$. By
Proposition 4.4.5 proved in Section 4.4.2, the codimension of
$\mathrm{Ind}_{1}$ in $X$ is $r+2$. Therefore, the result will not change if
we replace $X^{\prime}$ by
$X^{\prime\prime}=X-\mathrm{Ind}_{0}-\mathrm{Ind}_{1}$ and we use the same
notations for the restrictions of $\mathcal{O}_{X}$-modules to
$X^{\prime\prime}$. On $X^{\prime\prime}$, the vector bundle
$\Psi^{*}\mathcal{E}$ is exactly the kernel of $\phi$ and
$\Psi^{*}\mathcal{E}$ fits into the following short exact sequence

$0\to\Psi^{*}\mathcal{E}\to\mathcal{F}\to(\mathcal{F}/\mathcal{E})^{*}\otimes(\mathcal{F}/\mathcal{E})^{*}\to
0.$

Therefore,
$\Psi^{*}c_{1}(\mathcal{E})=c_{1}(\Psi^{*}\mathcal{E})=c_{1}(\mathcal{F})-2c_{1}((\mathcal{F}/\mathcal{E})^{*})=-(r+2)h-2(-(-(r+2)h+h))=-(3r+4)h$.
Since $c_{1}(\mathcal{E})=-h$, we find that $\Psi^{*}h=(3r+4)h$, as desired. ∎

######  Remark 4.4.2.

When $r=1$ and $n=5$, $Y$ is a cubic fourfold and $X$ is a hyper-Kähler
fourfold. Our result recovers the result of [Ame09] and [SV15, Proposition
21.4] which states $\Psi^{*}h=7h$.

######  Lemma 4.4.3.

Let $X^{0}=X-\mathrm{Ind}$ be the locus where $X$ is defined. We have
$(\Psi^{*}T_{X})|_{X^{0}}=T_{X^{0}}$.

###### Proof.

This result has been implicitly proven in [Voi04]. Let $\tau:\tilde{X}\to X$,
$\tilde{\Psi}:\tilde{X}\to X$ be the desingularisation of the indeterminacies
of the Voisin map $\Psi:X\dashrightarrow X$. Since $X$ is a $K$-trivial
variety, the exceptional divisor of $\tau$ coincides with the ramification
locus of $\tilde{\Psi}$ (see [Voi04, Lemma 4]). Therefore, the Voisin map
restricted to the defined domain $\Psi|_{X^{0}}:X^{0}\to X$ is étale, which
implies that the relative tangent bundle $T_{X^{0}/X}$ by the map
$\Psi|_{X^{0}}$ is zero. Hence the result. ∎

###### Proof of Theorem C.

We only consider the case $r=2$. Let $M\in CH_{0}(X)_{hom}$ be a class
belonging to the subring generated by $h$ and by $c_{i}(X)$. Let us assume $M$
is a monomial of the form $h^{k}\prod c_{i}(X)$. By Lemma 4.4.1 and Lemma
4.4.3, we have

$(\Psi^{*}M)|_{X^{0}}=10^{k}\cdot M|_{X^{0}},$ (4.24)

where $X^{0}=X-\mathrm{Ind}$ is the locus where $\Psi$ is defined. On the
other hand, by Theorem B, we have

$\Psi^{*}M=-8M.$ (4.25)

By (4.24) and (4.25), we find that $M|_{X^{0}}=0$, which means that $M$ is
supported on $\mathrm{Ind}$, as desired. ∎

####  4.4.2 Some Numerical Data

######  Proposition 4.4.4.

The first component, $\mathrm{Ind}_{0}$, has a codimension of $2$ in $X$, and
its Chow class in $X$ is expressed as:

$\left(\frac{1}{2}(r+2)(r+1)+2\right)c_{1}^{2}-(r+4)c_{2}.$

###### Proof.

With the same notations as in the proof of Lemma 4.4.1 in Section 4.4.1, Then
at point $x\in X$, the projectivation of the vector space $\mathcal{F}|_{x}$
is the intersection of the projective tangent spaces $T_{Y,y}$ for all $y\in
P_{x}$. We conclude that $\mathrm{Ind}_{0}$ is the locus in $X$ where
$\mathcal{G}$ is not surjective. The map $\mathcal{G}$ sends the subbundle
$\mathcal{E}$ to $0$. Hence, $\mathcal{G}$ factorizes through the following
morphism

$\bar{\mathcal{G}}:(V_{n+1}\otimes\mathcal{O}_{X})/\mathcal{E}\to\mathrm{Sym}^{2}\mathcal{E}^{*}.$

For $Y$ general, the locus where $\bar{\mathcal{G}}$ is not surjective is of
codimension
$\mathrm{rank}((V_{n+1}\otimes\mathcal{O}_{X})/\mathcal{E})-\mathrm{rank}(\mathrm{Sym}^{2}\mathcal{E}^{*})+1$,
which equals $2$ with the relation $n+1=\binom{r+3}{2}$ taken into account.

Now let us calculate the Chow class of $\mathrm{Ind}_{0}$ in $X$. We have
viewed $\mathrm{Ind}_{0}$ as the locus where the map
$\bar{\mathcal{G}}:(V_{n+1}\otimes\mathcal{O}_{X})/\mathcal{E}\to\mathrm{Sym}^{2}\mathcal{E}^{*}$
is not surjective. By the Porteous formula (see the Introduction part of
[HT84]), the class of $\mathrm{Ind}_{0}$ is represented by the degree $2$ part
of

$\frac{c((V_{n+1}\otimes\mathcal{O}_{X})/\mathcal{E})}{c(\mathrm{Sym}^{2}\mathcal{E}^{*})}=\frac{1}{c(\mathcal{E})c(\mathrm{Sym}^{2}\mathcal{E}^{*})}.$

Suppose $\mathcal{E}^{*}$ formally splits into $l_{1},\ldots,l_{r+1}$. Then

$c(\mathrm{Sym}^{2}\mathcal{E}^{*})=\prod_{1\leq i\leq j\leq
r+1}(1+l_{i}+l_{j}).$

The degree $1$ part of this expression is given by $\sum_{1\leq i\leq j\leq
r+1}(l_{i}+l_{j})=(r+2)c_{1}$. The degree $2$ part of this expression is given
by
$\frac{1}{2}\sum_{(i,j)\neq(k,l)\in\Delta_{r+1}}(l_{i}+l_{j})(l_{k}+l_{l})$,
where $\Delta_{r+1}=\\{(i,j):1\leq i\leq j\leq r+1\\}$. We calculate

$\displaystyle\sum_{(i,j)\neq(k,l)\in\Delta_{r+1}}(l_{i}+l_{j})(l_{k}+l_{l})$
$\displaystyle=$
$\displaystyle\sum_{(i,j),(k,l)\in\Delta_{r+1}}(l_{i}+l_{j})(l_{k}+l_{l})-\sum_{(i,j)\in\Delta_{r+1}}(l_{i}+l_{j})^{2}$
$\displaystyle=$
$\displaystyle(\sum_{(i,j)\in\Delta_{r+1}}(l_{i}+l_{j}))^{2}-\sum_{(i,j)\in\Delta_{r+1}}(l_{i}^{2}+l_{j}^{2})-2\sum_{(i,j)\in\Delta_{r+1}}l_{i}l_{j}$
$\displaystyle=$
$\displaystyle(r+2)^{2}c_{1}^{2}-(r+4)\sum_{i}l_{i}^{2}-2\sum_{i<j}l_{i}l_{j}$
$\displaystyle=$
$\displaystyle(r+2)^{2}c_{1}^{2}-(r+4)(c_{1}^{2}-2c_{2})-2c_{2}$
$\displaystyle=$ $\displaystyle(r^{2}+3r)c_{1}^{2}+2(r+3)c_{2}.$

Hence,

$c(\mathrm{Sym}^{2}\mathcal{E}^{*})=1+(r+2)c_{1}+\frac{1}{2}(r^{2}+3r)c_{1}^{2}+(r+3)c_{2}+\ldots$

On the other hand, we have $c(\mathcal{E})=1-c_{1}+c_{2}+\ldots$. Finally, by
a formal calculation, we find that the degree $2$ part of the expression
$\frac{1}{c(\mathcal{E})c(\mathrm{Sym}^{2}\mathcal{E}^{*})}$ is given by
$(\frac{1}{2}(r+2)(r+1)+2)c_{1}^{2}-(r+4)c_{2}$, as desired. ∎

######  Proposition 4.4.5.

The second component, $\mathrm{Ind}_{1}$, has a codimension of $r+2$ in $X$
for $r\geq 2$. For $r\leq 1$, the set $\mathrm{Ind}_{1}$ is empty.

###### Proof.

Recall that $\mathrm{Ind}_{1}$ is described as the locus of points $x\in X$
such that there is an $r+1$ space in $Y$ containing $P_{x}$. For $r\geq 2$,
the variety $F_{r+1}(Y)$ is nonempty. There is a natural incidence variety

$\mathcal{P}_{r+1,r}:=\\{(\lambda,x)\in F_{r+1}(Y)\times
F_{r}(Y):P_{\lambda}\supset P_{x}\\}.$

The variety $\mathcal{P}_{r+1,r}$ is a $\mathbb{P}^{r+1}$-bundle over
$F_{r+1}(Y)$, so $\dim\mathcal{P}_{r+1,r}=\dim F_{r+1}(Y)+(r+1)$. It is clear
that $\mathrm{Ind}_{1}$ is the image of the second projection map
$\mathcal{P}_{r+1,r}\to F_{r}(Y)$. Hence,
$\mathrm{codim}\,\mathrm{Ind}_{1}=\dim F_{r}(Y)-\dim F_{r+1}(Y)-(r+1)=r+2$.
For $r\leq 1$, the variety $F_{r+1}(Y)$ is empty. ∎

####  4.4.3 Discussions on Question 4.1.17

In this part, we would like to give some evidence on Question 4.1.17. Let
$X=F_{r}(Y)$ be the strict Calabi-Yau manifold constructed as in Section 4.1.1
for any $r\geq 2$. Let $\mathcal{P}=\\{(x,y)\in X\times Y:y\in P_{x}\\}$ be
the incidence correspondence of $X$ and $Y$. Let $\tau:\tilde{X}\to X$,
$\tilde{\Psi}:\tilde{X}\to X$ be the desingularization of the indeterminacy
locus of the Voisin map $\Psi:X\dashrightarrow X$.

