# Algebraic intermediate hyperbolicities

Antoine Etesse  Antoine Etesse
Aix Marseille Univ
CNRS, Centrale Marseille, I2M
Marseille
France<EMAIL_ADDRESS>,  Ariyan Javanpeykar  Ariyan Javanpeykar
Institut für Mathematik
Johannes Gutenberg-Universität Mainz
Staudingerweg 9, 55099 Mainz
Germany<EMAIL_ADDRESS>and  Erwan Rousseau  Erwan Rousseau
Institut Universitaire de France & Aix Marseille Univ
CNRS, Centrale Marseille, I2M
Marseille
France<EMAIL_ADDRESS>
###### Abstract.

We extend Lang’s conjectures to the setting of intermediate hyperbolicity and
prove two new results motivated by these conjectures. More precisely, we first
extend the notion of algebraic hyperbolicity (originally introduced by
Demailly) to the setting of intermediate hyperbolicity and show that this
property holds if the appropriate exterior power of the cotangent bundle is
ample. Then, we prove that this intermediate algebraic hyperbolicity implies
the finiteness of the group of birational automorphisms and of the set of
surjective maps from a given projective variety. Our work answers the
algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

###### Key words and phrases:

Hyperbolicity, moduli spaces of maps, ampleness, positivity, Lang conjectures

###### 2020 Mathematics Subject Classification:

32Q45 14E07 14J60 14G99 (11G35, 14G05, )

The third author was supported by the ANR project “FOLIAGE”, ANR-16-CE40-0008
and by the European Union’s Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie grant agreement No 75434

## Introduction

This paper is concerned with Lang’s conjectures on hyperbolic varieties.
Lang’s conjectures relate different notions of hyperbolicity for $X$ (see
[Lan86] and [Javb, §12] for a summary of his conjectures) from complex
analysis to algebraic geometry and number theory. The aim of this paper is to
prove several results motivated by these conjectures, and to extend Lang’s
conjectures on hyperbolic varieties to the more general setting of
“intermediate hyperbolicity” (see Section 1). The latter is hinted at in
Lang’s original paper, but not worked out anywhere in the literature in full.
In fact, in [Lan86, page 162] Lang explicitly says that “I omit the whole area
of intermediate hyperbolicity“. In this paper, we present his conjectures in
the intermediate setting, and prove several new results going in their
direction.

Let us briefly explain the notion of “intermediate hyperbolicity” from a
complex-analytic perspective. Recall that a complex space $X$ is _Kobayashi
hyperbolic_ if and only if the Kobayashi pseudometric $d_{X}$ on $X$ is a
metric (see [Kob98]). In the hope of understanding the property of being
Kobayashi hyperbolic for a complex space, Eisenmann [Eis70] extended
Kobayashi’s definition and introduced the notion of $p$-analytic hyperbolicity
for all $1\leq p\leq\dim(X)$ (see [Dem97, Definition 1.3.(ii)] for a precise
definition).

We note that $1$-analytic hyperbolicity and Kobayashi hyperbolicity coincide
for compact complex-analytic spaces. Furthermore, a complex space $X$ of
dimension $d=\dim(X)$ is $d$-analytically hyperbolic (as defined below) if and
only if it is “measure-hyperbolic” (as defined in Kobayashi’s book [Kob98]).
Moreover, if $X$ is $p$-analytically hyperbolic, then it is
$(p+1)$-analytically hyperbolic, so that we have a string of implications

$X\ \textrm{is}\ 1\textrm{-analytically hyperbolic}\implies\ldots\implies X\
\textrm{is}\ (\dim X)\textrm{-analytically hyperbolic}.$

In particular, the notions of $p$-analytic hyperbolicity interpolate between
the notions of Kobayashi hyperbolicity and measure hyperbolicity. This is why
we sometimes refer to the notion of $p$-hyperbolicity as being an
“intermediate hyperbolicity”. Kobayashi-Ochiai proved that a smooth projective
variety $X$ of general type (i.e., $\omega_{X}$ is big) is
$\dim(X)$-analytically hyperbolic, and Demailly generalized their result to
the setting of intermediate hyperbolicity by replacing the condition that
$\omega_{X}$ is big by a different positivity condition on
$\Lambda^{p}\Omega^{1}_{X}$. We state here a simpler version of his result for
the sake of convenience.

######  Theorem 0.1 (Demailly).

Let $X$ be a smooth projective variety over $\mathbb{C}$. If
$\bigwedge^{p}\Omega^{1}_{X}$ is ample, then $X$ is $p$-analytically
hyperbolic.

Motivated by Lang’s conjectures, the first aim of this paper is to prove
algebraic analogues of Demailly’s theorem (see Theorem 0.5 for a precise
statement). Our second aim is to investigate finiteness properties of
varieties which are hyperbolic in some intermediate sense (see Theorem 0.8).

###  0.1. An algebraic analogue of Demailly’s analytic theorem on intermediate
hyperbolicity

We introduce a notion of intermediate hyperbolicity inspired by Demailly’s
notion of algebraic hyperbolicity [Dem97] (see also [Java, Javb, JK20, JX,
Rou10]). Let us first introduce the following convenient terminology:

######  Definition 0.2 (Non-degenerate maps).

A rational map $f:Y\dashrightarrow X$ between projective varieties is said to
be non-degenerate if its image $f(Y)$ is of dimension $\min(\dim(X),\dim(Y))$.

Note that, if $\dim(Y)\leq\dim(X)$, a rational map $Y\dashrightarrow X$ of
projective varieties is non-degenerate if and only if it is generically finite
onto its image. If $\dim(Y)>\dim(X)$, such a rational map is non-degenerate if
and only if it is dominant.

######  Definition 0.3 ($p$-algebraic hyperbolicity modulo $\Delta$).

Let $X$ be a projective variety, let $\Delta$ be a closed subset of $X$, and
let $p$ be a positive integer. We say that $X$ is _$p$ -algebraically
hyperbolic modulo $\Delta$_ if, for every ample line bundle $L$ on $X$, there
is a real number $\alpha=\alpha(X,\Delta,L)$ such that, for every smooth
projective $p$-dimensional variety $Y$ with ample canonical bundle
$\omega_{Y}$ and every non-degenerate rational map $f:Y\dashrightarrow X$
whose image is not included in $\Delta$, the following inequality is satisfied

$(f^{*}L)\cdot(K_{Y})^{p-1}\leq\alpha\cdot K_{Y}^{p}.$

Following Lang’s terminology, we will say that $X$ is _pseudo-
$p$-algebraically hyperbolic_ if there is a proper closed subset
$\Delta\subsetneq X$ such that $X$ is $p$-algebraically hyperbolic modulo
$\Delta$. Also, we will say that $X$ is _$p$ -algebraically hyperbolic_ if $X$
is $p$-algebraically hyperbolic modulo the empty subset.

Note that the pull-back by $f$ of the line bundle $L$ is well-defined: since
$Y$ is smooth, the rational map $f$ is well-defined outside a closed subset
$F$ of codimension at least $2$, and one has the equality of the Picard groups
$\operatorname{Pic}(Y\setminus F)=\operatorname{Pic}(Y)$. Note also that a
projective variety $X$ is $1$-algebraically hyperbolic modulo the emptyset if
and only if it is algebraically hyperbolic in Demailly’s sense [JK20]. More
generally, a projective variety $X$ is pseudo-$1$-algebraically hyperbolic if
and only if it is pseudo-algebraically hyperbolic [Javb, §9].

