Periods of families of curves in threefolds

Hossein Movasati 111 Instituto de Matemática Pura e Aplicada, IMPA, Estrada
Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil, www.impa.br/$\sim$
hossein<EMAIL_ADDRESS>
###### Abstract

Clemens’ conjecture states that the the number of rational curve in a generic
quintic threefold is finite. If it is false we prove that certain periods of
rational curves in such a quintic threefold must vanish. Our method is based
on a generalization of a proof of Max Noether’s theorem using infinitesimal
variation of Hodge structures and its reformulation in terms of integrals and
Gauss-Manin connection.

##  1 Introduction

Clemens in [Cle84, page 300] in his study of Griffiths’ Abel-Jacobi mapping,
conjectures that the number of rational curves of degree $d$ in a generic
quintic threefold must be finite. The main motivation for this is a counting
of equations and parameters of a rational curve inside a quintic and its
verification for degree $1$ and $2$ curves by Sh. Katz. Since then the
conjecture is proved for $d\leq 11$. For an overview and many related
references see [Cot12, Cle21]. In this note we discuss a possible
generalization of a proof of Noether-Lefschetz theorem in the framework of
families of curves in quintic threefolds. It states that for a generic smooth
surface $X$ in $\mathbb{P}^{3}$, the only curves in $X$ are obtained by
intersecting $X$ with another surface. Equivalently, the Picard number of $X$
is equal to one. The infinitesimal variation of Hodge structures (IVHS)
developed by Ph. Griffiths and his coauthors in [CGGH83] gives a rigorous
proof of Noether-Lefschetz theorem. In [Mov21, Section 14.4] we have rewritten
this proof using integrals (periods). The main idea lies in the fact that the
restriction of holomorphic two forms over a curve is identically zero, and
hence its integration over the curve is zero. If we have a full family of such
surfaces and curves then we can make derivation of such integrals with respect
to the underlying parameters (this is the origin of Gauss-Manin connection and
IVHS) and we conclude that the integration of all elements of the primitive
cohomology of $X$ over the curve is zero. This concludes the proof. In the
present text we mimic this proof and this leads us to our main result.

######  Theorem 1.

If Clemens’ conjecture is false in degree $d$ then for a holomorphic family of
degree $d$ rational curves $Z_{s},\ s\in(\mathbb{C},0)$ in a generic quintic
threefold $X$ and any $\omega\in F^{2}H^{3}_{\rm dR}(X)$ we have

(1) $\int_{Z_{s}}\frac{\omega}{ds}=0,$

where $F^{\bullet}$ is the Hodge filtration of $H^{3}_{\rm dR}(X)$.

By an analytic family of algebraic cycles $Z_{s},s\in(\mathbb{C},0)$ in $X$ we
mean the following. We take an analytic variety $Z$, holomorphic maps
$f:Z\to(\mathbb{C},0)$ and $g:Z\to X$ such that $f$ is proper and for all
$s\in(\mathbb{C},0)$, $g$ restricted to $Z_{s}:=f^{-1}(s)$ is an injection.
The Gelfand-Leray form $\frac{\omega}{ds}$ in our context is only defined for
$\omega\in F^{2}H^{3}_{\rm dR}(X)=H^{30}\oplus H^{21}$, for which the pull-
back of $\omega$ to the two dimensional variety $Z$ by $f$ can be divided by
$df$. Similar to the case of Abel-Jacobi map, we also use the integration over
the path of homology between two algebraic cycles. According to [Cle84] this
goes back at least to [Wei62, page 333] which in turn must be inspired by
Picard’s intensive study of two dimensional integrals in [PS06]. The proof of
Theorem 1 actually implies that the Abel-Jacobi map attached to
$Z_{s},s\in(\mathbb{C},0)$ is identically zero, however, this is much weaker
statement than the vanishing 1, see Section 7. The integration over algebraic
cycles as in (1) appears first in Deligne’s study of absolute Hodge cycles,
see [DMOS82], and further applications of this in the study of Hodge loci has
been initiated in [Mov21, Chapters 18,19] [MV18].

The sketch of the proof of Theorem 1 is as follows. If a generic quintic
contains an infinite number of genus $g$ curves then we have to rigorously
define families of rational curves which vary inside families of quintic
threefolds. This is done in Section 2 for rational curves. This is the only
place which we consider $g=0$ and it is expected that Theorem 1 is true for
curves of any genus. For instance, in Section 7 we show that for curves which
are obtained by the intersection of $X$ with
$\mathbb{P}^{2}\subset\mathbb{P}^{4}$ we have automatically (1). We then study
integrals over homology paths between two curves and give a formula for its
derivation, see Section 3. Some of these integrals are identically zero. This
is described in Section 4. Derivating these integrals we get more vanishing
integrals and this finishes the proof in Section 5. In Section 6 we describe
how one can compute integrals (1) and as an example we perform this
computation for a well-known family of lines inside Fermat quintic. The
condition $X$ to be generic in Theorem 1 comes from our difficulties in
Section 2 to handle families of algebraic cycles. We expect that for any
family of rational curves inside any smooth quintic threefold, (1) is not
identically zero in $s$.

I presented the main ideas of the present paper in an informal online seminar
and I would like to thank the audience for their comments and questions. This
includes Ethan Cotterill who introduced me to the recent preprint [Mus21],
Roberto Villaflor with whom I had useful discussions regarding the Abel-Jacobi
map, Felipe Ramos who pointed out the correct definition of Hodge filtration
in relative de Rham cohomology in Section 4, and Younes Nikdelan, Jin Cao,
Jorge Duque. Finally, many thanks go to P. Deligne who wrote two useful
letters for the first draft of the present paper. These can be found in the
author’s webpage.