######  Proposition 4.4.6.

If the induced map

$\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{r+1}(Y)_{hom}$

is injective, then the divisor $\tilde{\Psi}(\tau^{-1}(\mathrm{Ind}_{0}))$ is
a constant-cycle subvariety in $X$.

###### Proof.

Let $x\in Ind_{0}$ be a general point representing an $r$-linear subspace
$P_{x}$ in $Y$. Then there is a unique $(r+2)$-linear subspace $\Theta_{x}$
that is tangent to $Y$ along $P_{x}$. Then the preimage $\tau^{-1}(x)$ is a
rational curve $\mathbb{P}^{1}_{x}$ parametrizing the $(r+1)$-linear subspaces
$H$ containing $P_{x}$ and contained in $\Theta_{x}$. Each $H$ intersects $Y$
with a double $P_{x}$ and a residual $r$-linear subspace $P_{x^{\prime}}$
represented by a point $x^{\prime}\in X$. The image
$\Gamma_{x}:=\tilde{\Psi}(\mathbb{P}^{1}_{x})$ is the rational curve in $X$
parametrizing all these residual planes $P_{x^{\prime}}$. Applying the
correspondence $\mathcal{P}_{*}$ on $\Gamma_{x}$, it is not hard to see that
$\mathcal{P}_{*}(\Gamma_{x})$ is the algebraic cycle represented by
$\Theta_{x}\cap Y$, and that this class is independent of the choice of
$x\in\mathrm{Ind}_{0}$ since it is a linear section of $Y$. By our assumption
that $\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{r+1}(Y)_{hom}$, the Chow classes
of the rational curves $\Gamma_{x}\in X$ are independent of the choice of
$x\in\mathrm{Ind}_{0}$. Notice that the divisor
$\tilde{\Psi}(\tau^{-1}(\mathrm{Ind}_{0}))$ is the closure of the union of
these rational curves $\Gamma_{x}$. Let $y_{x}\in\Gamma_{x}$ and
$y_{x^{\prime}}\in\Gamma_{x^{\prime}}$ and let $D\subset X$ be an ample
divisor such that $d=\deg(D.\Gamma_{x})=\deg(D.\Gamma_{x^{\prime}})$. Then
$d.y_{x}=D.\Gamma_{x}=D.\Gamma_{x^{\prime}}=d.y_{x^{\prime}}$. Since
$CH_{0}(X)_{hom}$ is torsion free, we find that $y_{x}=y_{x^{\prime}}$ in
$CH_{0}(X)$, as desired. ∎

Now let us focus on the case $r=2$.

######  Proposition 4.4.7.

In the case $r=2$, if the induced map

$\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{3}(Y)_{hom}$

is injective, then the first component $\mathrm{Ind}_{0}$ of the indeterminacy
locus is a constant-cycle subvariety.

###### Proof.

In the case $r=2$, Theorem B implies that
$\Psi_{*}(x_{1}-x_{2})=-8(x_{1}-x_{2})$. But Proposition 4.4.6 implies that
$\Psi_{*}(x_{1}-x_{2})=0$ since the divisor
$\tilde{\Psi}(\tau^{-1}(\mathrm{Ind}_{0}))$ is a constant-cycle subvariety in
$X$. Therefore, $x_{1}=x_{2}\in CH_{0}(X)$ since $CH_{0}(X)$ is torsion-free.
∎

######  Remark 4.4.8.

If we assume Conjecture 4.1.12, and if we suppose that the induced map

$\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{r+1}(Y)_{hom}$

is injective, then using the same argument as in Proposition 4.4.7, we can
actually show that $\mathrm{Ind}_{0}$ is a constant-cycle subvariety for any
$r\geq 2$. Notice that $\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{2}(Y)_{hom}$ is
_not_ injective in the hyper-Kähler case, since in that case, we know that
$CH_{2}(Y)_{hom}=0$ (see [BS83, Theorem 1 (ii)]) and [Huy24, Section 5.0.1]
while $CH_{1}(F_{1}(Y))_{hom}$ is big (see [SV15, Theorem 21.9]). This
phenomenon is also reflected by the fact that there is not any constant-cycle
divisor in a hyper-Kähler manifold of dimension $\geq 4$ (see [Voi16]) and
that $\mathrm{Ind}_{0}$ is not a constant-cycle subvariety in this case (see
[Ame09, Lemma 1], [Voi04]).

In the spirit of Proposition 4.4.6 and Proposition 4.4.7, it is reasonable to
propose the following

######  Question 4.4.9.

If $r\geq 2$, is the induced map

$\mathcal{P}_{*}:CH_{1}(X)_{hom}\to CH_{r+1}(Y)_{hom}$

injective?

In alignment with the principles of the generalized Bloch conjecture and the
generalized Hodge conjecture, the Chow group of $1$-cycles on a smooth
projective variety $X$ is expected to be “governed" by the quotient
$N^{1}/N^{2}H^{*}(X,\mathbb{Q})$, where $N^{i}$ denotes the Hodge coniveau
filtration on the cohomology $H^{*}(X,\mathbb{Q})$. Given that $X$ is a strict
Calabi-Yau manifold, the derivative of the period map

$\mathcal{D}:\mathrm{Def}(X)\to\mathbb{P}H^{N}(X,\mathbb{C}),$

that associates any point $b\in\mathrm{Def}(X)$ with the $1$-dimensional
subspace $H^{N,0}(X_{b})\subset H^{N}(X_{b},\mathbb{C})\cong
H^{N}(X,\mathbb{C})$, has as its image

$\mathrm{Hom}(H^{N,0}(X),H^{N-1,1}(X))\subset\mathrm{Hom}(H^{N,0}(X),H^{N}(X,\mathbb{C})/H^{N-1,1}(X)),$

according to Griffiths’ theory on period maps. Question 4.4.9 would likely
have an affirmative answer if the following conditions are met:

  * •

The map
$F_{r}:\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))\dashrightarrow\mathrm{Def}(X)$
is dominant, implying that for a general $Y$, the Fano variety of $r$-linear
spaces within $Y$, denoted by $X$, is also general in the moduli space of $X$.
Consequently, $N^{1}/N^{2}H^{N}(X,\mathbb{Q})=0$.

  * •

The induced map $[P]^{*}:H^{p+r,1+r}(Y)\to H^{p,1}(X)$ is surjective for
$1<p<N$. Considering that $[P]^{*}$ is injective (refer to, for example,
[Voi14, Lemma 4.6]), it is sufficient to compute the dimension of $H^{p,1}(X)$
and compare it to that of $H^{p+1,1+r}(Y)$. Notice that the dimension of
$H^{p+1,1+r}(Y)$ is well-established (see, for example, [Huy24, Section 1.1]
or [Voi03, Chapitre 18]).

The following provides some (partial) affirmative responses regarding the two
aforementioned desired conditions.

######  Theorem 4.4.10.

Let $X=F_{r}(Y)$ be the strict Calabi-Yau manifold as described in Section
4.1.1, with $r\geq 2$, then the map
$F_{r}:\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))\dashrightarrow\mathrm{Def}(X)$,
which assigns to
$Y\in\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))$ its Fano
variety of $r$-spaces $F_{r}(Y)$, is dominant and has a relative dimension of
$n^{2}+2n=\dim\mathrm{PSL}_{n+1}(\mathbb{C})$.

######  Remark 4.4.11.

Theorem 4.4.10 does not apply if $r=1$, due to the existence of non-projective
deformations of hyper-Kähler manifolds. This limitation is further highlighted
by our proof of Theorem 4.4.10, which crucially relies on the condition that
$r\geq 2$.

For the explicit calculations of Hodge numbers $h^{p,1}(X)$, we only manage to
get the result in the case $r=2$ due to the complexity of the computation.

######  Proposition 4.4.12.

In the case of $r=2$, we have the following

$h^{p,1}(X)=\left\\{\begin{array}[]{cc}1,&p=1\\\ 45,&p=3\\\ 120,&p=10\\\
0,&\textrm{others}.\end{array}\right.$

Notice that for $p>1$, the only non-zero $H^{p+2,3}(Y)$ for a cubic eightfold
$Y$ is $H^{5,3}(Y)$ whose dimension is $45$ (see [Huy24, Section 1.1]). Thus,
Proposition 4.4.12 indeed confirms the second condition mentioned above in the
case of $r=2$.

The proofs of Theorem 4.4.10 and Proposition 4.4.12 are detailed in the
following sections.

##### Proof of Theorem 4.4.10

###### Proof of Theorem 4.4.10.

For the brevity of the notation, in what follows, we will denote $G$ as the
Grassmannian $\mathrm{Gr}(r+1,n+1)$ and $S^{3}\mathcal{E}^{*}$ as the
symmetric product $\mathrm{Sym}^{3}\mathcal{E}^{*}$. Let

$\alpha:H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))\cong
H^{0}(G,S^{3}\mathcal{E}^{*})\to H^{0}(X,S^{3}\mathcal{E}^{*}|_{X})\cong
H^{0}(X,N_{X/G})$

be the restriction map composed with some canonical isomorphisms. Let

$\beta:H^{0}(X,N_{X/G})\to H^{1}(X,T_{X})$

be the connection map of the normal exact sequence. With the above notations,
we have the following two lemmas that we will prove later.

######  Lemma 4.4.13.

$H^{0}(X,N_{X/G})$ is identified with the tangent space of
$\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))$ at the point
$f$ representing the cubic hypersurface $Y_{0}$.

######  Lemma 4.4.14.

The map $\beta$ is surjective and $\dim\ker\beta=n^{2}+2n$.