Observe that we restrict to varieties $Y$ with ample canonical bundle. This is
analogous to the fact that one may test the algebraic hyperbolicity of a
projective variety on maps from curves of genus at least two (see the
introduction of [JK20] for a detailed explanation). Let us also emphasize on
the fact that the constant $\alpha$ appearing in the definition is independent
of $Y$: see Definition 0.6 where this hypothesis is relaxed.

Our first result gives an algebraic analogue of Demailly’s theorem (Theorem
0.1). To state it, we recall the definition of ampleness modulo a closed
subset. We refer the reader to Lazarsfeld’s books for a definition of the
augmented base locus [Laz04a, Laz04b].

######  Definition 0.4.

Let $X$ be a projective variety, let $\Delta\subset X$ be a closed subset, let
$E$ be a vector bundle on $X$, and let $p:\mathbb{P}(E^{\vee})\to X$ be the
natural projection. Then $E$ is _ample modulo $\Delta$_ if the augmented base
locus $B^{+}(\mathcal{O}_{\mathbb{P}(E^{\vee})}(1))$ is included in
$p^{-1}(\Delta)$.

In other words, $E$ is ample modulo $\Delta$, if and only if for any ample
line bundle $A$ on $\mathbb{P}(E^{\vee})$, there is an integer $m\geq 1$ such
that the base locus of $\mathcal{O}_{\mathbb{P}(E^{\vee})}(m)\otimes A^{-1}$
is included in $p^{-1}(\Delta)$.

Our first main result now reads as follows.

######  Theorem 0.5.

Let $X$ be a smooth projective variety, and let $\Delta\subset X$ be a proper
closed subset. If $\bigwedge^{p}\Omega^{1}_{X}$ is ample modulo $\Delta$, then
$X$ is $p$-algebraically hyperbolic modulo $\Delta$.

This algebraic analogue of Demailly’s analytic result fits in well with Lang’s
“pseudofied” notions of hyperbolicity (see [Lan86]). In fact, Lang defines a
variety to be pseudo-hyperbolic if it is hyperbolic modulo some proper closed
subset. He conjectured that pseudo-hyperbolicity (in any sense of the word
“hyperbolic”) coincides with being of general type (see [Lan86, page 161] or
the more recent [Javb, §12]). Demailly actually also proved a “pseudofied”
version of his theorem (Theorem 0.1). We omit a precise formulation of his
result here.

The algebraic hyperbolicity of a projective variety implies that moduli spaces
of maps from varieties are “bounded” (i.e. have only finitely many connected
components). We refer to [JK20] for precise statements. The following
definition extends the notion of boundedness introduced in [JK20] (see also
[Javb, JX]).

######  Definition 0.6 ($p$ algebraically bounded modulo $\Delta$).

Let $X$ be a projective variety, let $\Delta$ be a closed subset of $X$, and
let $p$ be a positive integer. We say that $X$ is _$p$ -algebraically bounded
modulo $\Delta$_ if, for every ample line bundle $L$ on $X$, every smooth
projective $p$-dimensional variety $Y$ with $\dim Y=p$, and every ample line
bundle $A$ on $Y$, there is a real number $C>0$ such that, for every non-
degenerate rational map $f:Y\dashrightarrow X$ whose image is not included in
$\Delta$, the following inequality is satisfied

$f^{*}L\cdot A^{p-1}\leq C.$

Following Lang’s terminology as before, we will say that $X$ is _pseudo-
$p$-algebraically bounded_ if there is a proper closed subset
$\Delta\subsetneq X$ such that $X$ is $p$-algebraically bounded modulo
$\Delta$. Also, we say that $X$ is _$p$ -algebraically bounded_ if it is
$p$-algebraically bounded modulo the empty subset. With this terminology at
hand, a projective variety is $1$-algebraically bounded over $k$ if and only
if it is bounded in the sense of [Javb, §10].

######  Remark 0.7.

Voisin introduced an algebraic analogue of (analytic) measure-hyperbolicity;
see [Voi03, Definition 2.20]. Indeed, she defines a variety $X$ to be
_algebraically measure hyperbolic_ if, for every ample line bundle $L$ on $X$,
there exists a constant $A>0$ such that, for any covering family of curves
$\pi:\mathcal{C}\to B$ (with generic fiber $C$ of genus $g$) and every
dominant map $\phi:\mathcal{C}\to X$ non-constant on the fibers, one has
$2g-2\geq A\cdot\deg\phi_{C}^{*}L.$ Conjecturally, if $X$ is a smooth
projective variety, then $X$ is $\dim X$-algebraically hyperbolic if and only
if it is algebraically measure hyperbolic (in Voisin’s sense). For example, by
[Voi03, Lemma 2.19], a variety of general type is algebraically measure-
hyperbolic, and in this paper we prove the similar (a priori different)
statement that a variety of general type is $\dim X$-algebraically hyperbolic;
see Theorem 0.5 in the case $p=\dim(X)$.

The notion of intermediate boundedness is weaker than the notion of
intermediate algebraic hyperbolicity (see Proposition 2.1), but we conjecture
that it is equivalent (see Section 1). Both notions allow for strings of
implications (see Proposition 2.2 and 2.3 for precise statements). However, we
emphasize that for intermediate algebraic boundedness this statement is only
proven under the additional assumption that the base field $k$ is uncountable.

These two intermediate notions of pseudo-hyperbolicity turn out to be strong
enough to force certain finiteness properties, as we show now.

###  0.2. Kobayashi-Ochiai’s finiteness theorem

For our final result, our starting point is Kobayashi-Ochiai’s finiteness
theorem for varieties of general type. Namely, if $X$ is a projective variety
of general type over $\mathbb{C}$ and $Y$ is a projective variety, then the
set of dominant rational maps $Y\dashrightarrow X$ is finite. As Lang’s
“intermediate” conjectures predict that a pseudo $p$ algebraically bounded
projective variety is of general type, our following result is in accordance
with Lang’s conjectures.

######  Theorem 0.8.

Let $p\geq 1$ be an integer and let $X$ be a projective pseudo
$p$-algebraically bounded variety over $\mathbb{C}$. Then the group of
birational automorphisms $\mathrm{Bir}(X)$ is finite and for every projective
variety $Y$, the set $\mathrm{Sur}(Y,X)$ of surjective morphisms $Y\to X$ is
finite.

We can use Theorem 0.5 and Theorem 0.8 to reprove part of Kobayashi-Ochiai’s
finiteness theorem. Indeed, if $X$ is a projective variety of general type,
then $X$ is $\dim(X)$ algebraically hyperbolic by Theorem 0.5 (since asking
for $\bigwedge^{\dim(X)}\Omega_{X}$ to be ample modulo some proper closed
subset $\Delta$ is equivalent to $X$ being of general type), hence $\dim(X)$
algebraically bounded by Proposition 2.1. In particular, by Theorem 0.8, we
deduce that $\mathrm{Bir}(X)$ is finite and $\mathrm{Sur}(Y,X)$ is finite for
every projective variety $Y$. (Note that we do not obtain the finiteness of
the set of dominant rational maps from $Y$ to $X$.)