##  2 Hilbert schemes

In the following we will discuss Hilbert schemes. We will not use scheme
structure of these objects; only the underlying analytic variety will be used.
Let ${\sf T}$ be an open subset of a Hilbert scheme parametrizing smooth
projective varieties $X\subset\mathbb{P}^{N}$ of dimension $2n+1$. Let also
${\sf S}$ be another component of a Hilbert scheme of projective varieties
$Z\subset\mathbb{P}^{N}$ of dimension $n$. We consider the incidence scheme

${\sf I}:=\\{(X,Z)\in{\sf T}\times{\sf S}|Z\subset X\\}.$

We fix an irreducible component $\tilde{\sf I}$ of ${\sf I}$, consider the
projections $\pi:{\sf I}\to{\sf T},\ \tilde{\sf I}\to{\sf T}$ and denote their
images by ${\sf V}$ and $\tilde{\sf V}$, respectively. From now on we denote
by $t$ a point in ${\sf T}$ and by $X_{t}$ the corresponding projective space.
Our main example is ${\sf
T}:=\mathbb{P}\left(\mathbb{C}[x_{0},x_{1},\ldots,x_{n+1}]_{a}\right)$ with
$n=3,\ a=5$ and the following with $N=4$ which is not actually a Hilbert
scheme but it will be enough for our purposes. A rational curve is given by
the image of a map

$f:\mathbb{P}^{1}\to\mathbb{P}^{N},[x:y]\mapsto[f_{0}(x,y):\cdots:f_{N}(x,y)],$

where $f_{i}(x,y)$ are homogeneous polynomials of degree $d$ in $x,y$. We may
expect that the quotient

(2)
$\mathbb{P}\left(\mathbb{C}[x,y]_{d}^{N+1}-\mathbb{C}[x,y]_{d}\mathbb{C}^{N+1}\right)/{\rm
PSL}(2,\mathbb{C}),$

exists as an algebraic variety over $\mathbb{C}$, where
$\mathbb{C}[x,y]\mathbb{C}^{N+1}$ corresponds to those $f$ such that its image
is a point. We may also expect that the universal family over ${\sf S}$
exists. This is

$\left(\mathbb{P}^{1}\times\mathbb{P}\left(\mathbb{C}[x,y]_{d}^{N+1}-\mathbb{C}[x,y]_{d}\mathbb{C}^{N+1}\right)\right)/{\rm
PSL}(2,\mathbb{C}),$

where the action of $A\in{\rm PSL}(2,\mathbb{C})$ in $\mathbb{P}^{1}$ is given
by the inverse of $A$, and hence, we have well-defined map

$F:{\sf P}\to\mathbb{P}^{N},\ \ \ \
[x:y],(f_{0},f_{1},\ldots,f_{N})\mapsto[f_{0}(x,y);f_{1}(x,y):\cdots:f_{N}(x,y)].$

We have a natural projection ${\sf P}\to{\sf S}$ such that its fibers are
rational curves and $F$ restricted to these fibers restores the map $f$. E.
Cotterill reminded me that the moduli space ${\sf S}$ has a natural
stratification given by the geometric genus of the rational curve. Each strata
might be isomorphic to a Hilbert scheme of rational curves of degree $d$ in
$\mathbb{P}^{N}$. It might be clarifying to write down the details of all
this. We avoid these details and simply define:

(3) ${\sf
S}:=\mathbb{P}\left(\mathbb{C}[x,y]_{d}^{N+1}-\mathbb{C}[x,y]_{d}\mathbb{C}^{N+1}\right),\
{\sf P}:=\mathbb{P}^{1}\times{\sf S}.$

In this way we have an action of ${\rm PSL}(2,\mathbb{C})$ on the incidence
scheme ${\sf I}$ such that it leaves the fibers of ${\sf I}\to{\sf T}$
invariant. In this example it is known that ${\sf I}$ is irreducible for
$d\leq 11$, however, for $d\geq 12$ it might have many irreducible components,
see [Cot12] and the references therein.

Figure 1: Incidence scheme

We take smooth points $t_{0}\in\tilde{\sf V},\ p\in\tilde{\sf I}$ with
$\pi(p)=t_{0}$ and $p$ is a regular point of $\pi$, that is, the derivative of
$\pi$ at $p$ is surjective. This follows from: Let
$\pi:(\mathbb{C}^{n},0)\to(\mathbb{C}^{m},0)$ be a surjective holomorphic map.
Then the locus of points $p\in(\mathbb{C}^{n},0)$ such that the derivative of
$\pi$ at $p$ is not surjective, is a proper analytic subset of
$(\mathbb{C}^{n},0)$, and hence its complement is dense. By implicit function
theorem, we can take coordinate system
$(t,s)\in(\mathbb{C}^{n}\times\mathbb{C}^{m},0)\cong(\tilde{\sf I},p)$ and
$t\in(\mathbb{C}^{n},0)\cong{\sf V}$ such that $\pi$ is just projection in the
$t$ coordinate, see Figure 1. We get a family $(X_{t},Z_{t,s})$ such that
$Z_{t,s}\subset X_{t}$.

######  Remark 1.

In the context of Hilbert schemes this family is flat. For simplicity, we
start with a pair $(X,Z)$ such that both $X$ and $Z$ are irreducible, and
hence, we can assume that $X_{t}$ and $Z_{t,s}$ are irreducible. In our main
example 3, since we have also the action of ${\rm PSL}(2,\mathbb{C})$ on the
fibers of $\tilde{\sf I}\to{\sf T}$ and this action has only discrete
(dimension zero) stabilizers, we replace $(\mathbb{C}^{m},0)$ with some linear
subspace $(\mathbb{C}^{m-3},0)$ which intersects the orbits of ${\rm
PSL}(2,\mathbb{C})$ in discrete sets.

If ${\sf V}$ is a proper subset of ${\sf T}$ ($\pi$ is not dominant) then a
generic $X$ does not contain any algebraic cycle $Z$. If ${\sf T}=\tilde{\sf
V}$ and $m=0$ for all components $\tilde{\sf I}$ of ${\sf I}$ then we have a
finite number of $Z$ inside a generic $X$. Note that $n=\dim(\tilde{\sf V})$
and $n+m$ is the dimension of $\tilde{\sf I}$. From now on we consider the
case $m>0$. For our main example this means that we assume that Clemens’
conjecture does not hold. Our main goal in this section is to describe
instances such that the following property holds:

######  Property 1.

We have

  1. 1.

The homology classes of $Z_{t,s_{1}},Z_{t,s_{2}}$ in $H_{*}(X_{t},\mathbb{Z})$
for $s_{1},s_{2}\in(\mathbb{C}^{m},0)$ and $t\in(\mathbb{C}^{n},0)$ are the
same. The homology path $\delta_{t}$ between $Z_{t,s_{1}}$ and $Z_{t,s_{2}}$
is supported in the family of algebraic cycles
$\cup_{s\in(\mathbb{C}^{m},0)}Z_{t,s}\subset X_{t}$.