Therefore, the map $\beta:H^{0}(X,N_{X/G})\to H^{1}(X,T_{X})$ identifies with
the tangent map of
$F_{r}:\mathbb{P}H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))\dashrightarrow\mathrm{Def}(X))$,
and Theorem 4.4.10 follows, knowing that $\mathrm{Def}(X)$ is smooth since $X$
is a strict Calabi-Yau manifold (see [Bog78, Tia87, Tod88] and [Ran92]). ∎

The proof of Lemma 4.4.13 and of Lemma 4.4.14 is a direct calculation using
the following theorem of Bott [Bot75, Dem76]. We use the following formulation
presented in [Man92, Proposition 2]. In the statement, $L_{\lambda}$
represents the Schur’s functor of weight $\lambda$.

######  Theorem 4.4.15 (Bott [Bot75, Dem76, Man92]).

Given decreasing weights $\lambda_{1}\in\mathbb{Z}^{n-r}$ and
$\lambda_{2}\in\mathbb{Z}^{r+1}$. Let
$\lambda=(\lambda_{1},\lambda_{2})\in\mathbb{Z}^{n+1}$. Let
$c(n+1)=(1,2,\ldots,n+1)$. If $\lambda-c(n+1)$ has common components, then
$H^{q}(\mathrm{Gr}(r+1,n+1),L_{\lambda_{1}}\mathcal{Q}\otimes
L_{\lambda_{2}}\mathcal{E})=0$ for any $q$ (and in this case, we say that
$\lambda-c(n+1)$ is irregular). Otherwise, we have

$H^{q}(\mathrm{Gr}(r+1,n+1),L_{\lambda_{1}}\mathcal{Q}\otimes
L_{\lambda_{2}}\mathcal{E})=\delta_{q,i(\lambda)}L_{\xi(\lambda)}V_{n+1}.$

Here $i(\lambda)$, $\xi(\lambda)$ are defined as follows. The weight
$\xi(\lambda)$ is defined as $(\lambda-c(n+1))^{\geq}+c(n+1)$, where
$(\lambda-c(n+1))^{\geq}$ is the decreasing integer sequence obtained by
permutation of the sequence $\lambda-c(n+1)$. The number $i(\lambda)$ is the
inversion number of the sequence $\lambda-c(n+1)$.

The following technical lemma, due to Debarre and Manivel [DM98, Lemma 3.9],
is also used frequently.

######  Lemma 4.4.16 (Debarre-Manivel [DM98]).

Let $V$ be a complex vector space, $m$ and $d$ be integers. For any
irreducible component $L_{\lambda}V$ of $\bigwedge^{j}\mathrm{Sym}^{d}V$, we
have

$|\lambda|_{>m}\geq j-\binom{m+d-1}{d}.$

Here, we write $\lambda$ as a decreasing sequence of integers
$(\lambda_{1},\ldots,\lambda_{\dim V})$ and
$|\lambda|_{>m}=\sum_{i>m}\lambda_{i}$.

###### Proof of Lemma 4.4.13.

Let $\mathcal{I}_{X}$ be the ideal sheaf of $X$ in $G$ that fits into the
following short exact sequence

$0\to\mathcal{I}_{X}\otimes S^{3}\mathcal{E}^{*}\to S^{3}\mathcal{E}^{*}\to
N_{X/G}\to 0.$

To prove Lemma 4.4.13, it suffices to prove that
$H^{0}(G,\mathcal{I}_{X}\otimes S^{3}\mathcal{E}^{*})$ is of dimension $1$
while $H^{1}(G,\mathcal{I}_{X}\otimes S^{3}\mathcal{E}^{*})=0$. By the Koszul
resolution

$\ldots\to\bigwedge^{i+1}S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}\to\bigwedge^{i}S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}\to\ldots\to S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}\to\mathcal{I}_{X}\otimes S^{3}\mathcal{E}^{*}\to 0,$

it suffices to prove

  * (i)

$\dim H^{0}(G,S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}^{*})=1$ and
$H^{1}(G,S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}^{*})=0$.

  * (ii)

$H^{i-d}(G,\bigwedge^{i}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}^{*})=0$ for
any $i\geq 2$ and for any $d=0,1,2$.

To prove (i), by the Littlewood-Richardson rule [Man92, Section 2.1.1], we
find $S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}^{*}=\mathcal{O}_{G}\oplus
L_{(3,0,\ldots,0,-3)}\mathcal{E}$. Since $n-r\geq 3$ in our setting, the
sequence $(0,\ldots,0,3,0,\ldots,0,-3)-c(n+1)$ is irregular. Hence,
$H^{q}(G,S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*})=H^{q}(G,\mathcal{O}_{G})$. The latter has dimension $1$
when $q=0$ and $0$ otherwise. This proves (i).

To prove (ii), let
$L_{\lambda}\mathcal{E}\subset\bigwedge^{i}S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}$ be an irreducible component. Then there is a weight
$\mu$ such that $L_{\lambda}\mathcal{E}\subset L_{\mu}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}$. By the Littleword-Richardson rule, for any $m$, we have
$|\lambda|_{>m}\geq|\mu|_{>m}-3$. Therefore, by Lemma 4.4.16, we find

$|\lambda|_{>m}\geq i-\binom{m+2}{3}-3.$ (4.26)

By Theorem 4.4.15, $H^{i-d}(L_{\lambda}\mathcal{E})\neq 0$ implies that
$(-1,-2,\ldots,-(n-r),\lambda_{1}-(n-r)-1,\ldots,\lambda_{r+1}-(n+1))$ is
regular and that $i-d$ is its inversion number. Equivalently, it means that
there exists $0\leq h\leq r+1$, such that

$\lambda_{h}-(n-r)-h\geq 0,$ (4.27) $\lambda_{h+1}-(n-r)-(h+1)\leq-(n-r+1),$
(4.28) $i-d=h(n-r).$ (4.29)

Here, we use the convention that $\lambda_{0}=+\infty$ and that
$\lambda_{r+2}=-\infty$ to simplify the notations. We divide into three cases
and conclude none of them is possible.

  * •

If $h=r+1$, then by (4.29), $i=(r+1)(n-r)+d>\mathrm{rank}S^{3}\mathcal{E}$.
Hence, $H^{i-d}(\bigwedge^{i}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}^{*})=0$.

  * •

If $0<h<r+1$, then we have $i=h(n-r)+d$ by (4.29), and $|\lambda|_{>h}\leq
h(r+1-h)$ by (4.28). Combining with (4.26), we obtain

$h(n-r)+d-\binom{h+2}{3}-3\leq h(r+1-h),$

or equivalently,

$n-2r-1\leq\frac{1}{6}(h-1)(h-2)+\frac{3-d}{h}=:\phi_{d}(h).$ (4.30)

It is not hard to see that

$\max_{0<h<r+1}\phi_{d}(h)\leq\left\\{\begin{array}[]{cc}3-d,&r=2\\\
\frac{1}{6}(r-1)(r-2)+1,&r\geq 3.\end{array}\right.$

A direct comparition with (4.30) gives a contradiction when $r\geq 2$.

  * •

If $h=0$, then $i=d$ by (4.29). But we have assumed $i\geq 2$, so there is
only the case $i=d=2$ to discuss. In this case, (4.28) shows that
$\lambda_{1}\leq 0$. We will show that this is impossible. In fact, the
irreducible components of $\bigwedge^{2}S^{3}\mathcal{E}$ are
$L_{(3,3)}\mathcal{E}$ and $L_{(5,1)}\mathcal{E}$. By the Littlewood-
Richardson rule, the $\lambda_{1}$ in $L_{(3,3)}\mathcal{E}\otimes
S^{3}\mathcal{E}^{*}$ would be $\geq 1$ and that in
$L_{(5,1)}\mathcal{E}\otimes S^{3}\mathcal{E}^{*}$ would be $\geq 3$,
contradicting the constraints that $\lambda_{1}\leq 0$.

∎

###### Proof of Lemma 4.4.14.

Let us consider the normal exact sequence

$0\to T_{X}\to T_{G|X}\to N_{X/G}\to 0.$

Since $X$ is strict Calabi-Yau, $H^{0}(X,T_{X})=0$. Hence, the kernel of
$\beta:H^{0}(X,N_{X/G})\to H^{1}(X,T_{X})$ is $H^{0}(X,T_{G|X})$ and the
cokernel is contained in $H^{1}(X,T_{G|X})$. Hence, to prove Lemma 4.4.14, it
suffices to show that $H^{0}(X,T_{G|X})\cong H^{0}(G,T_{G})$ and that
$H^{1}(X,T_{G|X})=0$. Considering the short exact sequence

$0\to\mathcal{I}_{X}\otimes T_{G}\to T_{G}\to T_{G|X}\to 0,$

and knowing that $H^{1}(G,T_{G})=0$, it suffices to show that
$H^{i}(G,\mathcal{I}_{X}\otimes T_{G})=0$ for $i=0,1,2$. Now we take the
Koszul resolution of $\mathcal{I}_{X}\otimes T_{G}$, noticing that
$T_{G}=\mathcal{E}^{*}\otimes\mathcal{Q}$:

$\ldots\to\bigwedge^{i+1}S^{3}\mathcal{E}\otimes\mathcal{E}^{*}\otimes\mathcal{Q}\to\bigwedge^{i}S^{3}\mathcal{E}\otimes\mathcal{E}^{*}\otimes\mathcal{Q}\to\ldots\to
S^{3}\mathcal{E}\otimes\mathcal{E}^{*}\otimes\mathcal{Q}\to\mathcal{I}_{X}\otimes
E^{*}\otimes\mathcal{Q}\to 0.$

Therefore, it suffices to show that
$H^{i+d}(G,\bigwedge^{i}S^{3}\mathcal{E}\otimes\mathcal{E}^{*}\otimes\mathcal{Q})=0$
for any $i\geq 1$ and any $d\in\\{-1,0,1\\}$. To prove this, let
$L_{\lambda}\mathcal{E}\subset\bigwedge^{i}S^{3}\mathcal{E}\otimes\mathcal{E}^{*}$
be an irreducible component. By the Littlewood-Richardson rule and Lemma
4.4.16, we have

$|\lambda|_{>m}\geq i-\binom{m+2}{3}-1.$ (4.31)

In order that the sequence
$(1,0,\ldots,0,\lambda_{1},\ldots,\lambda_{r+1})-c(n+1)=(0,-2,-3,\ldots,-(n-r),\lambda_{1}-(n-r)-1,\ldots,\lambda_{r+1}-(n+1))$
to be regular, and that
$H^{i+d}(G,\bigwedge^{i}S^{3}\mathcal{E}\otimes\mathcal{E}^{*}\otimes\mathcal{Q})\neq
0$, one of the following two cases needs to happen:

  * (a)

There exists $0\leq h\leq r+1$ such that

$\lambda_{h}-(n-r)-h\geq 1,$ (4.32) $\lambda_{h+1}-(n-r)-(h+1)\leq-(n-r)-1,$
(4.33) $i+d=h(n-r).$ (4.34)

  * (b)

There exists $0\leq h\leq r$ such that

$\lambda_{h}-(n-r)-h\geq 1,$ (4.35) $\lambda_{h+1}-(n-r)-(h+1)=-1,$ (4.36)
$\lambda_{h+2}-(n-r)-(h+2)\leq-(n-r)-1,$ (4.37) $i+d=h(n-r)+(n-r-1).$ (4.38)

We will show that in both cases, the cohomology is always zero.