Theorem 0.8 provides an “intermediate” version of the finiteness of
$\operatorname{Aut}(X)$ and $\mathrm{Bir}(X)$ proven for a pseudo-bounded
projective variety $X$ in [JK20, JX]. Also, it generalizes Matsumura’s
finiteness theorem that a variety of general type has only finitely many
automorphisms, and provides algebraic analogues of arithmetic finiteness
results proven in [Java].

Kobayashi asked about analytic analogues of Matsumura’s finiteness theorem for
$\mathrm{Aut}(X)$. In fact, in [Kob93] Kobayashi asked whether a $\dim(X)$
analytically hyperbolic projective variety has only finitely many
automorphisms. The answer is expected to be positive, as such a variety is
expected to be of general type. For recent progress on Kobayashi’s problem we
refer the reader to [Ete]. Our result (Theorem 0.8) can be interpreted as
providing a positive answer to the algebraic analogue of Kobayashi’s question.

###  0.3. Outline of paper

In Section 1, we state conjectures relating several intermediate notions of
hyperbolicity. We take the opportunity to introduce several _new_ arithmetic
notions of intermediate hyperbolicity which will be the topic of future works.
In Section 2, we state and prove basic results concerning intermediate
algebraic hyperbolicity and intermediate algebraic boundedness. In particular,
our definitions and terminologies are (à posteriori) motivated and explained
by Proposition 2.5. In Section 3, we prove our criteria ensuring pseudo
$p$-algebraic hyperbolicity (Theorem 0.5). Finally, in Section 4, we prove our
finiteness results (Theorem 0.8) for pseudo $p$-algebraically bounded
varieties.

######  Acknowledgements.

The second named author thanks Junyi Xie for explaining Hanamura’s work on
birational self-maps (Section 4.2). The second named author is grateful to the
IHES and the University of Paris-Saclay for their hospitality. Part of this
work was done during the visit of the three authors at the Freiburg Institute
for Advanced Studies; they thank the Institute for providing an excellent
working environment.

######  Conventions.

Throughout this paper, we will let $k$ denote an algebraically closed field of
characteristic zero. A _variety over $k$_ is a finite type separated integral
(i.e. irreducible and reduced) scheme over $k$. Unless stated otherwise
explicitly (e.g. if we want to restrict ourselves to complex projective
varieties as in Theorem 0.8), all our varieties will be defined over $k$.

##  1\. Lang’s intermediate algebraic and arithmetic conjectures

In this section, we pursue the intermediate $p$-hyperbolicity analogues of
Lang’s conjectures in the algebraic and arithmetic setting (thereby leaving
out the complex-analytic analogues which are discussed in [Ete]). We will
build on Lang’s original conjectures [Lan86] and the extensions of his
conjectures summarized in [Javb, §12].

Throughout this section, let $k$ be an algebraically closed field of
characteristic zero. Given a proper scheme $X$ over $k$, we refer to [Javb,
§7] for the definition of pseudo-Mordellicity, to [Javb, §9] for the
definition of pseudo-algebraic hyperbolicity, and to [Javb, §10] for the
definition of pseudo-boundedness. The notions of $p$-algebraic hyperbolicity
and $p$-algebraic boundedness are defined in the introduction (see Definition
0.3 and Definition 0.6). To state the general conjecture for varieties of
general type, we will need one additional definition. To state this
definition, we refer to [Javb, §3] for the notion of a model.

######  Definition 1.1 ($p$-Mordellicity modulo $\Delta$).

Let $p\geq 0$ be an integer, let $X$ be a proper variety over $k$, and let
$\Delta\subset X$ be a closed subset. Then, we say that $X$ is _$p$ -Mordellic
modulo $\Delta$ over $k$_ if, for every finitely generated subfield $K\subset
k$, every model $\mathcal{X}$ for $X$ over $K$, and every $p$-dimensional
smooth projective geometrically connected variety $Y$ over $K$, the set of
non-degenerate rational maps $f:Y\dashrightarrow\mathcal{X}$ whose images are
not included in $\Delta$ is finite.

######  Definition 1.2.

Let $p\geq 0$. A proper variety $X$ over $k$ is _$p$ -Mordellic over $k$_ if
$X$ is $p$-Mordellic modulo the empty subset over $k$.

######  Definition 1.3.

Let $p\geq 0$. A proper variety $X$ over $k$ is _pseudo- $p$-Mordellic over
$k$_ if there is a proper closed subset $\Delta\subsetneq X$ such that $X$ is
$p$-Mordellic modulo $\Delta$ over $k$.

Note that $X$ is Mordellic over $k$ (as defined in [Javb, §3]) if and only if
$X$ is $0$-Mordellic modulo the empty subset over $k$. Indeed, a
$0$-dimensional smooth projective geometrically connected variety $Y$ over $K$
is isomorphic to $\operatorname{Spec}K$, so that the set of morphisms
$Y\to\mathcal{X}$ equals the set of $K$-rational points of $\mathcal{X}$. Note
also that the fact that we allow for $p=0$ in the above definition is an
artifact of the arithmetic setting; finiteness of “points” is a reasonable
property to impose (and study) over finitely generated fields of
characteristic zero.

Let $X$ be a projective variety over $k$, and let $\Delta\subset X$ be a
closed subset. Assume that for every algebraically closed field extension
$L/k$ of finite transcendence degree, the variety $X_{L}$ is Mordellic modulo
$\Delta_{L}$ over $L$. Then, for every $p$, the variety $X$ is $p$-Mordellic
modulo $\Delta$ over $k$. Indeed, the set of rational maps
$f:Y\dashrightarrow\mathcal{X}$ with $f(Y)\not\subset\Delta$ equals the set of
$K(Y)$-rational points of $\mathcal{X}\setminus\Delta$, and the latter is
finite by the Mordellicity assumption on the varieties $X_{L}$.

For varieties of general type, we expect all notions (including the
intermediate ones) of pseudo-hyperbolicity to coincide. The following
conjecture provides a precise statement.

######  Conjecture 1.4 (Lang’s intermediate pseudo-conjectures).

Let $X$ be a projective variety over $k$, and let $p\geq 1$ be an integer.
Then the following are equivalent.

  1. (1)

The variety $X$ is of general type.

  2. (2)

There is a proper closed subset $\Delta\subsetneq X$ such that every
subvariety $Y\subset X$ of dimension at least $p$ with $Y\not\subset\Delta$ is
of general type.

  3. (3)

The variety $X$ is pseudo-Mordellic over $k$.

  4. (4)

The variety $X$ is pseudo-$p$-Mordellic over $k$.

  5. (5)

The variety $X$ is pseudo-algebraically hyperbolic over $k$.

  6. (6)

The variety $X$ is pseudo-$p$-algebraically hyperbolic over $k$.

  7. (7)

The variety $X$ is pseudo-bounded over $k$.

  8. (8)

The variety $X$ is pseudo-$p$-algebraically bounded over $k$.