  2. 2.

For $s$ fixed and $t_{1},t_{2}\in(\mathbb{C}^{n},0)$, the homology class of
$Z_{t_{1},s}$ is mapped to the homology class of $Z_{t_{2},s}$ under the
monodromy map $H_{*}(X_{t_{1}},\mathbb{Z})\to H_{*}(X_{t_{2}},\mathbb{Z})$.

Even though the varieties $X_{t}$ are $C^{\infty}$ isomorphic, the algebraic
cycles $Z_{t_{1},s},Z_{t_{2},s}$ might not be isomorphic topologically. For
instance, $Z_{t_{1},s}$ might be a curve with only one nodal singularity and
$Z_{t_{2},s}$ a curve with only one cuspidal singularity.

######  Proposition 1.

If ${\sf S}$ parameterizes rational curves of degree $d$ as in (3) then
Property 1 holds.

###### Proof.

We have an small open smooth set $U$ of $\tilde{\sf I}$ with the coordinate
system $(t,s)$. We get a holomorphic family
$f_{t,s}:\mathbb{P}^{1}\to\mathbb{P}^{N},[x:y]\mapsto[f_{0}(x,y):\cdots:f_{N}(x,y)]$,
where $f_{i}(x,y)$ are homogeneous polynomials of degree $d$ in $x,y$
depending holomorphically in $(t,s)$. Such $f_{i}$’s are just the ingredients
of the projection of $(t,s)$ under $U\to{\sf S}$ (recall the definition of
${\sf S}$ in 3). By definition the image of $f_{t,s}$ is $Z_{t,s}$. We get map
$f_{t,s,*}:H_{2}(\mathbb{P}^{1},\mathbb{Z})\to H_{2}(X_{t},\mathbb{Z})$, and
since $\mathbb{Z}$ is discrete, we conclude the desired property. By our
considerations in Remark 1, for fixed $t$ the family of algebraic cycles
$Z_{t,s}$ is not constant in $s$. ∎

From now on we replace $(\mathbb{C}^{m},0)$ with a one dimensional subspace
and assume that $m=1$.

######  Definition 1.

Let $X$ be a smooth projective variety and $Z_{1},Z_{2}\subset X$ be two
algebraic cycles. We say that $Z_{1}$ is strongly algebraically equivalent to
$Z_{2}$ if we have a holomorphic family of algebraic cycles
$Z_{s},s\in(\mathbb{C},0)$ such that

  1. 1.

$Z_{1}$ and $Z_{2}$ are two members of this family.

  2. 2.

The homology path $\delta$ between $Z_{1}$ and $Z_{2}$ is supported in the
family of algebraic cycles $\cup_{s\in(\mathbb{C},0)}Z_{s}\subset X$.

######  Remark 2.

From the first item of the above definition it follows that the homology
classes of $Z_{1},Z_{2}$ in $H_{*}(X,\mathbb{Z})$ are the same, see for
instance [Ful98, Proposition 19.1.1, page 373] for this implication. It is
natural to expect that the second item holds automatically, however, this does
not seem to be the case. The proof in the mentioned reference is not adaptable
to provide this stronger statement.

Recall Property 1. For fixed $t$ and $s_{1},s_{2}\in(\mathbb{C}^{m},0)$,
$Z_{1,t}:=Z_{t,s_{1}}$ and $Z_{2,t}:=Z_{t,s_{2}}$ are strongly algebraically
equivalent. Definition 1 combines topological and algebraic ingredients,
whereas the the classical definition of algebraic equivalence in algebraic
geometry is purely algebraic. It would be instructive to compare these
definitions.

Figure 2: Gauss-Manin connection

######  Remark 3.

A more general definition is as follows: Let $X$ be a smooth projective
variety and and $Z_{1},Z_{2}\subset X$ be two irreducible subvariety of $X$.
We say that $Z_{1}$ and $Z_{2}$ are flat algebraically equivalent if we have
have holomorphic maps $f:Z\to(\mathbb{C}^{m},0)$ and $g:Z\to X$ such that $f$
is flat and $g$ restricted to the fibers of $f$ is an injection. Duco van
Straten kindly reminded me [January 14, 2019] that “the requirement of
flatness of a cycle is not a reasonable condition at all. Flatness is just too
special. There is the notion of families of cycles used by Barlet in analytic
and by Kollar in algebraic geometry[see [Kol95], page 45-46 and [Bar75]]. It
is very ugly and algebraically hard to use, but geometrically reasonable.” For
our purpose we need Proposition 1 which is very specific to rational curves.
One might expect that if two irreducible algebraic cycle $Z_{1},Z_{2}\subset
X$ are flat algebraically equivalent then they are homologically equivalent.
It is not known to the author whether this is true for rational curves.

######  Remark 4.

We take two transversal sections $\Sigma_{1}$ and $\Sigma_{2}$ in $({\sf
I},p)$ to the fiber of $\pi$ such that under projection $\pi$ they are
biholomorphic to $(\mathbb{C}^{n},0)$. For instance,
$\Sigma_{i}:=\\{s=s_{i}\\}$ for two values $s_{1}$ and $s_{2}$ of $s$. We have
a family $Z_{1,t},Z_{2,t}\subset X_{t},\ t\in(\mathbb{C}^{n},0)$ such that
$Z_{1,t}:=Z_{t,s_{1}}$ and $Z_{2,t}:=Z_{t,s_{2}}$ are strongly algebraically
equivalent, see Figure 2.

##  3 Gauss-Manin connection

In this section we write down a formula which can be formulated in terms of
the Gauss-Manin connection in relative cohomology, however, for the sake of
simplicity and avoiding unnecessary notations, we present it as derivating
integrals over non closed homology classes.

Let us consider a projective variety $X$ of dimension $2n+1$ and two algebraic
cycles $Z_{1},Z_{2}\subset X$ of dimension $n$ such that their homology class
$[Z_{1}]$ and $[Z_{2}]$ are equal in $H_{2n}(X,\mathbb{Z})$, that is, $Z_{1}$
is homologous to $Z_{2}$. Let $Y=Z_{1}\cup Z_{2}$ and $\delta\in
H_{2n+1}(X,Y)$ be the homology between $Z_{1}$ and $Z_{2}$, that is, it is
mapped to $[Z_{2}]-[Z_{1}]\in H_{2n}(Y,\mathbb{Z})$ under the boundary map
$H_{2n+1}(X,Y)\to H_{2n}(Y)$.