For Case (a),

  * •

If $h=r+1$, then (4.34) shows that
$i=(r+1)(n-r)-d>\mathrm{rank}S^{3}\mathcal{E}$, we have
$\bigwedge^{i}S^{3}\mathcal{E}=0$.

  * •

If $h=0$, then (4.34) shows that $i=-d$. Since we suppose $i\geq 1$, only the
case $i=-d=1$ remains to be checked. In this case, (4.33) becomes
$\lambda_{1}\leq 0$. However, by the Littlewood-Richardson rule, in the
decomposition of $S^{3}\mathcal{E}\otimes\mathcal{E}^{*}$, $\lambda_{1}\geq
2$.

  * •

If $0<h<r+1$, then $|\lambda|_{>h}\leq h(r+1-h)$ by (4.33). Combining (4.31)
and (4.34), we get

$h(n-r)-d-\binom{h+2}{3}-1\leq h(r+1-h),$

and this inequality can be shown to be impossible using similar method as in
(4.30).

For Case (b), we have $|\lambda|_{>h}\leq n-r+h+(r-h)(h+1)=n+(r-h)h$.
Combining with (4.31) and (4.38), we get

$h(n-r)+(n-r-1)-1-\binom{h+2}{3}-1\leq n+(r-h)h,$

or equivalently,

$\binom{h+2}{3}-h(h+n-2r)+r+3\geq 0.$ (4.39)

It is easy to check that (4.39) is impossible if $r=2,h=2$, or if $r\geq
3,0<h\leq r$. In fact, define $\phi(h)=\binom{h+2}{3}-h(h+n-2r)+r+3$. Then
$\phi(0)=r+3>0\geq\phi(1)=-n+3r+3$. Since $\phi(h)$ is a cubic function, it
suffices to show that $\phi(r)<0$ for $r\geq 2$ and that $\phi(1)<0$ for
$r\geq 3$. The latter is easy to check. For the former, we have

$\phi(r)=\binom{r+2}{3}-r(n-r)+r+3=r(-\frac{1}{3}r^{2}-r-\frac{5}{3}+\frac{3}{r})<0$

for any $r\geq 2$. The remaining cases that have not been checked are
$r=2,h=1$ or $h=0$. To this ends, we consider $|\lambda|_{>h+1}$, by (4.31),
(4.37) and (4.38), we get

$(h+1)(n-r)-3-\binom{h+3}{3}\leq(r-h)(h+1).$ (4.40)

If $r=2,h=1$, then the left-hand-side of (4.40) is $7$ whereas the right-hand-
side is $2$—the inequality (4.40) does not hold. If $h=0$, the left-hand-side
of (4.40) is $n-r-4$ whereas the right-hand-side is $r$—the inequality (4.40)
does not hold either. We have eliminated all the possibilities and Lemma
4.4.14 is proven. ∎

##### Proof of Proposition 4.4.12

###### Proof of Proposition 4.4.12.

By Hodge symmetry, let us calculate instead the dimension of
$H^{q}(X,\Omega_{X})$ for any $q$. Consider the conormal exact sequence

$0\to
S^{3}\mathcal{E}|_{X}\to(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X}\to\Omega_{X}\to
0,$

we need to know the cohomology groups of $S^{3}\mathcal{E}|_{X}$ and of
$S^{3}\mathcal{E}|_{X}$.

Calculation of $H^{i}(X,S^{3}\mathcal{E}|_{X})$: by the Koszul resolutioin, we
need to calculate the cohomology groups of
$\bigwedge^{k}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E}$. By the calculations
using SageMath, the only non-zero cohomology groups of the form
$H^{j}(G,\bigwedge^{k}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})$ are listed as
below.

  * •

$h^{7}(G,\bigwedge^{2}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})=10$.

  * •

$h^{7}(G,\bigwedge^{3}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})=55$.

  * •

$h^{14}(G,\bigwedge^{6}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})=10$.

  * •

$h^{21}(G,\bigwedge^{9}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})=1$.

  * •

$h^{21}(G,\bigwedge^{10}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})=220$.

The above calculation gives us the following consequences.

  * •

There is an exact sequence

$0\to H^{4}(X,S^{3}\mathcal{E}|_{X})\to
H^{7}(G,\bigwedge^{3}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\to
H^{7}(G,\bigwedge^{2}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\to
H^{5}(X,S^{3}\mathcal{E}|_{X})\to 0.$

The calculation using SageMath, together with Theorem 4.4.15, shows that
$H^{7}(G,\bigwedge^{3}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\cong
H^{7}(G,L_{(10,1,1)}\mathcal{E})\cong L_{(3,1,\ldots,1)}V_{10}\cong
S^{2}V_{10}\otimes\det V_{10}$, that
$H^{7}(G,\bigwedge^{2}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\cong
H^{7}(G,L_{(8,1)}\mathcal{E})\cong L_{(1,1,\ldots,1,0)}V_{10}\cong
V_{10}^{*}\otimes\det V_{10}$, and that the map
$H^{7}(G,\bigwedge^{3}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\to
H^{7}(G,\bigwedge^{2}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})$ is viewed as
$\lrcorner\sigma_{f}:S^{2}V_{10}\otimes\det V_{10}\to V_{10}^{*}\otimes\det
V_{10}$, where $\sigma_{f}$ is the global section of
$H^{0}(\mathbb{P}^{9},\mathcal{O}(3))\cong
S^{3}H^{0}(\mathbb{P}^{9},\mathcal{O}(1))=S^{3}V_{10}^{*}$, and a generic
choice of $\sigma_{f}$ gives surjective contraction map

$\lrcorner\sigma_{f}:S^{2}V_{10}\otimes\det V_{10}\to V_{10}^{*}\otimes\det
V_{10}.$

Therefore, we conclude that $h^{4}(X,S^{3}\mathcal{E}|_{X})=45$ and that
$h^{5}(X,S^{3}\mathcal{E}|_{X})=0$.

  * •

$h^{8}(X,S^{3}\mathcal{E}|_{X})=h^{14}(G,\bigwedge^{6}S^{3}\mathcal{E}\otimes
S^{3}\mathcal{E})=10$.

  * •

There is an exact sequence

$0\to H^{11}(X,S^{3}\mathcal{E}|_{X})\to
H^{21}(G,\bigwedge^{10}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\to
H^{21}(G,\bigwedge^{9}S^{3}\mathcal{E}\otimes S^{3}\mathcal{E})\to
H^{12}(X,S^{3}\mathcal{E}|_{X})\to 0.$

But $H^{12}(X,S^{3}\mathcal{E}|_{X})=0$ since $\dim X=11$. Hence,
$h^{11}(X,S^{3}\mathcal{E}|_{X})=219$.

  * •

$H^{i}(X,S^{3}\mathcal{E}|_{X})=0$ for other $i$.

Calculation of $H^{i}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})$: by the
Koszul resolution, we need to calculate the cohomology group of
$\bigwedge^{k}S^{3}\mathcal{E}\otimes\mathcal{E}\otimes\mathcal{Q}^{*}$. This
is done using SageMath. The only non-zero cohomology groups of the form
$H^{j}(G,\bigwedge^{k}S^{3}\mathcal{E}\otimes\mathcal{E}\otimes\mathcal{Q}^{*})$
are listed as follows.

  * •

$h^{1}(G,\bigwedge^{0}S^{3}\mathcal{E}\otimes\mathcal{E}\otimes\mathcal{Q}^{*})=1$.

  * •

$h^{15}(G,\bigwedge^{7}S^{3}\mathcal{E}\otimes\mathcal{E}\otimes\mathcal{Q}^{*})=10$.

  * •

$h^{21}(G,\bigwedge^{10}S^{3}\mathcal{E}\otimes\mathcal{E}\otimes\mathcal{Q}^{*})=99$.

The above calculation gives us the following consequences.

  * •

$h^{1}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})=1$.

  * •

$h^{8}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})=10$.

  * •

$h^{11}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})=99$.

  * •

$h^{i}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})=0$ for other $i$.

Calculation of $H^{q}(X,\Omega_{X})$: We use the conormal exact sequence

$0\to
S^{3}\mathcal{E}|_{X}\to(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X}\to\Omega_{X}\to
0$

and the above calculation to obtain:

  * •

$h^{1}(X,\Omega_{X})=1$.

  * •

$h^{3}(X,\Omega_{X})=45$.

  * •

There is an exact sequence

$0\to H^{7}(X,\Omega_{X})\to H^{8}(X,S^{3}\mathcal{E}|_{X})\to
H^{8}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})\to H^{8}(X,\Omega_{X})\to
0.$

By chasing the diagram, we find that $H^{8}(X,S^{3}\mathcal{E}|_{X})\cong
H^{14}(G,L_{(10,9,2)}\mathcal{E})\cong L_{(3,1,\ldots,1)}V_{10}$ and
$H^{8}(X,\Omega_{X})\cong H^{15}(G,L_{(0,\ldots,0,-1)}\mathcal{Q}\otimes
L_{(10,9,3)}\mathcal{E})\cong L_{(3,1,\ldots,1)}V_{10}$ and the map in between
is given by the identity map. Therefore,
$H^{7}(X,\Omega_{X})=H^{8}(X,\Omega_{X})=0$.