  9. (9)

The variety $X$ is $\dim(X)$-Mordellic over $k$.

  10. (10)

The variety $X$ is $\dim(X)$-algebraically-hyperbolic over $k$.

  11. (11)

The variety $X$ is $\dim(X)$-algebraically-bounded over $k$.

Note that (1) is independent of $p$, so that part of this conjecture is
reduntant. For example, $(5)$ is equivalent to $(6)$ with $p=1$. Nonetheless,
we chose to present the conjecture in this way to facilitate discussing known
results in the following remark.

######  Remark 1.5 (What do we know about Conjecture 1.4 ?).

The following statements hold.

  1. (1)

Obviously, $(2)\implies(1)$.

  2. (2)

If $\dim X=1$, then Conjecture 1.4 holds by Faltings’s proof of Mordell’s
conjecture and the classical finiteness theorem of De Franchis-Severi for
Riemann surfaces. More generally, if $X$ is a closed subvariety of an abelian
variety, then conjecture 1.4 holds by Faltings’s proof of Mordell-Lang [Fal94]
and the work of Ueno, Bloch-Ochiai-Kawamata [Kaw80] and Yamanoi [Yam15] on
closed subvarieties of abelian varieties.

  3. (3)

If $\dim X=2$, then $(3)\implies(2)$, $(5)\implies(2)$, and $(7)\implies(2)$.
This is explained in [Javb].

  4. (4)

By [BJK], we have that $(5)\implies(7)$.

  5. (5)

In this paper we prove $(6)\implies(8)$ (Proposition 2.1), and thus
$(10)\implies(11)$. We also show that $(1)\implies(10)$ (and thus
$(1)\implies(11)$). See Theorem 0.5.

  6. (6)

Assuming $k$ is uncountable, we show that $(8)\implies(11)$; see Proposition
2.2.

  7. (7)

We show that, if $\Lambda^{p}\Omega^{1}_{X}$ is ample modulo some proper
closed subset, then $X$ satisfies $(1)$, $(2)$, $(6)$, $(8)$, $(10)$, and
$(11)$ (see Theorem 0.5).

  8. (8)

We show that, assuming $X$ satisfies $(10)$ or $(11)$, then
$\mathrm{Bir}_{k}(X)$ is finite and $\mathrm{Sur}_{k}(Y,X)$ is finite for
every $Y$ (see Theorem 0.8).

Part of the above conjecture already appears in Lang’s original paper. For
example, the equivalence of $(1)$, and $(3)$ is stated explicitly in [Lan86].
Moreover, the equivalence of $(1)$, $(5)$, and $(7)$ is implicit in Lang’s
original conjectures (see [Javb, §12]). However, the other conjectured
equivalences are new.

To conclude this section, we push Lang’s conjectures further, and state the
general conjecture for “intermediate” exceptional loci.

######  Conjecture 1.6 (Lang’s intermediate conjectures for exceptional loci).

Let $X$ be a projective variety over $k$, let $\Delta\subset X$ be a closed
subset, and let $p$ be a positive integer. Then the following statements are
equivalent.

  1. (1)

Every subvariety $Y$ of $X$ of dimension at least $p$ with
$Y\not\subset\Delta$ is of general type.

  2. (2)

The variety $X$ is Mordellic modulo $\Delta$ over $k$.

  3. (3)

The variety $X$ is $p$-Mordellic modulo $\Delta$ over $k$.

  4. (4)

The variety $X$ is $p$-algebraically hyperbolic modulo $\Delta$ over $k$.

  5. (5)

The variety $X$ is $p$-algebraically bounded modulo $\Delta$ over $k$.

Note that Conjecture 1.6 (for all $X$ and $p$) implies Conjecture 1.4 (for all
$X$ and $p$).

######  Remark 1.7 (What do we know about this conjecture?).

By Proposition 2.1, we have that $(3)\implies(4)$. We also show that, if
$\Lambda^{p}\Omega_{X}$ is ample modulo $\Delta$, then $(1),(3)$, and $(4)$
hold; see Theorem 0.5.

##  2\. Algebraic hyperbolicity and algebraic boundedness

In this section, we prove a few basic facts concerning intermediate algebraic
hyperbolicity and algebraic boundedness, justifying in particular our
definitions. Let us first record the relationship between intermediate
algebraic hyperbolicity and intermediate algebraic boundedness in the
following proposition:

######  Proposition 2.1.

Let $X$ be a projective variety and let $\Delta\subset X$ be a proper closed
subset of $X$. If $X$ is $p$-algebraically hyperbolic modulo $\Delta$, then
$X$ is $p$-algebraically bounded modulo $\Delta$.

###### Proof.

Let $Y$ be a smooth projective variety of dimension $p$, and consider a
ramified cover $\psi:\tilde{Y}\to Y$ with $\tilde{Y}$ smooth and
$K_{\tilde{Y}}$ ample (see e.g. [Laz04a]). Denote $R$ the ramification divisor
so that one has the following equality of divisors:

$K_{\tilde{Y}}=\psi^{*}K_{Y}+R.$

Let $f:Y\dashrightarrow X$ be non-degenerate rational map, whose image is not
included in $\Delta$. Fix $L\to X$ an ample line bundle on $X$ and $A\to Y$ an
ample line bundle on $Y$. By hypothesis on $X$, there exists a constant
$\alpha$ independent of $f$ and $Y$ such that the following inequality is
satisfied

$(\psi\circ f)^{*}L\cdot K_{\tilde{Y}}^{p-1}\leq\alpha K_{\tilde{Y}}^{p}.$

Let $m$ be an integer such that $mK_{\tilde{Y}}-\psi^{*}A$ is ample: $m$ does
depend on $Y$ but not on $f$. Then one has the following inequality:

$(\psi\circ f)^{*}L\cdot\left(\psi^{*}A\right)^{p-1}\leq\alpha
m^{p-1}K_{\tilde{Y}}^{p}.$

Using the projection formula, one deduces that $X$ is indeed $p$-algebraically
bounded, as the bound on the right does not depend on the rational map $f$. ∎

Let us now prove that the notions of intermediate algebraic hyperbolicity and
intermediate algebraic boundedness form a string of implications, starting
with the case of algebraic boundedness:

######  Proposition 2.2.

Assume that $k$ is uncountable. Let $X$ be a projective variety, let
$\Delta\subset X$ be a closed subset, and let $p\in\mathbb{N}_{\geq 1}$. If
$X$ is $p$-algebraically bounded modulo $\Delta$, then $X$ is
$(p+1)$-algebraically bounded modulo $\Delta$.

###### Proof.

Fix $L$ an ample line bundle on $X$, $Y$ a smooth projective variety of
dimension $p+1$ and $A$ a very ample line bundle on $Y$. We argue by
contradiction. Thus, suppose that there exists a sequence of non-degenerate
rational maps $f_{i}:Y\dashrightarrow X$ whose image are not included in
$\Delta$ such that $f_{i}^{*}L\cdot A^{p}$ tends to infinity as $i$ tends to
infinity. As $k$ is uncountable, one can choose a smooth ample divisor
$H\subset Y$ taken in the linear system $|A|$ such that, for any
$i\in\mathbb{N}_{\geq 1}$, the rational map $(f_{i})_{|H}:H\dashrightarrow X$
is a well-defined non-degenerate map whose image is not included in $\Delta$.
Since $f_{i}^{*}L\cdot A^{p}=(f_{i})_{|H}^{*}L\cdot A_{|H}^{p-1}$, this
contradicts the $p$-algebraic boundedness of $X$. ∎

Replacing “algebraic boundedness“ by “algebraic hyperbolicity“, we also have
the following; note that the uncoutability hypothesis in the statement has
been dropped.