We consider a one parameter holomorphic family $(X_{t},Y_{t}),\
t\in(\mathbb{C},t_{0})$ of such objects, that is we have a proper submersion
$\pi:{\sf X}\to(\mathbb{C},t_{0})$ and ${\sf Y}={\sf Z}_{1}\cup{\sf
Z}_{2}\subset{\sf X}$ with the properties of fibers as above. From this we
mainly mean the family $(X_{t},Z_{1,t}\cup Z_{2,t})$ constructed in
Proposition 1, see also Remark 4. The subindex $t$, for instance
$\delta_{t}\in H_{2n+1}(X_{t},Y_{t})$, will be used in order to emphasize the
dependence in $t$. For a complex manifold $M$, $\Omega^{i}_{M^{\infty}}$ and
$\Omega^{i}_{M}$ denotes the set of global $C^{\infty}$ and holomorphic
$i$-forms in $M$, respectively. Let $\omega\in\Omega^{2n+1}_{{\sf
X}^{\infty}}$ and $\alpha\in\Omega^{2n}_{{\sf Y}^{\infty}}$ such that

$d\omega\in\pi^{-1}\Omega^{1}_{(\mathbb{C},t_{0})}\wedge\Omega^{2n+1}_{{\sf
X}^{\infty}},\ \omega|_{{\sf
Y}}-d\alpha\in\pi^{-1}\Omega^{1}_{(\mathbb{C},t_{0})}\wedge\Omega^{2n}_{{\sf
Y}^{\infty}}.$

In the following theorem we have assumed that the integrals depends
holomorphically in $t$. One can achieve this by considering algebraic de Rham
cohomology and it is the case when $X_{t}$’s are hypersurfaces.

######  Theorem 2.

We have

(4) $\frac{\partial}{\partial
t}\int_{\delta_{t}}\omega=\int_{\delta_{t}}\frac{d\omega}{dt}+\int_{Z_{2,t}}\frac{\omega|_{{\sf
Y}}-d\alpha}{dt}-\int_{Z_{1,t}}\frac{\omega|_{{\sf
Y}}-d\alpha}{dt}+\frac{\partial}{\partial
t}\left(\int_{Z_{2,t}-Z_{1,t}}\alpha\right).$

###### Proof.

We define $D_{t}$ to be the locus in ${\sf X}$ which $\delta_{t}$ sweeps:

$D_{t}=\cup_{s\in[0,1]}\delta_{\gamma_{t}(s)}\subset{\sf X}$

such that $\gamma$ is a path in $(\mathbb{C},t_{0})$ connecting $t_{0}$ to
$t$, see Figure 2. Note that $D_{t}$ is oriented in a canonical way such that

$\partial D_{t}:=(\pi|_{{\sf Z}_{1}})^{-1}(\gamma)+\delta_{t}-(\pi|_{{\sf
Z}_{2}})^{-1}(\gamma)-\delta_{t_{0}}=\gamma_{1}+\delta_{t}-\gamma_{2}-\delta_{t_{0}},$

see Figure 2. By Stokes theorem we have

$\displaystyle\int_{\delta_{t}}\omega$ $\displaystyle=$
$\displaystyle\int_{D_{t}}d\omega+\int_{\delta_{t_{0}}}\omega+\int_{\gamma_{2}}(\omega|_{{\sf
Z}_{2}}-d\alpha)-\int_{\gamma_{1}}(\omega|_{{\sf
Z}_{1}}-d\alpha)+\int_{\gamma_{2}-\gamma_{1}}d\alpha.$

Taking the differential $\frac{\partial}{\partial t}$ of the above equality
and using the Fubini’s theorem we get the desired result. ∎

##  4 A kind of Hodge cycle

For a smooth projective variety $X$, the integration of
$F^{\frac{n}{2}+1}H^{n}_{\rm dR}(X)$ over an algebraic cycle of dimension
$\frac{n}{2}$ in $X$ is zero, and this simple fact leads us to the notion of
Hodge cycle and then the Hodge conjecture, see [MV21, Section 5.13]. In this
section we follow a similar argument to distinguish between algebraic
equivalence of algebraic cycles in the sense of Definition 1 and homology
between algebraic cycles. For a pair of algebraic varieties $Y\subset X$, the
relative algebraic de Rham cohomology is defined as the hypercohomology of
complexes $H^{m}_{\rm dR}(X,Y):={\mathbb{H}}^{m}(X,\Omega^{\bullet}_{X,Y})$,
where $\Omega^{m}_{X,Y}:=\Omega^{m}_{X}\times\Omega^{m-1}_{Y}$ is equipped
with $d:\Omega^{m}_{X,Y}\to\Omega^{m+1}_{X,Y},\
d(\omega,\alpha):=(d\omega,\omega|_{Y}-d\alpha)$. A naive definition of the
Hodge filtration $F^{i}H^{m}_{\rm
dR}(X,Y):={\mathbb{H}}^{m}(X,\Omega^{\bullet\geq i}_{X,Y})$ is not correct and
it turns out that the correct definition compatible with long exact sequences
is: $F^{i}H^{m}_{\rm dR}(X,Y):={\rm
Image}({\mathbb{H}}^{m}(X,F^{i}\Omega^{\bullet}_{X,Y})\to{\mathbb{H}}^{m}(X,\Omega^{\bullet}_{X,Y}))$,
where

(5) $(F^{i}\Omega^{\bullet}_{X,Y})^{m}:=\left\\{\begin{array}[]{cc}0&m<i\\\
\Omega^{m}_{X}&m=i\\\ \Omega^{m}_{X,Y}&m>i\end{array}\right.,$

see [PS08, Theorem 3.22]. We will consider a good covering $\\{U_{i}\\}_{i\in
I}$ of $X$ and consider elements of cohomologies relative to this covering.
For more details see [MV21].

######  Proposition 2.

Let $X$ be a smooth projective variety of dimension $2n+1$ and let
$Z_{1},Z_{2}$ be two subvarieties of $X$ of dimension $n$. If $Z_{1}$ and
$Z_{2}$ are strongly algebraically equivalent (in the sense of Definition 1)
then

(6) $\int_{\delta}F^{n+2}H^{2n+1}_{\rm dR}(X,Y)=0,$

where $\delta$ is the homology between $Z_{1}$ and $Z_{2}$.