  * •

There is an exact sequence

$0\to H^{10}(X,\Omega_{X})\to H^{11}(X,S^{3}\mathcal{E}|_{X})\to
H^{11}(X,(\mathcal{E}\otimes\mathcal{Q}^{*})|_{X})\to H^{11}(X,\Omega_{X})\to
0.$

But $h^{11}(X,\Omega_{X})\cong h^{1}(X,K_{X})\cong h^{1,0}(X)=0$. Hence,
$h^{10}(X,\Omega_{X})=219-99=120$. This coincides with our expectation since
$h^{10}(X,\Omega_{X})=h^{1}(X,\Omega_{X}^{10})=h^{1}(X,T_{X})$ and
$h^{1}(X,T_{X})=120$ by our calculation in Theorem 4.4.10.

  * •

$h^{i}(X,\Omega_{X})=0$ for other $i$.

∎

##  Appendix A Computational Verification in Chapter 4

###  A.1 Computational Verification in Section 4.3.3

This appendix provides a comprehensive breakdown of the calculations
underpinning the results introduced in Section 4.3.3, employing Mathematica
for computational support.

####  A.1.1 Computational verification using Mathematica I

$\boldsymbol{\text{(*Define the variables*)}}\\\
\boldsymbol{\text{li}=\text{Array}[l,3];\text{(*l1, l2, l3*)}}\\\
\boldsymbol{\text{mj}=\text{Array}[m,6];\text{(*m1, m2, m3, m4, m5, m6*)}}\\\
\boldsymbol{\text{(*Create the product expression*)}}\\\
\boldsymbol{\text{productExpr}=\text{Product}[\text{li}[[i]]+\text{mj}[[j]],\\{i,1,3\\},\\{j,1,6\\}];}\\\
\boldsymbol{\text{sym1}=\text{SymmetricReduction}[\text{productExpr},\text{mj},\\{\text{d1},\text{d2},\text{d3},\text{d4},\text{d5},\text{d6}\\}][[1]];}\\\
\boldsymbol{\text{sym2}=\text{SymmetricReduction}[\text{sym1},\text{li},\\{\text{c1},\text{c2},\text{c3}\\}][[1]]}$

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^{2}-80\text{d2}\text{d6}+30(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})+31(-\text{d3}\text{d5}+4\text{d2}\text{d6})+12(-\text{d1}\text{d2}\text{d5}+14\text{d2}\text{d6}-5(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-3(-\text{d3}\text{d5}+4\text{d2}\text{d6}))+12(-\text{d4}^{2}+6\text{d2}\text{d6}-2(-\text{d3}\text{d5}+4\text{d2}\text{d6}))+5(-\text{d1}\text{d3}\text{d4}+32\text{d2}\text{d6}-10(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-11(-\text{d3}\text{d5}+4\text{d2}\text{d6})-3(-\text{d1}\text{d2}\text{d5}+14\text{d2}\text{d6}-5(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-3(-\text{d3}\text{d5}+4\text{d2}\text{d6}))-4(-\text{d4}^{2}+6\text{d2}\text{d6}-2(-\text{d3}\text{d5}+4\text{d2}\text{d6})))+2(-\text{d2}^{2}\text{d4}+53\text{d2}\text{d6}-20(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-18(-\text{d3}\text{d5}+4\text{d2}\text{d6})-7(-\text{d1}\text{d2}\text{d5}+14\text{d2}\text{d6}-5(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-3(-\text{d3}\text{d5}+4\text{d2}\text{d6}))-6(-\text{d4}^{2}+6\text{d2}\text{d6}-2(-\text{d3}\text{d5}+4\text{d2}\text{d6}))-2(-\text{d1}\text{d3}\text{d4}+32\text{d2}\text{d6}-10(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-11(-\text{d3}\text{d5}+4\text{d2}\text{d6})-3(-\text{d1}\text{d2}\text{d5}+14\text{d2}\text{d6}-5(-\text{d1}^{2}\text{d6}+2\text{d2}\text{d6})-3(-\text{d3}\text{d5}+4\text{d2}\text{d6}))-4(-\text{d4}^{2}+6\text{d2}\text{d6}-2(-\text{d3}\text{d5}+4\text{d2}\text{d6})))))+\text{c3}^{3}(\text{d3}^{3}-93\text{d3}\text{d6}+36(-\text{d4}\text{d5}+3\text{d3}\text{d6})+36(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})+15(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6}))+6(-\text{d2}^{2}\text{d5}+24\text{d3}\text{d6}-6(-\text{d4}\text{d5}+3\text{d3}\text{d6})-9(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})-2(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})))+6(-\text{d1}\text{d4}^{2}+24\text{d3}\text{d6}-9(-\text{d4}\text{d5}+3\text{d3}\text{d6})-6(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})-2(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})))+3(-\text{d2}\text{d3}\text{d4}+60\text{d3}\text{d6}-22(-\text{d4}\text{d5}+3\text{d3}\text{d6})-22(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})-8(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6}))-3(-\text{d2}^{2}\text{d5}+24\text{d3}\text{d6}-6(-\text{d4}\text{d5}+3\text{d3}\text{d6})-9(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})-2(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})))-3(-\text{d1}\text{d4}^{2}+24\text{d3}\text{d6}-9(-\text{d4}\text{d5}+3\text{d3}\text{d6})-6(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})-2(-\text{d1}\text{d3}\text{d5}+15\text{d3}\text{d6}-4(-\text{d4}\text{d5}+3\text{d3}\text{d6})-4(-\text{d1}\text{d2}\text{d6}+3\text{d3}\text{d6})))))+\text{c2}\text{c3}^{2}(\text{d3}^{2}\text{d4}-80\text{d4}\text{d6}+30(-\text{d5}^{2}+2\text{d4}\text{d6})+31(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})+12(-\text{d1}\text{d4}\text{d5}+14\text{d4}\text{d6}-5(-\text{d5}^{2}+2\text{d4}\text{d6})-3(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))+12(-\text{d2}^{2}\text{d6}+6\text{d4}\text{d6}-2(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))+5(-\text{d2}\text{d3}\text{d5}+32\text{d4}\text{d6}-10(-\text{d5}^{2}+2\text{d4}\text{d6})-11(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})-3(-\text{d1}\text{d4}\text{d5}+14\text{d4}\text{d6}-5(-\text{d5}^{2}+2\text{d4}\text{d6})-3(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))-4(-\text{d2}^{2}\text{d6}+6\text{d4}\text{d6}-2(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})))+2(-\text{d2}\text{d4}^{2}+53\text{d4}\text{d6}-20(-\text{d5}^{2}+2\text{d4}\text{d6})-18(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})-7(-\text{d1}\text{d4}\text{d5}+14\text{d4}\text{d6}-5(-\text{d5}^{2}+2\text{d4}\text{d6})-3(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))-6(-\text{d2}^{2}\text{d6}+6\text{d4}\text{d6}-2(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))-2(-\text{d2}\text{d3}\text{d5}+32\text{d4}\text{d6}-10(-\text{d5}^{2}+2\text{d4}\text{d6})-11(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})-3(-\text{d1}\text{d4}\text{d5}+14\text{d4}\text{d6}-5(-\text{d5}^{2}+2\text{d4}\text{d6})-3(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6}))-4(-\text{d2}^{2}\text{d6}+6\text{d4}\text{d6}-2(-\text{d1}\text{d3}\text{d6}+4\text{d4}\text{d6})))))+\text{c1}\text{c3}^{2}(\text{d3}\text{d4}^{2}-80\text{d5}\text{d6}+31(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})+12(-\text{d2}\text{d3}\text{d6}+10\text{d5}\text{d6}-3(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))+12(-\text{d1}\text{d5}^{2}+11\text{d5}\text{d6}-2(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))+5(-\text{d2}\text{d4}\text{d5}+35\text{d5}\text{d6}-11(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})-3(-\text{d2}\text{d3}\text{d6}+10\text{d5}\text{d6}-3(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))-4(-\text{d1}\text{d5}^{2}+11\text{d5}\text{d6}-2(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})))+2(-\text{d3}^{2}\text{d5}+50\text{d5}\text{d6}-18(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})-7(-\text{d2}\text{d3}\text{d6}+10\text{d5}\text{d6}-3(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))-6(-\text{d1}\text{d5}^{2}+11\text{d5}\text{d6}-2(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))-2(-\text{d2}\text{d4}\text{d5}+35\text{d5}\text{d6}-11(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})-3(-\text{d2}\text{d3}\text{d6}+10\text{d5}\text{d6}-3(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6}))-4(-\text{d1}\text{d5}^{2}+11\text{d5}\text{d6}-2(-\text{d1}\text{d4}\text{d6}+5\text{d5}\text{d6})))))+\text{c3}^{2}(\text{d4}^{3}-90\text{d6}^{2}+36(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})+15(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2}))+6(-\text{d2}\text{d5}^{2}+30\text{d6}^{2}-9(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})-2(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})))+6(-\text{d3}^{2}\text{d6}+20\text{d6}^{2}-6(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})-2(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})))+3(-\text{d3}\text{d4}\text{d5}+60\text{d6}^{2}-22(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})-8(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2}))-3(-\text{d2}\text{d5}^{2}+30\text{d6}^{2}-9(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})-2(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})))-3(-\text{d3}^{2}\text{d6}+20\text{d6}^{2}-6(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})-2(-\text{d2}\text{d4}\text{d6}+15\text{d6}^{2}-4(-\text{d1}\text{d5}\text{d6}+6\text{d6}^{2})))))$