######  Proposition 2.3.

Let $X$ be a projective variety, let $\Delta\subset X$ be a closed subset, and
let $p\in\mathbb{N}_{\geq 1}$. If $X$ is $p$-algebraically hyperbolic modulo
$\Delta$, then $X$ is $(p+1)$-algebraically hyperbolic modulo $\Delta$.

###### Proof.

Fix $L\to X$ an ample line on $X$, and suppose that $X$ is $p$-algebraically
hyperbolic modulo $\Delta$. One must prove that there exists a constant
$\alpha$ such that for any smooth projective variety $Y$ canonically polarized
of dimension $(p+1)$ and any non-degenerate rational map $f:Y\dashrightarrow
X$ whose image is not included in $\Delta$, one has the following inequality

$f^{*}L\cdot K_{Y}^{p}\leq\alpha K_{Y}^{p+1}.$

Fix $f:Y\dashrightarrow X$ a rational map as above. By the work of Demailly
and Angehrn-Siu (see e.g. [Laz04b][10.2]), there exists a natural number
$m\in\mathbb{N}$ independent of $Y$ such that $mK_{Y}$ is very ample. Take
$H\in|mK_{Y}|$ a smooth hypersurface such that $f_{|H}:H\dashrightarrow X$
remains a non-degenerate rational map whose image is not included in $\Delta$.
By the adjunction formula, one has the following equality of linear
equivalence classes of divisors

$K_{H}=(m+1)(K_{Y})_{|H}.$

Since $X$ is $p$-algebraically hyperbolic modulo $\Delta$, there exists a
constant $\beta$, independent of $f_{|H}$ and $H$, such that one has the
following inequality

$f_{|H}^{*}L\cdot K_{H}^{p-1}\leq\beta K_{H}^{p}.$

This inequality can be rewritten as follows

$f^{*}L\cdot K_{Y}^{p}\leq\beta(m+1)K_{Y}^{p+1},$

with $m$ and $\beta$ independent of $f$ and $Y$, so that the proof is
complete. ∎

We end this section with a classic yet fundamental application of the work of
Kollár and Matsusaka [KM83], which justifies our definition and terminology of
intermediate algebraic boundedness. Before so, let us recall the notion of
Hilbert polynomials of rational maps. Let $f:Y\dashrightarrow X$ be a rational
map. Denote $\operatorname{Graph}(f)$ the (closure of the) graph of $f$, and
recall that a _Hilbert polynomial of $f$_ is obtained as follows. Fix $A$ an
ample line bundle on $Y$ and $L$ an ample line bundle on $X$. The Hilbert
polynomial of $f$, computed with respect to the ample line bundles $A\to Y$
and $L\to X$, is by definition the Hilbert polynomial of the projective
variety $\operatorname{Graph}(f)$ computed with respect to the ample line
bundle $A\boxtimes L_{|\operatorname{Graph}(f)}$. This is the unique
polynomial $P_{f}\in\mathbb{Z}[X]$ such that for any $m\in\mathbb{N}$, $m\gg
1$, the following equality is satisfied

$P_{f}(m)=\chi\left(\operatorname{Graph}(f),\left(A\boxtimes
L_{|\operatorname{Graph}(f)}\right)^{m}\right)=\dim\mathrm{H}^{0}\left(\operatorname{Graph}(f),\left(A\boxtimes
L_{|\operatorname{Graph}(f)}\right)^{m}\right),$

where the last equality follows from the ampleness of $A\boxtimes
L_{|\operatorname{Graph}(f)}$. In the case where the map $f:Y\dashrightarrow
X$ is actually a morphism, one easily sees that the Hilbert polynomial of $f$
computed with respect to $A\to Y$ and $L\to X$ is also the Hilbert polynomial
of $Y$ computed with respect to the ample line bundle $A\otimes f^{*}L$. This
practical way of interpretating Hilbert polynomials of morphisms can be
carried over for rational maps, provided one assumes $Y$ to be smooth (locally
factorial is actually enough):

######  Lemma 2.4.

Let $Y,X$ be projective varieties, and suppose that $Y$ is smooth. Let
$f:Y\dashrightarrow X$ be a rational map, and denote by $P_{f}$ its Hilbert
polynomial computed with respect to the ample line bundles $A\to Y$ and $L\to
X$. Then for any $m\gg 1$, one has the following equality:

$P_{f}(m)=\chi(Y,(A\otimes f^{*}L)^{m})=\dim\mathrm{H}^{0}(Y,(A\otimes
f^{*}L)^{m}).$

###### Proof.

Recall that the pull-back by $f$ of the line bundle $L$ is well-defined: since
$Y$ is smooth, the rational map $f$ is well-defined outside a closed subset
$F$ of codimension at least $2$, and one has the equality of the Picard groups
$\operatorname{Pic}(Y\setminus F)=\operatorname{Pic}(Y)$. Let
$\pi_{1}:\operatorname{Graph}(f)\to Y$ (resp.
$\pi_{2}:\operatorname{Graph}(f)\to X$) be the first (resp. second)
projection, and observe that the pull-back $f^{*}L$ is nef. Indeed, let
$m\in\mathbb{N}_{\geq 1}$ be such that $L^{m}$ is globally generated, and let
$y\in Y$. Pick any $x\in f(y):=\pi_{2}(\pi_{1}^{-1}(y))$, and take
$s\in\mathrm{H}^{0}(X,L^{m})$ such that $s(x)\neq 0$. Then the section
$f^{*}s\in\mathrm{H}^{0}(Y,f^{*}L^{m})$ induced by $f$ does not vanish at $y$.
Therefore, the line bundle $A\otimes f^{*}L$ is ample. To conclude, it is now
enough to show the equality

$\dim\mathrm{H}^{0}(\operatorname{Graph}(f),\left(A\boxtimes
L\right)^{m}_{|\operatorname{Graph}(f)})=\dim\mathrm{H}^{0}(Y,(A\otimes
f^{*}L)^{m})$

for any $m\in\mathbb{N}$. Observe that one has the following equality of
(isomorphism classes of) line bundles on $\operatorname{Graph}(f)$:

$\pi_{1}^{*}(A\otimes f^{*}L)=\left(A\boxtimes
L\right)_{|\operatorname{Graph}(f)}.$

Therefore, there is a natural injective map of vector spaces induced by
$\pi_{1}$

$\begin{array}[]{lll}\mathrm{H}^{0}(Y,(A\otimes
f^{*}L)^{m})&\longrightarrow&\mathrm{H}^{0}(\operatorname{Graph}(f),\left(A\boxtimes
L\right)^{m}_{|\operatorname{Graph}(f)})\\\
s&\longmapsto&s\circ\pi_{1}\end{array}.$