###### Proof.

The homology path $\delta$ between $Z_{1}$ and $Z_{2}$ is supported in the
family of algebraic cycles $\cup_{s\in(\mathbb{C},0)}Z_{s}$ which is the image
of a holomorphic map $g:Z\to X$ with $\dim(Z)=n+1$. By definition,
$F^{n+2}H^{2n+1}_{\rm dR}(X,Y)$ contains only differential $i$-forms with
$i>n+1$. Therefore, the pull-back of this piece of the Hodge filtration to $Z$
is identically zero, and hence, its restriction to ${\rm Image}(g)$ and its
integration over $\delta$ is zero. ∎

######  Remark 5.

The vanishing (6) follows the same principle which has produced the Hodge
conjecture, see [MV21, Proposition 5.12, Section 5.13]. One might speculate
(similar to the Hodge conjecture) that its inverse is true, that is, if
$Z_{1}$ and $Z_{2}$ as above are homologous and we have (6) then they are
strongly algebraically equivalent. Note that there might not exist closed
cycles $\delta\in H_{2n+1}(X,\mathbb{Q})$ such that (6) happens, in other
words we might have $H^{2n+1}(X,\mathbb{Q})\cap\left(H^{n,n+1}\oplus
H^{n+1,n}\right)=0$. Therefore, our speculation does not follow from
reformulation of the generalized Hodge conjecture by A. Grothendieck.

######  Proposition 3.

Under the canonical map $H^{2n+1}_{\rm dR}(X,Y)\to H^{2n+1}_{\rm dR}(X)$, the
piece of Hodge filtration $F^{i}H^{2n+1}_{\rm dR}(X,Y)$ is mapped
isomorphically to $F^{i}H^{2n+1}_{\rm dR}(X)$ for $i\geq n+1$. For $i\leq n$,
the kernel of $F^{i}H^{2n+1}_{\rm dR}(X,Y)\to F^{i}_{\rm dR}H^{2n+1}(X)$ is
one dimensional.

###### Proof.

We have the long exact sequence

(7) $H^{2n}_{\rm dR}(X)\to H^{2n}_{\rm dR}(Y)\to H^{2n+1}_{\rm dR}(X,Y)\to
H^{2n+1}_{\rm dR}(X)\to H^{2n+1}(Y)=0$

and hence $H^{2n+1}_{\rm dR}(X,Y)\to H^{2n+1}_{\rm dR}(X)$ is surjective.
Moreover, $H^{2n}_{\rm dR}(Y)$ is of dimension $2$ generated by the cohomology
classes $[Z_{1}]$ and $[Z_{2}]$. After writing the above long exact sequence
in homology and using the fact that $Z_{1}$ and $Z_{2}$ are homologous in $X$
through $\delta$, we observe that the integration of the images $x_{1}$ and
$-x_{2}$ of $[Z_{1}]$ and $-[Z_{2}]$ in $H^{2n+1}_{\rm dR}(X,Y)$ over $\delta$
are the same and over the image of $H_{2n+1}(X)\to H_{2n+1}(X,Y)$ are zero.
This implies that the image of $H^{2n}_{\rm dR}(Y)\to H^{2n+1}_{\rm dR}(X,Y)$
is one dimensional generated by $x_{1}=-x_{2}$. We take a good covering ${\cal
U}$ of $(X,Y)$. It turns out that an element of $H^{2n+1}_{\rm dR}(X,Y)$
relative to this covering is of the form

(8)
$\omega=(\omega^{0},0)+(\omega^{1},\alpha^{0})+\cdots+(\omega^{n},\alpha^{n-1})+(\omega^{n+1},\alpha^{n})+(\omega^{n+2},0)+\cdots+(\omega^{2n+1},0)$
$(\omega^{j},\alpha^{j-1})\in C^{2n+1-j}({\cal U},\Omega^{j}_{X,Y}),$

We have $\alpha^{j}=0,\ \ j=n+1,\cdots,2n$ because $\dim(Y)=n$. This implies
the statement on the pieces of Hodge filtration for $i\geq n+2$. For $i=n+1$
we note that $\alpha^{n}=0$ by our definition of Hodge filtration in (5). ∎

Note that using the long exact sequences of the pair $(X,Y)$ both in homology
and cohomology we have

(9) $\int_{\delta}\tilde{\alpha}=\int_{Z_{2}-Z_{1}}\alpha,\ \ \ \ \
\forall\alpha\in H^{2n}_{\rm dR}(Y),$

and $\tilde{\alpha}$ is the image of $\alpha$ under $H^{2n}_{\rm dR}(Y)\to
H^{2n+1}_{\rm dR}(X,Y)$. Let $Z_{s},s\in(\mathbb{C},0)$ be a family of
algebraic cycles as in Introduction. We fix points $s_{0},s\in(\mathbb{C},0)$,
and a path $\gamma$ which connects $s_{0}$ to $s$ in $(\mathbb{C},0)$. Let $n$
be the complex dimension of the fibers and $Y_{s}=Z_{s}\cup Z_{s_{0}}$. We
consider $\omega\in\check{F}^{n+1}H^{2n+1}_{\rm dR}(X)$ and

$I(s):=\int_{Z_{s_{0}}}^{Z_{s}}\omega:=\int_{f^{-1}(\gamma)}\omega.$

Since $\omega$ is a closed differential $2n+1$, this integral does only
depends on the homotopy class of $\gamma$. By the definition of Hodge
filtration, the Cech cohomology representation of $\omega\in
F^{n+1}H^{2n+1}(X,Y_{s})$ in (8) starts with $(\omega^{n+1},\alpha^{n})$.
Moreover, its pull-back by $g$ contains only the term
$(\omega^{n+1},\alpha^{n})$. By definition of $\check{F}^{n+1}H^{2n+1}_{\rm
dR}(X)$ we have $\alpha^{n}=0$ and define the Gelfand-Leray form in our
context as

$\frac{\omega}{df}:=\frac{\omega^{n+1}}{df}\in H^{2n}_{\rm dR}(Z_{s}).$

######  Proposition 4.

We have

$I^{\prime}(s)=\int_{Z_{s}}\frac{\omega}{df}.$

###### Proof.

We use Fubini’s theorem:

$I(s)=\int_{f^{-1}(\gamma)}\frac{\omega}{df}\wedge
ds=\int_{\gamma}\left(\int_{Z_{s}}\frac{\omega}{df}\right)ds.$

∎

######  Remark 6.

If instead of $(\mathbb{C},0)$ we use an algebraic (compact) curve $C$, $I(s)$
is a multi valued function with branching points in the set of atypical values
of $f$. We denote it by $A\subset C$ (it also includes the singularities of
$C$). The difference of two branches of $I$ is the period
$\int_{f^{-1}(\gamma)}\omega$, where $\gamma$ is a closed path in $C\backslash
A$. In this case Proposition 4 implies that $I^{\prime}(s)$ is a meromorphic
function on $C$.