####  A.1.2 Computational verification using Mathematica II

$\boldsymbol{\text{(*Define the Chern classes of the symmetric product*)}}\\\
\boldsymbol{}\\\
\boldsymbol{\text{S1}[\text{c1$\\_$},\text{c2$\\_$},\text{c3$\\_$}]\text{:=}4\text{c1};}\\\
\boldsymbol{\text{S2}[\text{c1$\\_$},\text{c2$\\_$},\text{c3$\\_$}]\text{:=}5\text{c1}{}^{\wedge}2+5\text{c2};}\\\
\boldsymbol{\text{S3}[\text{c1$\\_$},\text{c2$\\_$},\text{c3$\\_$}]\text{:=}2\text{c1}{}^{\wedge}3+11\text{c1}\text{c2}+7\text{c3};}\\\
\boldsymbol{}\\\ \boldsymbol{\text{(*Define the Euler classes up to degree 3
in ci*)}}\\\ \boldsymbol{}\\\
\boldsymbol{\text{e1}=\text{d6}{}^{\wedge}3+\text{c1}\text{d5}\text{d6}{}^{\wedge}2+\text{c1}{}^{\wedge}2\text{d4}\text{d6}{}^{\wedge}2+\text{c2}(\text{d5}{}^{\wedge}2\text{d6}-2\text{d4}\text{d6}{}^{\wedge}2)+\text{c1}\text{c2}(\text{d4}\text{d5}\text{d6}-3\text{d3}\text{d6}{}^{\wedge}2)}\\\
\boldsymbol{+\text{c3}(\text{d5}{}^{\wedge}3-6\text{d3}\text{d6}{}^{\wedge}2+3(-\text{d4}\text{d5}\text{d6}+3\text{d3}\text{d6}{}^{\wedge}2))+\text{c1}{}^{\wedge}3\text{d3}\text{d6}{}^{\wedge}2;}\\\
\boldsymbol{\text{e2}=(\text{C1}-\text{c1}){}^{\wedge}6+(\text{C1}-\text{c1}){}^{\wedge}5\text{S1}[\text{c1},\text{c2},\text{c3}]+(\text{C1}-\text{c1}){}^{\wedge}4\text{S2}[\text{c1},\text{c2},\text{c3}]+}\\\
\boldsymbol{(\text{C1}-\text{c1}){}^{\wedge}3\text{S3}[\text{c1},\text{c2},\text{c3}];}\\\
\boldsymbol{\text{e3}=8(\text{C1}-\text{c1}){}^{\wedge}3+4(\text{C1}-\text{c1}){}^{\wedge}2\text{c1}+2(\text{C1}-\text{c1})\text{c2}+\text{c3};}\\\
\boldsymbol{}\\\ \boldsymbol{\text{(*The class of }[\text{Tilde }F]\text{ up
to degree }3\text{*)}}\\\ \boldsymbol{}\\\
\boldsymbol{\text{final}=\text{e1}\text{e2}\text{e3};}\\\
\boldsymbol{\text{final}=\text{Expand}[\text{final}];}\\\
\boldsymbol{\text{Collect}[\text{final},\\{\text{c1},\text{c2},\text{c3}\\}]}$

$8\text{C1}^{9}\text{d6}^{3}+\text{c3}^{3}(7\text{C1}^{3}\text{d5}^{3}-21\text{C1}^{3}\text{d4}\text{d5}\text{d6}+21\text{C1}^{3}\text{d3}\text{d6}^{2})+\text{c2}^{3}(10\text{C1}^{5}\text{d5}^{2}\text{d6}-20\text{C1}^{5}\text{d4}\text{d6}^{2})+\text{c1}^{9}(4\text{C1}^{2}\text{d4}\text{d6}^{2}+24\text{c2}\text{d4}\text{d6}^{2})+\text{c3}^{2}(57\text{C1}^{6}\text{d5}^{3}-171\text{C1}^{6}\text{d4}\text{d5}\text{d6}+171\text{C1}^{6}\text{d3}\text{d6}^{2}+7\text{C1}^{3}\text{d6}^{3})+\text{c3}(8\text{C1}^{9}\text{d5}^{3}-24\text{C1}^{9}\text{d4}\text{d5}\text{d6}+24\text{C1}^{9}\text{d3}\text{d6}^{2}+57\text{C1}^{6}\text{d6}^{3})+\text{c2}^{2}(42\text{C1}^{7}\text{d5}^{2}\text{d6}-84\text{C1}^{7}\text{d4}\text{d6}^{2}+10\text{C1}^{5}\text{d6}^{3}+\text{c3}(10\text{C1}^{5}\text{d5}^{3}-30\text{C1}^{5}\text{d4}\text{d5}\text{d6}+19\text{C1}^{4}\text{d5}^{2}\text{d6}+30\text{C1}^{5}\text{d3}\text{d6}^{2}-38\text{C1}^{4}\text{d4}\text{d6}^{2}))+\text{c1}^{8}(-24\text{C1}^{3}\text{d4}\text{d6}^{2}+28\text{c3}\text{d4}\text{d6}^{2}+4\text{C1}^{2}\text{d5}\text{d6}^{2}+\text{c2}^{2}(24\text{d4}\text{d5}\text{d6}-72\text{d3}\text{d6}^{2})+\text{c2}(4\text{C1}^{2}\text{d4}\text{d5}\text{d6}-12\text{C1}^{2}\text{d3}\text{d6}^{2}-148\text{C1}\text{d4}\text{d6}^{2}+24\text{d5}\text{d6}^{2}))+\text{c2}(8\text{C1}^{9}\text{d5}^{2}\text{d6}-16\text{C1}^{9}\text{d4}\text{d6}^{2}+42\text{C1}^{7}\text{d6}^{3}+\text{c3}^{2}(19\text{C1}^{4}\text{d5}^{3}-57\text{C1}^{4}\text{d4}\text{d5}\text{d6}+7\text{C1}^{3}\text{d5}^{2}\text{d6}+57\text{C1}^{4}\text{d3}\text{d6}^{2}-14\text{C1}^{3}\text{d4}\text{d6}^{2})+\text{c3}(42\text{C1}^{7}\text{d5}^{3}-126\text{C1}^{7}\text{d4}\text{d5}\text{d6}+57\text{C1}^{6}\text{d5}^{2}\text{d6}+126\text{C1}^{7}\text{d3}\text{d6}^{2}-114\text{C1}^{6}\text{d4}\text{d6}^{2}+19\text{C1}^{4}\text{d6}^{3}))+\text{c1}^{7}(52\text{C1}^{4}\text{d4}\text{d6}^{2}-24\text{C1}^{3}\text{d5}\text{d6}^{2}+4\text{C1}^{2}\text{d6}^{3}+\text{c2}^{2}(-148\text{C1}\text{d4}\text{d5}\text{d6}+24\text{d5}^{2}\text{d6}+444\text{C1}\text{d3}\text{d6}^{2}-36\text{d4}\text{d6}^{2})+\text{c3}(4\text{C1}^{2}\text{d5}^{3}-12\text{C1}^{2}\text{d4}\text{d5}\text{d6}+12\text{C1}^{2}\text{d3}\text{d6}^{2}-196\text{C1}\text{d4}\text{d6}^{2}+28\text{d5}\text{d6}^{2})+\text{c2}(-24\text{C1}^{3}\text{d4}\text{d5}\text{d6}+4\text{C1}^{2}\text{d5}^{2}\text{d6}+72\text{C1}^{3}\text{d3}\text{d6}^{2}+334\text{C1}^{2}\text{d4}\text{d6}^{2}-148\text{C1}\text{d5}\text{d6}^{2}+24\text{d6}^{3}+\text{c3}(24\text{d5}^{3}-44\text{d4}\text{d5}\text{d6}-12\text{d3}\text{d6}^{2})))+\text{c1}^{6}(-40\text{C1}^{5}\text{d4}\text{d6}^{2}+52\text{C1}^{4}\text{d5}\text{d6}^{2}-24\text{C1}^{3}\text{d6}^{3}+\text{c2}^{3}(12\text{d4}\text{d5}\text{d6}-36\text{d3}\text{d6}^{2})+\text{c3}^{2}(28\text{d5}^{3}-84\text{d4}\text{d5}\text{d6}+84\text{d3}\text{d6}^{2})+\text{c2}^{2}(342\text{C1}^{2}\text{d4}\text{d5}\