In the other direction, any global section
$\tilde{s}\in\mathrm{H}^{0}(\operatorname{Graph}(f),\left(A\boxtimes
L\right)^{m}_{|\operatorname{Graph}(f)})$ induces a global section
$s\in\mathrm{H}^{0}(Y\setminus F,(A\otimes f^{*}L)^{m})$, and Riemann’s
extension theorem (or an algebraic variant of it) allows to conclude. ∎

The terminology algebraic boundedness now comes from the following classic
application of the work of Kollár and Matsusaka [KM83] (see [Laz04b][Thm
6.3.29]):

######  Proposition 2.5.

Let $X$ be a projective variety and let $\Delta\subset X$ be a proper closed
subset of $X$. Suppose that $X$ is $p$-algebraically bounded modulo $\Delta$.
Then for any smooth projective variety $Y$ of dimension $p$, the coefficients
of the Hilbert polynomials of the non-degenerate rational morphisms from $Y$
to $X$, whose image are not included in $\Delta$, are uniformly bounded.

###### Proof.

Fix $A\to Y$ and $L\to X$ two ample line bundles, and let $f:Y\dashrightarrow
X$ be a non-degenerate rational map whose image is not included in $\Delta$.
By Lemma 2.4, one has to bound independently of $f$ the coefficients of the
Hilbert polynomial of $Y$ computed with respect to $A\otimes f^{*}L$. By the
work of Kollár and Matsusaka [KM83], it suffices to bound the intersection
numbers $(A+f^{*}L)^{p}$ and $(A+f^{*}L)^{p-1}\cdot K_{Y}$ independently of
$f$.

Recall that the hypothesis of $p$-algebraic boundedness gives us a constant
$C$ independent of $f$ such that the following inequality holds

$f^{*}L\cdot A^{p-1}\leq C.$

By the numerical criterion of bigness (see e.g. [Laz04a][Thm 2.2.15]), one
deduces that for $r>p\frac{C}{A^{p}}$, the line bundle $rA-f^{*}L$ is big. In
particular, a large enough multiple of this line bundle is effective. One now
shows by induction on $i\geq 1$ the existence of a constant $C_{i}$
independent of $f$ such that the following inequality holds

$(f^{*}L)^{i}A^{p-i}\leq C_{i}.$

One knows that it is satisfied for $i=1$. Suppose that it is satisfied for
$1\leq i<p$. Since the line bundle $f^{*}L$ is nef (see e.g. the proof of
Lemma 2.4), the line bundle $f^{*}L+A$ is ample. From the effectivity of a
large enough multiple of $rA-f^{*}L$, one then deduces the following
inequality

$(rA-f^{*}L)\cdot(f^{*}L+A)^{i}\cdot A^{p-i-1}\geq 0.$

This in turn implies the following inequality

$(f^{*}L)^{i+1}\cdot
A^{p-i-1}\leq(rA-f^{*}L)\cdot\left(\sum\limits_{k=0}^{i-1}\binom{k}{i}(f^{*}L)^{k}\cdot
A^{p-1-k}\right)+rf^{*}L^{i}\cdot A^{p-i}.$

By induction hypothesis, each term appearing in the (developped) sum on the
right can be bounded by a constant independent of $f$, so that the induction
is complete.

It is now clear that the above implies that one can bound $(f^{*}L+A)^{p}$
independently of $f$. As for $(f^{*}L+A)^{p-1}\cdot K_{Y}$, one picks $m$ such
that $mA-K_{Y}$ is ample ($m$ is obviously independent of $f$), so that

$(f^{*}L+A)^{p-1}\cdot K_{Y}\leq m(f^{*}L+A)^{p-1}\cdot A$

and the above allows to conclude. ∎

##  3\. From positivity modulo $\Delta$ to intermediate algebraic
hyperbolicity

In this section, we prove the following criterion for intermediate algebraic
hyperbolicity, wich must be seen as the algebraic analogue of Demailly’s
Theorem 0.1:

######  Theorem 3.1.

Let $X$ be a smooth projective variety, and let $\Delta\subset X$ be a proper
closed subset. If $\bigwedge^{p}\Omega^{1}_{X}$ is ample modulo $\Delta$, then
$X$ is $p$-algebraically hyperbolic modulo $\Delta$.

###### Proof.

Fix $L$ an ample line bundle on $X$, and let $Y$ be a smooth projective
$p$-dimensional variety with $K_{Y}$ ample. Let also $f:Y\dashrightarrow X$ be
a non-degenerate rational map such that $f(Y)\not\subset\Delta$. To prove the
theorem, one needs to show that there exists a constant $\alpha$ independent
of $f$ and $Y$ such that the following inequality is satisfied

$\displaystyle f^{*}L\cdot K_{Y}^{p-1}\leq\alpha K_{Y}^{p}.$

Let $\pi_{X}\colon\mathbb{P}(\bigwedge^{p}T_{X})\to X$ be the natural
projection onto $X$, and let $\mathcal{O}_{\mathbb{P}(\bigwedge^{p}TX)}(1)$ be
the dual of the tautological line bundle on $\mathbb{P}(\bigwedge^{p}T_{X})$.
As the rational map $f$ is non-degenerate, it induces via its differential a
rational map

$\tilde{f}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
30.60619pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y\simeq\mathbb{P}(\bigwedge^{p}T_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
54.60619pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
54.60619pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}\big{(}\bigwedge^{p}T_{X}\big{)},}$}}}}}}}\ignorespaces}}}}\ignorespaces\
(y,[v_{1}\wedge\dotsc\wedge
v_{p}])\mapsto\big{(}f(y),[df_{y}(v_{1})\wedge\dotsc\wedge
df_{y}(v_{p})]\big{)}.$

Observe that $\tilde{f}$ is well defined outside a closed subset $F$ of
codimension at least two. Indeed, writing $\tilde{f}$ in trivializations, one
sees that the indeterminacy locus of $\tilde{f}$ comes either from the
indeterminacy locus of $f$, which has codimension at least two as $Y$ is
smooth, or from the indeterminacy locus of a rational map from $Y$ into the
projective space $\mathbb{P}\left(\bigwedge^{p}k^{\dim(X)}\right)$, which is
also of codimension at least two by smoothness of $Y$. In particular, as
$\operatorname{Pic}(Y)\simeq\operatorname{Pic}(Y\setminus F)$ by smoothness of
$Y$, the pull-back by $\tilde{f}$ of any line bundle on
$\mathbb{P}(\bigwedge^{p}TX)$ is well-defined.