##  5 Proof of Theorem 1

We will use Griffiths description of the cohomology of hypersurfaces in
[Gri69]. Let ${\sf T}$ be the parameter space of smooth hypersurfaces of
degree $d$ in $\mathbb{P}^{2n+2}$, $F=\sum_{t\in{\sf T}}t_{\alpha}x^{\alpha}$
be the universal homogeneous polynomial of degree $d$ in
$x=(x_{0},x_{1},\cdots,x_{2n+2})$ and ${\sf
X}:=\\{F=0\\}\subset\mathbb{P}^{2n+2}\times{\sf T}$. We define

$\omega_{P}={\rm Resi}\frac{P\Omega}{F^{k}},\ \ \deg(P)=k\cdot d-(2n+3),\ \
\Omega:=\sum_{i=0}^{2n+2}(-1)^{i}x_{i}\widehat{dx_{i}}$

For the case $(2n+3)|d$ we also define $\omega:=\omega_{1}$ ($1$ refers to the
constant polynomial $1$). This gives us $\omega_{P}\in H^{2n+1}(X)$, and since
$H^{2n+1}_{\rm dR}(X,Y)\to H^{2n+1}_{\rm dR}(X)$ is surjective it comes from
an element in $H^{2n+1}_{\rm dR}(X,Y)$ which we denote it again by
$\omega_{P}$. By Proposition 3, for $\deg P=kd-(2n+3),\ k\leq n+1$ this
element is unique as $F^{n+1}H^{2n+1}_{\rm dR}(X,Y)\to F^{n+1}H^{2n+1}_{\rm
dR}(X)$ is an isomorphism.

Consider the case $n=1$ and $d=5$. Assume that Clemens’ conjecture is not
true. It follows that for a component $\tilde{\sf I}$ of ${\sf I}$, the map
$\pi:\tilde{\sf I}\to{\sf T}$ in Section 2 is surjective, and hence, ${\sf
V}={\sf T}$. Moreover, the irreducible components of fibers of $\pi$ have
dimension $\geq 3$ (recall that they are ${\rm PSL}(2,\mathbb{C})$-invariant).
We take $t_{0}\in{\sf T}$ which is still a generic point of ${\sf T}$, and we
cannot assume that $X_{t_{0}}$ is the Fermat variety. We get a holomorphic
family of algebraic curves $Z_{1,t},Z_{2,t}\subset X_{t},\ \ t\in({\sf T},0)$
such that $Z_{1,t}$ and $Z_{2,t}$ are strongly algebraically equivalent. We
use Proposition 2 and we get

(10) $\int_{\delta}\omega=0.$

Note that by Proposition 3, $F^{3}H^{3}(X,Y)$ is one dimensional and it is
generated by $\omega$. We write $\omega$ in a covering and it is of the form
$(\omega^{3},0)$, where $\omega_{3}=\\{\omega_{3,i}\\}$. Note that the element
$\omega^{3}$ restricted to ${\sf Y}$ is identically zero, and hence in Theorem
2 the corresponding $\alpha$ can be taken $0$. Since $\omega$ is
$D_{X,Y}$-closed, we have $d\omega=dt\wedge*$ and $\delta\omega=dt\wedge*$.
Consider a derivation $\frac{\partial}{\partial t}$ in ${\sf T}$, for instance
take $F=F_{0}+tx^{i}$, where $x^{i}$ is a homogeneous monomial of degree $d$
in $x$. Since $\tilde{\sf I}\to{\sf T}$ is surjective, we can have such
derivations for all monomials $x^{i}$ of degree $d$. We have

(11) $d\left(\frac{P\Omega}{F^{k}}\right)=dt\wedge\frac{-k\frac{\partial
F}{\partial t}P\Omega}{F^{k+1}}$

where $d$ is the differential in the variety $(\mathbb{P}^{4}\times{\sf
T})\backslash{\sf X}$. This can be seen easily after passing to an affine
coordinate. Taking residue we have

$d\omega_{P}=dt\wedge\omega_{-k\frac{\partial F}{\partial t}P}$

We make the derivation of (10) with respect to $t$, use Theorem 2 and conclude
that

(12) $\int_{\delta}\omega_{P}=0,\ \ \forall P\in\mathbb{C}[x]_{5}.$

In the Cech cohomology representation of $\omega_{P}$ as above, it is of the
form $(\omega^{3},0)+(\omega^{2},0)$ because

$D_{X,Y}(\omega^{3},0)=(d\omega^{3},0)+(\delta\omega^{3},0).$

Therefore, $\omega_{P}$ is in $F^{2}H^{3}_{\rm dR}(X)$. We use Proposition 4
and conclude that $\int_{Z_{s}}\frac{\omega_{P}}{ds}=0$.

######  Remark 7.

A more constructive way to prove Clemens’ conjecture, is to give an example of
a smooth quintic threefold with isolated rational curves. In this paper we
have generalized a proof of Noether’s theorem, and T. Shioda in [Shi81] has
given an explicit example of a surface of prime degree and with Picard rank
equal to one, and hence, he has given a constructive proof of Noether’s
theorem. The following quintic is a natural generalization of Shioda’s example

$x_{0}^{d}+x_{1}x_{2}^{d-1}+x_{2}x_{3}^{d-1}+x_{3}x_{4}^{d-1}+x_{4}x_{1}^{d-1}=0,\
\ d=5,$

and since $5$ is a prime number, we may expect that Clemens’ conjecture holds
for this quintic

##  6 Computing period of curves

In order to gather evidences that the periods (1) do not vanish identically in
$s$, one may try to compute them for well-known families of rational curves
inside smooth quintic threefolds, see for instance [CDVV12, Wal12, Mus21]. In
this section, we describe how to compute the period (1) and perform this for
one family of rational curves.