text{d6}-148\text{C1}\text{d5}^{2}\text{d6}-1026\text{C1}^{2}\text{d3}\text{d6}^{2}+258\text{C1}\text{d4}\text{d6}^{2}+12\text{d5}\text{d6}^{2})+\text{c3}(-24\text{C1}^{3}\text{d5}^{3}+72\text{C1}^{3}\text{d4}\text{d5}\text{d6}-72\text{C1}^{3}\text{d3}\text{d6}^{2}+559\text{C1}^{2}\text{d4}\text{d6}^{2}-196\text{C1}\text{d5}\text{d6}^{2}+28\text{d6}^{3})+\text{c2}(52\text{C1}^{4}\text{d4}\text{d5}\text{d6}-24\text{C1}^{3}\text{d5}^{2}\text{d6}-156\text{C1}^{4}\text{d3}\text{d6}^{2}-278\text{C1}^{3}\text{d4}\text{d6}^{2}+342\text{C1}^{2}\text{d5}\text{d6}^{2}-148\text{C1}\text{d6}^{3}+\text{c3}(-148\text{C1}\text{d5}^{3}+248\text{C1}\text{d4}\text{d5}\text{d6}+28\text{d5}^{2}\text{d6}+144\text{C1}\text{d3}\text{d6}^{2}-48\text{d4}\text{d6}^{2})))+\text{c1}^{5}(-20\text{C1}^{6}\text{d4}\text{d6}^{2}-40\text{C1}^{5}\text{d5}\text{d6}^{2}+52\text{C1}^{4}\text{d6}^{3}+\text{c2}^{3}(-38\text{C1}\text{d4}\text{d5}\text{d6}+12\text{d5}^{2}\text{d6}+114\text{C1}\text{d3}\text{d6}^{2}-24\text{d4}\text{d6}^{2})+\text{c3}^{2}(-196\text{C1}\text{d5}^{3}+588\text{C1}\text{d4}\text{d5}\text{d6}-588\text{C1}\text{d3}\text{d6}^{2}-7\text{d4}\text{d6}^{2})+\text{c3}(52\text{C1}^{4}\text{d5}^{3}-156\text{C1}^{4}\text{d4}\text{d5}\text{d6}+156\text{C1}^{4}\text{d3}\text{d6}^{2}-838\text{C1}^{3}\text{d4}\text{d6}^{2}+559\text{C1}^{2}\text{d5}\text{d6}^{2}-196\text{C1}\text{d6}^{3})+\text{c2}^{2}(-326\text{C1}^{3}\text{d4}\text{d5}\text{d6}+342\text{C1}^{2}\text{d5}^{2}\text{d6}+978\text{C1}^{3}\text{d3}\text{d6}^{2}-652\text{C1}^{2}\text{d4}\text{d6}^{2}-38\text{C1}\text{d5}\text{d6}^{2}+12\text{d6}^{3}+\text{c3}(12\text{d5}^{3}-28\text{d4}\text{d5}\text{d6}+12\text{d3}\text{d6}^{2}))+\text{c2}(-40\text{C1}^{5}\text{d4}\text{d5}\text{d6}+52\text{C1}^{4}\text{d5}^{2}\text{d6}+120\text{C1}^{5}\text{d3}\text{d6}^{2}-100\text{C1}^{4}\text{d4}\text{d6}^{2}-326\text{C1}^{3}\text{d5}\text{d6}^{2}+342\text{C1}^{2}\text{d6}^{3}+\text{c3}(342\text{C1}^{2}\text{d5}^{3}-467\text{C1}^{2}\text{d4}\text{d5}\text{d6}-196\text{C1}\text{d5}^{2}\text{d6}-651\text{C1}^{2}\text{d3}\text{d6}^{2}+34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))+\text{c2}(-36\text{C1}^{8}\text{d4}\text{d5}\text{d6}+56\text{C1}^{7}\text{d5}^{2}\text{d6}+108\text{C1}^{8}\text{d3}\text{d6}^{2}-70\text{C1}^{7}\text{d4}\text{d6}^{2}-178\text{C1}^{6}\text{d5}\text{d6}^{2}+240\text{C1}^{5}\text{d6}^{3}+\text{c3}^{2}(81\text{C1}^{2}\text{d5}^{3}-264\text{C1}^{2}\text{d4}\text{d5}\text{d6}+21\text{C1}\text{d5}^{2}\text{d6}+306\text{C1}^{2}\text{d3}\text{d6}^{2}-42\text{C1}\text{d4}\text{d6}^{2})+\text{c3}(240\text{C1}^{5}\text{d5}^{3}-1030\text{C1}^{5}\text{d4}\text{d5}\text{d6}+700\text{C1}^{4}\text{d5}^{2}\text{d6}+1650\text{C1}^{5}\text{d3}\text{d6}^{2}-1381\text{C1}^{4}\text{d4}\text{d6}^{2}-65\text{C1}^{3}\text{d5}\text{d6}^{2}+81\text{C1}^{2}\text{d6}^{3})))+\text{c1}(8\text{C1}^{9}\text{d5}\text{d6}^{2}-36\text{C1}^{8}\text{d6}^{3}+\text{c3}^{3}(-21\text{C1}^{2}\text{d5}^{3}+63\text{C1}^{2}\text{d4}\text{d5}\text{d6}-63\text{C1}^{2}\text{d3}\text{d6}^{2})+\text{c2}^{3}(10\text{C1}^{5}\text{d4}\text{d5}\text{d6}-28\text{C1}^{4}\text{d5}^{2}\text{d6}-30\text{C1}^{5}\text{d3}\text{d6}^{2}+56\text{C1}^{4}\text{d4}\text{d6}^{2})+\text{c3}^{2}(-310\text{C1}^{5}\text{d5}^{3}+930\text{C1}^{5}\text{d4}\text{d5}\text{d6}-930\text{C1}^{5}\text{d3}\text{d6}^{2}+7\text{C1}^{3}\text{d5}\text{d6}^{2}-21\text{C1}^{2}\text{d6}^{3})+\text{c3}(-36\text{C1}^{8}\text{d5}^{3}+108\text{C1}^{8}\text{d4}\text{d5}\text{d6}-108\text{C1}^{8}\text{d3}\text{d6}^{2}+57\text{C1}^{6}\text{d5}\text{d6}^{2}-310\text{C1}^{5}\text{d6}^{3})+\text{c2}^{2}(42\text{C1}^{7}\text{d4}\text{d5}\text{d6}-178\text{C1}^{6}\text{d5}^{2}\text{d6}-126\text{C1}^{7}\text{d3}\text{d6}^{2}+356\text{C1}^{6}\text{d4}\text{d6}^{2}+10\text{C1}^{5}\text{d5}\text{d6}^{2}-28\text{C1}^{4}\text{d6}^{3}+\text{c3}(-28\text{C1}^{4}\text{d5}^{3}+103\text{C1}^{4}\text{d4}\text{d5}\text{d6}-65\text{C1}^{3}\text{d5}^{2}\text{d6}-141\text{C1}^{4}\text{d3}\text{d6}^{2}+130\text{C1}^{3}\text{d4}\text{d6}^{2}))+\text{c2}(8\text{C1}^{9}\text{d4}\text{d5}\text{d6}-36\text{C1}^{8}\text{d5}^{2}\text{d6}-24\text{C1}^{9}\text{d3}\text{d6}^{2}+72\text{C1}^{8}\text{d4}\text{d6}^{2}+42\text{C1}^{7}\text{d5}\text{d6}^{2}-178\text{C1}^{6}\text{d6}^{3}+\text{c3}^{2}(-65\text{C1}^{3}\text{d5}^{3}+202\text{C1}^{3}\text{d4}\text{d5}\text{d6}-21\text{C1}^{2}\text{d5}^{2}\text{d6}-216\text{C1}^{3}\text{d3}\text{d6}^{2}+42\text{C1}^{2}\text{d4}\text{d6}^{2})+\text{c3}(-178\text{C1}^{6}\text{d5}^{3}+591\text{C1}^{6}\text{d4}\text{d5}\text{d6}-310\text{C1}^{5}\text{d5}^{2}\text{d6}-705\text{C1}^{6}\text{d3}\text{d6}^{2}+620\text{C1}^{5}\text{d4}\text{d6}^{2}+19\text{C1}^{4}\text{d5}\text{d6}^{2}-65\text{C1}^{3}\text{d6}^{3})))$