Observe now that the rational map $f$ induces also the following non-trivial
morphism of line bundles on $Y\setminus F$:

$\overline{f}\colon\left|\begin{array}[]{rcl}(K_{Y}^{\vee})_{|Y\setminus
F}&\longrightarrow&\left(\tilde{f}^{*}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}T_{X})}(-1)\right)_{|Y\setminus
F}\\\ \big{(}y,v_{1}\wedge\dotsc\wedge
v_{p}\big{)}&\longmapsto&\big{(}y,df_{y}(v_{1})\wedge\dotsc\wedge
df_{y}(v_{p})\big{)}.\\\ \end{array}\right..$

It extends to a non-trivial morphism of line bundles on $Y$ by Riemann’s
extension theorem (or an algebraic variant of it). In particular, since

$\operatorname{Hom}(K_{Y}^{\vee},\tilde{f}^{*}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}T_{X})}(-1))\simeq\mathrm{H}^{0}(Y,K_{Y}\otimes\tilde{f}^{*}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}T_{X})}(-1)),$

one deduces that the divisor
$K_{Y}\otimes\tilde{f}^{*}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}T_{X})}(-1)$ is
effective. On the other hand, as $\bigwedge^{p}\Omega_{X}^{1}$ is ample modulo
$\Delta$, there exists an integer $m>0$ such that

$\operatorname{Bs}\big{(}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}TX)}(m)\otimes\pi_{X}^{*}L^{-1}\big{)}\subset\pi_{X}^{-1}(\Delta).$

Note that $m$ is independent of $f$ and $Y$. Since $f(Y)\not\subset\Delta$ (by
assumption), the pull-back

$\tilde{f}^{*}\big{(}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}TX)}(m)\otimes\pi_{X}^{*}L^{-1}\big{)}=\tilde{f}^{*}\mathcal{O}_{\mathbb{P}(\bigwedge^{p}TX)}(m)\otimes
f^{*}L^{-1}$

remains effective. Denote $E:=f^{*}L^{-1}\otimes K_{Y}^{\otimes m}$ so that
one has the equality:

(3.1) $\displaystyle f^{*}L+E=mK_{Y}.$

By the above, $E$ is an effective divisor, as it the sum of two effective
divisors.

One now concludes the proof as follows. Take $r\in\mathbb{N}$ such that
$rK_{Y}$ is very ample, and let $H$ be a general complete intersection of
$p-1$ hypersurfaces in the linear system $|rK_{Y}|$. Then one has the
following equality

$(f^{*}L)\cdot K_{Y}^{p-1}=\frac{1}{r^{p-1}}\deg(f^{*}L_{|H}).$

For a general $H$, the restricted divisor $E_{|H}$ remains effective, so that
it follows from (3.1) that

$(f^{*}L)\cdot
K_{Y}^{p-1}\leq\frac{1}{r^{p-1}}\deg\left((mK_{Y})_{|H})\right)=mK_{Y}^{p}.$

This finishes the proof, as $m$ is independent of $f$ and $Y$.

∎

##  4\. Finiteness results for intermediate pseudo bounded and pseudo
hyperbolic varieties

In this section, we investigate the finiteness of the sets of surjective
morphisms and birational automorphisms for pseudo $p$-algebraically bounded
varieties and pseudo $p$-algebraically hyperbolic varieties. In the first two
parts, we prove such finiteness results for $\dim(X)$ algebraically bounded
varieties. In a third and last part, we state the same finiteness results for
$p$-algebraically bounded and $p$-algebraically hyperbolic varieties: even
though the statements are immediate from Proposition 2.2 and Proposition 2.3,
we choose to put them in a separate section to emphasize on the fact that the
hypothesis “$k$ uncountable“ is à priori important in the case of intermediate
algebraic boundedness, while it is not for intermediate algebraic
hyperbolicity.

###  4.1. Surjective morphisms to $\dim(X)$-algebraically bounded varieties

Crucial to our proofs below is the (obvious) non-uniruledness of a $\dim(X)$
algebraically bounded variety. We record this observation in the following
lemma.

######  Lemma 4.1.

Let $X$ be a projective variety. If $X$ is $\dim(X)$-algebraically bounded,
then $X$ is non-uniruled. ∎

If $X$ and $Y$ are projective varieties, we let
$\underline{\operatorname{Hom}}(Y,X)$ be the scheme parametrizing morphisms
from $Y$ to $X$. If $\mathrm{Hilb}(Y\times X)$ is the Hilbert scheme of
$Y\times X$, then the morphism
$\underline{\operatorname{Hom}}(Y,X)\to\mathrm{Hilb}(Y\times X)$ mapping a
morphism $f:Y\to X$ to its graph $\operatorname{Graph}(f)\subset Y\times X$ is
an open immersion [Nit05, Theorem 6.6]. We also let
$\underline{\mathrm{Sur}}(Y,X)$ be the open and closed subscheme of
$\underline{\operatorname{Hom}}(Y,X)$ parametrizing surjective morphisms from
$Y$ to $X$.

Before stating and proving our results, recall that the scheme
$\underline{\mathrm{Sur}}(Y,X)$ is of finite type if and only if it has
finitely many connected components. That is, $\underline{\mathrm{Sur}}(Y,X)$
is of finite type if and only if, for every ample line bundle $L$ on $X$ and
every ample line bundle $A$ on $Y$, there is an integer $n\geq 1$ and
polynomials $\Phi_{1},\ldots,\Phi_{n}$ in $\mathbb{Q}[t]$ such that, for every
surjective morphism $f:Y\to X$, the Hilbert polynomial of $f$ with respect to
the ample line bundle $A\boxtimes L_{|\operatorname{Graph}(f)}$ lies in the
finite set $\\{\Phi_{1},\ldots,\Phi_{n}\\}$. From slight adaptations of
Proposition 2.2 and Proposition 2.5, we obtain the following lemma.

######  Lemma 4.2.

Let $X$ be a projective $\dim X$-algebraically bounded variety. Then for every
smooth projective variety $Y$ of dimension $\dim(Y)\geq\dim(X)$, the scheme
$\underline{\mathrm{Sur}}(Y,X)$ is of finite type.

###### Proof.

If $k$ is uncountable, this follows immediately from Proposition 2.2 and
Proposition 2.5. In a greater generality, one can proceed as in Proposition
2.2 and Proposition 2.5 once the following is noticed. Keep the notations of
Proposition 2.2, and suppose that $p\geq\dim(X)$. The uncountability
hypothesis was used to find a smooth ample divisor $H$ in the linear system
$|A|$ such that for any $i\in\mathbb{N}$, the rational maps $(f_{i})_{|H}$
remain non-degenerate. Suppose now that the $f_{i}:Y\rightarrow X$ are instead
surjective morphisms. It is then clear that for any smooth ample divisor $H$,
the restriction maps $(f_{i})_{|H}:H\to Y$ remain surjective, as for any $y\in
Y$, $f^{-1}(\\{y\\})$ is positive dimensional, and thus intersects $H$. ∎

To prove the rigidity of surjective morphisms, we will appeal to a theorem of
Hwang-Kebekus-Peternell [HKP06]. Their result relates the infinitesimal
deformation space of a surjective morphism $Y\to X$ to the infinitesimal
automorphisms of a suitable cover of $X$. For this reason, we investigate
first the discreteness of $\operatorname{Aut}(X)$, where
$\operatorname{Aut}(X)$ denotes the locally finite type group scheme of
automorphisms of $X$. Interestingly, to prove the rigidity of
$\operatorname{Aut}(X)$, we will appeal to the boundedness of
$\underline{\mathrm{Sur}}(Y,X)$ (for every smooth projective variety $Y$)
proved above.