Let $X\subseteq\mathbb{P}^{m+1}$ be a smooth degree $d$ hypersurface given by
$X=\\{F=0\\}$ and

$\Omega=\sum_{i=0}^{m+1}(-1)^{i}x_{i}dx_{0}\wedge\cdots\widehat{dx_{i}}\cdots\wedge
dx_{m+1}.$

Let $P$ be a polynomial of degree $d(q+1)-m-2)$ with $q\in\\{0,1,\ldots,m\\}$.
By the computations in [CG80] we know that

(13) ${\rm
Resi}\left(\frac{P\Omega}{F^{q+1}}\right)=\frac{(-1)^{m}}{q!}\left\\{\frac{P\Omega_{J}}{F_{J}}\right\\}_{|J|=q}\in
H^{q}({\cal U},\Omega_{X}^{m-q}),$

where $\Omega_{J}:=\iota_{\frac{\partial}{\partial
x_{j_{q}}}}(\cdots\iota_{\frac{\partial}{\partial x_{j_{0}}}}(\Omega)\cdots)$,
$F_{J}:=F_{j_{0}}\cdots F_{j_{q}}$, $F_{j}:=\frac{\partial F}{\partial
x_{i}}$, $j_{0}<j_{1}<\cdots<j_{q}$ and ${\cal U}=\\{U_{i}\\}_{i=0}^{m+1}$ is
the Jacobian covering restricted to $X$ given by $U_{i}=\\{F_{i}\neq 0\\}\cap
X$. We have also

$\Omega_{J}=(-1)^{j_{0}+j_{1}+\cdots+j_{q}+\binom{q+2}{2}}\sum_{l=0}^{m-q}(-1)^{l}x_{k_{l}}\widehat{dx_{k_{l}}},$

where $k_{0}<k_{1}<\cdots<k_{m-q}$ is obtained form $0,1,2,\ldots,m+1$ by
removing $j_{0},j_{1},\ldots,j_{q}$, see for instance [Vil18]. We apply this
theorem for the Fermat variety of dimension $m=3$ and degree $d=5$ and with
$q=1$. In this case the Jacobian covering is the usual covering by
$U_{i}:=\\{x_{i}\not=0\\}$. For a homogeneous polynomial $P$ of degree $5$ up
to constant independent of $X$, we have

$\omega_{P}:={\rm
Resi}\left(\frac{P\Omega}{F^{2}}\right)=\left\\{\sum_{j_{2}\not=j_{0},j_{1}}\frac{(-1)^{j_{2}^{*}}Px_{j_{2}}}{x_{j_{0}}^{4}x_{j_{1}}^{4}}dx_{j_{3}}\wedge
dx_{j_{4}}\right\\}\in H^{1}({\cal U},\Omega_{X}^{2}),$

where the sum runs over
$j_{2}\in\\{0,1,\ldots,4\\}\backslash\\{j_{0},j_{1}\\}$ with
$\\{j_{0},j_{1},\ldots,j_{4}\\}=\\{0,1,\ldots,4\\}$, and $j_{2}^{*}$ is the
position of $j_{2}$ in $0,1,2,3,4$ with $j_{0},j_{1}$ removed (counting from
zero). Let $[x_{0}(s,t):x_{1}(s,t):\cdots:x_{4}(s,t)],\ \ s\in(\mathbb{C},0),\
t\in\mathbb{P}^{1}$ be a family of rational curves inside the Fermat quintic
threefold. We write $x_{i}(s,t)=x_{i}(t)+sy_{i}(t)+O(s)$ and $dx_{j_{3}}\wedge
dx_{j_{4}}=(x_{j_{3}}^{\prime}y_{j_{4}}-x_{j_{4}}^{\prime}y_{j_{3}})dt\wedge
ds+O(s)$. We conclude that

$\int_{Z_{s}}\frac{\omega_{P}}{ds}=\sum_{j_{0},j_{1}}{\rm
Resi}_{x_{j_{0}}=0}\left(\frac{P}{x_{j_{0}}^{4}x_{j_{1}}^{4}}\sum_{j_{2}\not=j_{0},j_{1}}(-1)^{j_{2}^{*}}x_{j_{2}}(x_{j_{3}}^{\prime}y_{j_{4}}-x_{j_{4}}^{\prime}y_{j_{3}})dt\right)$

where ${\rm Resi}_{x_{j_{0}=0}}$ refers to sum of the residues around the
points of $x_{j_{0}}=0$ in $\mathbb{P}^{1}$. Note that for
$x_{i}=x_{i}(\frac{at+b}{ct+d})$ with $a,b,c,d$ depending on $s$, the
expression $x_{j_{2}}^{\prime}y_{j_{3}}-x_{j_{3}}^{\prime}y_{j_{2}}$ is
identically zero because $\frac{\partial}{\partial
s}x_{i}(s,t)=x_{i}^{\prime}(t)\frac{\partial}{\partial
s}\left(\frac{at+b}{ct+d}\right)$. This is natural because the action of ${\rm
PSL}(2,\mathbb{C})$ on $\mathbb{P}^{1}$ does not produce families of rational
curves. Note also that $y_{0}x_{0}^{4}+y_{1}x_{1}^{4}+\cdots+y_{4}x_{4}^{4}$.
Families of lines in the Fermat quintic fourfold has been studied in [AK91].
It has $50$ one dimensional families of the form

$[x:y]\mapsto[x:-\zeta y:ay:by:cy],\ \zeta^{5}=1,\ a^{5}+b^{5}+c^{5}=0.$

We set $x=t,y=1$ and write this as
$x(t,s)=[t:-\zeta:1:s:(-s^{5}-1)^{\frac{1}{5}}],\ s\in(\mathbb{C},0)$. We have
$y=[0:0:0:1:0],x^{\prime}=[1:0:0:0:0]$ and so the only non-trivial possibility
for $(j_{3},j_{4})$ is $(0,3)$. Therefore, we have
$(j_{0},j_{1},j_{2})=(1,2,4),(1,4,2),(2,4,1)$ and hence up to some $2\pi i$
factor