## Bibliography

  * [AP92] A. Alzati and G. P. Pirola, On the holomorphic length of a complex projective variety, Arch. Math. (Basel) 59 (1992), no. 4, 398–402. MR 1179468.
  * [Ame09] E. Amerik, A computation of invariants of a rational self-map, Ann. Fac. Sci. Toulouse Math. 18 (2009), no. 3, 445–457
  * [ABR11] E. Amerik, F. Bogomolov, and M. Rovinsky, _Remarks on Endomorphisms and Rational Points,_ Compositio Mathematica 147, no. 6 (2011): 1819–42.
  * [AC08] E. Amerik and F. Campana, _Pure and Applied Mathematics Quarterly_ , Volume 4, Number 2 (Special Issue: In honor of Fedor Bogomolov, Part 1 of 2) 509—545, 2008.
  * [AV08] E. Amerik, C. Voisin, _Potential density of rational points on the variety of lines of a cubic fourfold_ , Duke Math. Journal, Vol 145, n0 2, 379-408, (2008).
  * [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, _Geometry of Algebraic Curves, Volume I_ , 1985, Springer.
  * [Bai23] C. Bai, _On Abel–Jacobi maps of Lagrangian families,_ Math. Z. 304, 34 (2023).
  * [Bai24] C. Bai, _On some birational invariants of hyper-Kähler manifolds_ , International Mathematics Research Notices, Volume 2024, Issue 10, May 2024, Pages 8628–8637.
  * [Bai24II] C. Bai, _On the geometry of the higher dimensional Voisin maps_ , ArXiv: 2404.10138.
  * [BDELU17] F. Bastianelli, P. De Poi, L. Ein, R. Lazarsfeld, B. Ullery, _Measures of irrationality for hypersurfaces of large degree_ , Compos. Math. 153 (2017), no. 11, 2368-2393.
  * [BCD] Bastianelli, Cortini, and De Poi, _The gonality theorem of Noether for hypersurfaces_ , J. Algebraic Geom., 23 (2014), no. 2, 313–339.
  * [Baz17] I. Bazhov, _On the Chow group of zero-cycles of Calabi-Yau hypersurfaces_ , manuscripta mathematica 152(1), 2017.
  * [Bea83] A. Beauville, _Variétés kählériennes dont la première classe de Chern est nulle_ , J. Differential Geometry 18(1983), 755-782.
  * [Bea07] A. Beauville. On the splitting of the Bloch-Beilinson filtration, in Algebraic cycles and motives (vol. 2), London Math. Soc. Lecture Notes 344, 38-53; Cambridge University Press (2007).
  * [BD85] A. Beauville, R. Donagi, _The variety of lines of a cubic fourfold_ , C. R. Acad. Sci. Paris 301 Séries I, (1985), 703-706.
  * [Blo80] S. Bloch, _Lectures on algebraic cycles,_ Duke Univ. Math. Ser. IV (1980).
  * [BS83] S. Bloch, and V. Srinivas, _Remarks on correspondences and algebraic cycles,_ Amer. J. Math., 105(5):1235–1253, 1983.
  * [Bog78] F. A. Bogomolov, _Hamiltonian Kähler manifolds_ , Dokl. Akad. Nauk SSSR 243(1978), 1101-1104.
  * [Bot75] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1975), 203–248.
  * [Bri77] J. Briançon. _Description de $Hilb^{n}\mathbb{C}\\{x,y\\}$_, Invent. Math., 41:45–89, 1977.
  * [CG72] H. Clemens, P. Griffiths, _The intermediate Jacobian of the cubic threefold_ , Ann. of Math. (2) 95 (1972), 281–356.
  * [DM98] O. Debarre, L. Manivel, _Sur la variété des espaces linéaires contenus dans une intersection complète_ , Math. Ann. 312 (1998), 549-574.
  * [dCM02] M. A. de Cataldo, L. Migliorini, _The Chow groups and the motive of the Hilbert scheme of points on a surface_ , J. algebra 251 (2002), 824-848.
  * [Del71] P. Deligne, _La conjecture de Weil pour les surfaces K3_ , Inventiones mathematicae, 15(3):206–226, 1971.
  * [Dem76] B. Demazure, _A very simple proof of Bott’s theorem_ , Invent. Math. 33, 271-272, (1976).
  * [DM98] R. Donagi, E. Markman, _Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles_ , Integrable Systems and Quantum Groups (Montecatini Terme 1993), Lecture Notes in Math., 1620, pp. 1-119. Springer, Berlin-Heidelberg, 1998.
  * [EL20] L. Ein, R. Lazarsfeld, _The Konno invariant of some algebraic varieties_ , European Journal of Mathematics volume 6, 420–429 (2020).
  * [Enr49] F. Enriques, Federigo, _Le Superficie Algebriche_ , Nicola Zanichelli, Bologna, 1949, MR 0031770.
  * [Fuj87] A. Fujiki, _On the de Rham Cohomology Group of a Compact Kahler Symplectic Manifold_ , Adv. Stud. Pure Math. 10 (1987), 105-165.
  * [Ful79] W. Fulton, J. Hansen, _A connectedness theorem for projective varieties with applications to intersections and singularities of mappings_ , Annals of Mathematics. 110 (1): 159–166, 1979.
  * [Ful84] W. Fulton, _Intersection Theory_ , Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 2. Springer (1984).
  * [GK20] F. Gounelas, A. Kouvidakis, _Geometry of Lines on a Cubic Four-Fold_ , International Mathematics Research Notices, 2020.
  * [GHS03] T. Graber, J. Harris, J. Starr, _Families of rationally connected varieties_ , J. Amer. Math. Soc. 16 (2003), p. 57-67.
  * [GL14] D. Greb, C. Lehn, _Base manifolds for Lagrangian fibrations on hyperkähler manifolds_ , Int. Math. Res. Not. IMRN 2014, no. 19, 5483–5487.
  * [Gri68] P. Griffiths, _Periods of integrals on algebraic manifolds, I, II_ , Amer. J. Math. 90, 568-626, 805-865, 1968.
  * [Gro69] A. Grothendieck, _Hodge’s general conjecture is false for trivial reasons_ , Inst. Hautes Études Sci. Publ. Math. 29, 95-103, 1969.
  * [HT84] J. Harris, L. W. Tu, _On symmetric and skew-symmetric determinantal varieties_ , Topology, Volume 23, Issue 1, 1984, Pages 71-84.
  * [Huy14] D. Huybrechts, _Curves and cycles on K3 surfaces (with an appendix by C. Voisin)_ , Algebraic Geometry 1 (2014), 69-106.
  * [Huy99] D. Huybrechts, _Compact hyperkähler manifolds: basic results_ , Inventiones mathematicae volume 135, pages 63–113 (1999)
  * [Huy24] D. Huybrechts, _The geometry of cubic hypersurfaces_ , to appear in Cambridge University Press.
  * [HX22] D. Huybrechts, C. Xu, _Lagrangian fibrations of hyperkaehler fourfolds_ , Journal IJM 21 (2022), 921-932
  * [Hwa08] J. Hwang, _Base manifolds for fibrations of projective irreducible symplectic manifolds_ , Invent. Math. 174 (2008), no. 3, 625–644.
  * [IM08] A. Iliev, L. Manivel, _Cubic hypersurfaces and integrable systems_ , Amer. J. Math. 130 (2008), no. 6, 1445–1475.
  * [Kod64-68] K. Kodaira, (1964), _On the structure of compact complex analytic surfaces. I-IV_ , American Journal of Mathematics, 86 (4): 751–798, 88 (3): 682–721, 90 (1): 55–83, 90 (4): 1048–1066. 1964,1966, 1968.
  * [Kol96] J. Kollár, _Rational curves in algebraic varieties_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge (A Series of Modern Surveys in Mathematics), Springer-Verlag Berlin Heidelberg, 1996.
  * [LSV17] R. Laza, G. Saccà, C. Voisin, _A hyper-Kähler compactification of the Intermediate Jacobian fibration associated with a cubic fourfold_ , Acta Mathematica, 218:1 (2017), 55-135.
  * [Lin16] H.-Y. Lin, _On the Chow group of zero-cycles of a generalized Kummer variety_ , Advances in Mathematics 298, 448-472, 2016.
  * [Man92] L. Manivel, _Théorèmes d’annulation pour les fibrés associés à un fibré ample_ , Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 19, no 4 (1992), p. 515-565.
  * [Mat99] D. Matsushita, _On fibre space structures of a projective irreducibe symplectic manifold_ , Topology, 38 (1999), 79-83.
  * [Mat15] D. Matsushita, _On base manifolds of lagrangian fibrations_ , Sci. China Math, 58(2015), no. 3, 531-542.
  * [Mat16] D. Matsushita, _On Deformations of Lagrangian Fibrations_ , K3 Surfaces and Their Moduli. Progress in Mathematics, vol 315. Birkhäuser, Cham.
  * [Mar20] O. Martin, _Zero-cycles and measures of irrationality for abelian varieties_ , PhD Thesis, 2020.
  * [Nik80] V.V. Nikulin, _Integral Symmetric Bilinear Forms and Some of their Applications,_ Math. USSR Izvestija Vol. 14 (1980), No. 1, 103–167.
  * [Otw99] A. Otwinowska, _Remarques sur les groupes de Chow des hypersurfaces de petit degré_ , C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 51–56.
  * [Ran92] Z. Ran, _Deformations of manifolds with torsion or negative canonical bundle_ , J. Alg. Geom. 1(1992), 279-291.
  * [Roi80] A. Roitman, _The torsion of the group of 0-cycles modulo rational equivalence_ , Ann. Math. 111, 553–569, 1980.
  * [SV15] M. Shen, Ch. Vial, _The Fourier transforms for certain hyper-Kähler fourfolds_ , Memoirs of the American Mathematical Society, Volume: 240; 2015;
  * [Stap17] D. Stapleton, _The degree of irrationality of very general hypersurfaces in some homogeneous spaces_ , Ph.D. thesis, Stony Brook University, 2017.
  * [Tia87] G. Tian, _Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric_ , Mathematical Aspects of String Theory (S.T.Yau, ed.), World Scientific, Singa- pore, 1987, 629-646.
  * [Tod88] A.A.Todorov, _The Weil-Petersson geometry of the moduli space of $SU(n\geq 3)$ (Calabi-Yau) manifolds_, Inst. Hautes Études Sci., Nov. 1988.
  * [vGV16] B. van Geemen, C. Voisin, _On a conjecture of Matsushita_ , Int. Math. Res. Notices (2016) Vol. 2016, 3111-3123.
  * [Voi92] C. Voisin, _Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes_ , Complex projective geometry (Trieste, 1989/Bergen, 1989), 294-303, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992.
  * [Voi03] C. Voisin. _Théorie de Hodge et géométrie algébrique complexe_ , Cours spécialisés, SMF 2003.
  * [Voi04] C. Voisin, _Intrinsic pseudo-volume forms and K-correspondences_ , Proceedings of the Fano Conference, (Eds A.Collino, A. Conte, M. Marchisio), Publication de l’Université de Turin (2004).
  * [Voi04II] C. Voisin, _Remarks on filtrations on Chow groups and the Bloch conjecture_ , Annali di Matematica, 2004.
  * [Voi08] C. Voisin. On the Chow ring of certain algebraic hyper-Kähler manifolds. Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 613-649.
  * [Voi10] Claire Voisin, _Hodge loci_ , Handbook of moduli, 3:507–546, 2010.
  * [Voi13] C. Voisin, _The generalized Hodge and Bloch conjectures are equivalent for general complete hypersurfaces_ , J. Alg. Geom. 22, 141-174, 2013.
  * [Voi14] C. Voisin, _Chow rings, decomposition of the diagonal and the topology of families_ , Annals of Math. Studies 187, Princeton University Press 2014.
  * [Voi15] C. Voisin, _Unirational threefolds with no universal codimension $2$ cycle_, Inventiones mathematicae: Volume 201, Issue 1 (2015), Page 207-237.
  * [Voi16] C. Voisin, _Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties_ , in “K3 Surfaces and Their Moduli”, Proceedings of the Schiermonnikoog conference 2014, C. Faber, G. Farkas, G. van der Geer, Editors, Progress in Math 315, Birkhäuser (2016), 365-399.
  * [Voi16II] C. Voisin, _The Hodge conjecture. Open problems in mathematics_ , 521-543, Springer, (2016).
  * [Voi17] C. Voisin, _Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-Kähler manifolds_ , The Abel Symposium Abelsymposium 2017: Geometry of Moduli, pp 295-326.
  * [Voi18] C. Voisin, _Chow ring and the gonality of general abelian varieties_ , Ann. H. Lebesgue 1 (2018), 313–332.
  * [Voi21] C. Voisin, _On fibrations and measures of irrationality of hyper-Kähler manifolds_ , la Revista de la Unión Matemática Argentina, volume spécial MCA 2021.
  * [Voi22I] C. Voisin, _Triangle varieties and surface decomposition of hyper-Kähler manifolds_ , Recent Developments in Algebraic Geometry: To Miles Reid for his 70th Birthday, edited by Hamid Abban; Gavin Brown; Alexander Kasprzyk; Shigefumi Mori, London Mathematical Society Lecture Note Series Cambridge University Press.
  * [Voi22II] C. Voisin, _On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties_ , Advances in Mathematics, Volume 396, 2022.
  * [Yang19] R. Yang, _On irrationality of hypersurfaces in $\mathbb{P}^{n+1}$_, Proc. Amer. Math. Soc. 147 (2019), 971–976.