$\int_{Z_{s}}\frac{\omega_{P}}{ds}={\rm
Resi}_{x_{1}=0}\left(\frac{P}{x_{1}^{4}x_{2}^{4}}x_{4}dt\right)+{\rm
Resi}_{x_{1}=0}\left(\frac{P}{x_{1}^{4}x_{4}^{4}}(-x_{2})dt\right)+{\rm
Resi}_{x_{2}=0}\left(\frac{P}{x_{2}^{4}x_{4}^{4}}(-x_{1})dt\right).$

The third residue is zero as $x_{2}=-\zeta$ is constant. We set
$P=x_{1}^{3}x_{2}^{2}$ and as $x_{2}$ and $x_{4}$ are independent of $t$ and
we have

$\int_{Z_{s}}\frac{\omega_{P}}{ds}=\frac{x_{4}}{x_{2}^{2}}-\frac{x_{2}^{3}}{x_{4}^{4}}=\frac{(-s^{5}-1)^{\frac{1}{5}}}{\zeta^{2}}+\frac{\zeta^{3}}{(-s^{5}-1)^{\frac{4}{5}}}.$

It would be also nice to make the computations for the following family of
conics described in [Mus21]:

$a^{2}(x_{0}+x_{1})=b^{2}(x_{2}+x_{3}),\ bx_{4}=c(x_{0}+x_{1}),\
b(x_{0}^{2}+x_{1}^{2})=\pm
ia(x_{2}^{2}+x_{3}^{2})=0,a^{10}+b^{10}-4b^{5}c^{5}=0.$

##  7 Abel-Jacobi map

Let $Z_{s},\ s\in{\sf S}$ be a family of curves in a smooth threefold $X$. We
fix a curve $Z_{s_{0}}$ and the Abel-Jacobi map is defined as

${\sf S}\to{\rm Jac}(X):=\frac{(F^{2}H^{3}_{\rm
dR}(X))^{\vee}}{H_{3}(X,\mathbb{Z})},\
s\mapsto\int_{Z_{s_{0}}}^{Z_{s}}F^{2}H^{3}_{\rm dR}(X),$

where the integration takes over the homology $\delta$ between $Z_{s_{0}}$ and
$Z$: $\partial\delta=Z_{s}-Z_{s_{0}}$. For a proof of the existence of such a
homology path see [Ful98, Proposition 19.1.1, page 373], however, note that
from this proof it is not clear whether $\delta$ is supported in
$\cup_{s\in{\sf S}}Z_{s}$ or not. If the parameter space ${\sf
S}=\mathbb{P}^{1}$ is the rational curve, it is well-known that there is no
non constant holomorphic map from $\mathbb{P}^{1}$ to a complex compact tori,
and hence, the Abel-Jacobi map is zero in this case. This implies that for
some homology $\delta$ between $Z_{s_{0}}$ and $Z_{s}$ we have in
$\int_{\delta}F^{2}=0$, but not necessarily $\delta$ is supported in
$\cup_{s\in\mathbb{P}^{1}}Z_{s}$. There are examples of this situation for
which we have this stronger statement.

######  Proposition 5.

Let $Z\subset X$ be an irreducible divisor in $X$ with a singular set of
codimension $\geq 2$ and such that a desingularization $\tilde{Z}$ of $Z$
satisfies $H_{1}(\tilde{Z},\mathbb{Q})=0$. Then for any rational function $f$
in $Z$ and family of curves $Z_{s}:=f^{-1}(s),\ s\in\mathbb{P}^{1}$ we have
$\int_{Z_{s_{0}}}^{Z_{s}}F^{2}=0$, where the integration takes place over
$f^{-1}(\gamma)$ and $\gamma$ is any path in $\mathbb{P}^{1}$ connecting
$s_{0}$ to $s$.

###### Proof.

The pull-back $\pi^{*}\omega$ of $\omega\in H^{3}_{\rm dR}(X)$ by the
desingularization map $\pi:\tilde{Z}\to Z\subset X$ is an element in
$H^{3}_{\rm dR}(\tilde{Z})$ which in turn by hard Lefschetz theorem is
isomorphic to $H^{1}_{\rm dR}(\tilde{Z})$. By our hypothesis this is zero and
so $\pi^{*}\omega=D\eta$, for some $2$-cocycle
$\eta\in\oplus_{i=0}^{2}C^{i}({\cal U},\Omega^{2-i}_{\tilde{Z}})$. Here, we
are using the notation of hypercohomology relative to a good covering ${\cal
U}$ of $\tilde{Z}$. Let $\tilde{f}=f\circ\pi$ and
$\tilde{Z}_{s}:=\tilde{f}^{-1}(s)$. Since a desingularization is a
biholomorphism over a Zariski open subset and no component of a fiber of $f$
is inside the singular set of $Z$, it is enough to prove that
$\int_{\tilde{Z}_{s_{0}}}^{\tilde{Z}_{s}}F^{2}=0$. By Stokes theorem we have

$\int_{\tilde{Z}_{s_{0}}}^{\tilde{Z}_{s}}\pi^{*}\omega=\int_{\tilde{Z}_{s}}\eta-\int_{\tilde{Z}_{s_{0}}}\eta$

The map $\mathbb{P}^{1}\to\mathbb{C},s\mapsto\int_{Z_{s}}\eta$ is holomorphic,
and hence, constant. ∎

######  Remark 8.

Note that if $\dim(X)>2$ then by the long exact sequence of
$0\to\mathbb{Z}\to{\cal O}_{X}\to{\cal O}_{X}^{*}\to 0$ and $H^{1}(X,{\cal
O}_{X})=H^{2}(X,{\cal O}_{X})=0$, we know that $H^{1}(X,{\cal
O}_{X}^{*})\cong\mathbb{Z}$ and any divisor is a hypersurface section (not
necessarily transversal section) of $X$. By Lefschetz theorem a transversal
(and hence smooth) hyperplane section $Z$ of $X$ satisfies
$H_{1}(Z,\mathbb{Q})=0$.

######  Remark 9.

In the proof of Theorem 1 we have the vanishing integral (12) for $\delta$
which is supported in the family of cycles $Z_{s},\ s\in\mathbb{C}$. This
implies that in this context the Abel-Jacobi map is zero. Theorem 1 is
stronger than this.

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