######  Lemma 4.3.

Let $X$ be a projective $\dim(X)$-algebraically bounded variety. Then
$\mathrm{Aut}(X)$ is zero-dimensional.

###### Proof.

Since $X$ is non-uniruled (Lemma 4.1), the connected component
$A:=\mathrm{Aut}^{0}(X)$ of the identity of $\mathrm{Aut}(X)$ is an abelian
variety. For $a$ in $A$ and $x$ in $X$, let $a\cdot x$ denote the action of
$A$ on $X$. Let $\psi:\tilde{X}\to X$ be a resolution of singularities of $X$.
Consider the sequence of surjective morphisms $f_{n}:A\times\tilde{X}\to X$
given by

$f_{n}(a,\tilde{x})=(na)\cdot\psi(x).$

Since the degree of the (finite étale) morphism $[n]:A\times X\to A\times X$
equals $n^{2\dim A}$ and thus increases with $n$, one sees that the Hilbert
polynomials of the morphisms $f_{n}:A\times\tilde{X}\to X$ are pairwise
distinct. In particular, the scheme
$\underline{\mathrm{Sur}}(A\times\tilde{X},X)$ is not of finite type. Since
$A\times\tilde{X}$ is smooth, this contradicts Lemma 4.2. ∎

We now record the basic fact that a finite surjective cover of a
$p$-algebraically bounded projective variety remains $p$-algebraically
bounded:

######  Lemma 4.4.

Let $Z\to X$ be a finite surjective morphism of projective varieties. If $X$
is $p$-algebraically bounded, then $Z$ is $p$-algebraically bounded.

###### Proof.

This is a straightforward consequence of the projection formula, and the fact
that the pull-back of an ample by a finite morphism remains ample. ∎

We now prove the desired finiteness of surjective morphisms $Y\to X$, assuming
$X$ is $\dim(X)$ algebraically bounded.

######  Theorem 4.5.

If $X$ is a $\dim(X)$-algebraically bounded projective variety and $Y$ is a
projective variety, then $\underline{{\mathrm{Sur}}}(Y,X)$ is finite.

###### Proof.

Observe first that one can always suppose that $Y$ is smooth. Indeed, if one
takes $\tilde{Y}\to Y$ a resolution of singularities of $Y$, and if one knows
that $\underline{\operatorname{Sur}}(\tilde{Y},X)$ is finite, then one
immediately deduces the finiteness of $\operatorname{Sur}(Y,X)$. Furthermore,
by Lemma 4.4, one can always suppose that $X$ is normal: indeed, every
surjective morphism from $Y$ smooth to $X$ factors uniquely through the
normalization $\tilde{X}\to X$, where the normalization map is a surjective
and finite morphism.

By Lemma 4.2, the scheme $\underline{\mathrm{Sur}}(Y,X)$ is of finite type.
Thus, it suffices to show that $\underline{\mathrm{Sur}}(Y,X)$ is zero-
dimensional. To do so, let $f:Y\to X$ be a surjective morphism. As $X$ is non-
uniruled (Lemma 4.1) and normal, by the theorem of Hwang-Kebekus-Peternell
[HKP06], there is a finite surjective morphism $Z\to X$ and a morphism $Y\to
Z$ such that $f:Y\to X$ factors as

$Y\to Z\to X,$

with $\mathrm{Aut}^{0}(Z)$ surjecting onto the connected component of $f$ in
$\underline{\mathrm{Sur}}(Y,X)$. Now, as $Z\to X$ is finite surjective and $X$
is $\dim X$ algebraically bounded, it follows from Lemma 4.4 that $Z$ is $\dim
X=\dim Z$ algebraically bounded. In particular, it follows from Lemma 4.3 that
$\mathrm{Aut}^{0}(Z)$ is trivial. As $\operatorname{Aut}^{0}(Z)$ surjects onto
the connected component of $f$ in $\underline{\mathrm{Sur}}(Y,X)$, it follows
that the latter is trivial, which finishes the proof. ∎

###  4.2. Birational selfmaps of $\dim(X)$ algebraically bounded varieties

Let $X$ be a projective integral variety. We define $\mathrm{Bir}(X)$ to be
the subscheme of the Hilbert scheme $\mathrm{Hilb}(X\times X)$ parametrizing,
roughly speaking, closed subschemes $Z\subset X\times X$ such that $Z$ is
integral and both projections $Z\to X$ are birational (see [Han87, Definition
1.8] for a more precise formulation). Note that the (abstract) group
$\mathrm{Bir}(X)$ of birational selfmaps $X\dashrightarrow X$ is in bijection
with the set of $k$-points of the $k$-scheme $\mathrm{Bir}(X)$.

Hanamura shows that the scheme $\mathrm{Bir}(X)$ can be endowed with the
structure of a group scheme structure, assuming that $X$ is a (terminal)
minimal model (see [Han87, §3]). We will use a slight extension of his result.

In fact, in [PS14, §4] Prokhorov and Shramov show that a proper non-uniruled
integral variety over $k$ has a pseudo-minimal model. Hanamura’s main result
on the scheme $\mathrm{Bir}(X)$ for minimal models $X$ is easily seen to
extend to pseudo-minimal models by following his proof closely. Indeed,
Hanamura’s proof relies on the fact that on a minimal model every pseudo-
automorphism is an automorphism. This property also holds for pseudo-minimal
models by [PS14, Corollary 4.7]. In particular, Hanamura’s work gives the
following statement (see [Han87, §3]).

######  Theorem 4.6 (Hanamura).

If $X$ is a pseudo-minimal model, then the scheme $\mathrm{Bir}(X)$ can be
endowed with the structure of a group scheme (over $k$) such that
$\mathrm{Bir}^{0}(X)$ is isomorphic to $\mathrm{Aut}^{0}(X)$.

We use Hanamura’s structure result to prove the following finiteness result.

######  Proposition 4.7.

Let $X$ be a projective variety. If $X$ is $\dim(X)$-algebraically bounded,
then $\mathrm{Bir}(X)$ is finite.

###### Proof.

As $X$ is $\dim(X)$ algebraically bounded, one knows that $X$ is non-uniruled
(Lemma 4.1). Therefore, by the work of Prokhorov-Shramov [PS14, §4], the
projective variety $X$ has a pseudo-minimal model. Let $Y$ be a pseudo-minimal
model for $X$. Note that $\mathrm{Bir}(X)=\mathrm{Bir}(Y)$ (since $X$ and $Y$
are birational). Now, by Theorem 4.6, the scheme $\mathrm{Bir}(Y)$ can be
endowed with the structure of a group scheme in such a way that
$\mathrm{Bir}^{0}(Y)=\mathrm{Aut}^{0}(Y)$. Since $X$ is $\dim(X)$
algebraically bounded and $Y$ is birational to $X$, it follows that $Y$ is
also $\dim(X)$ algebraically bounded. In particular, $\operatorname{Aut}(Y)$
is trivial (Lemma 4.3) and thus Hanamura’s group scheme $\mathrm{Bir}(Y)$ is
zero-dimensional. Finally, to conclude that $\mathrm{Bir}(X)$ is finite, it
suffices to show that $\mathrm{Bir}(Y)$ is of finite type. This boundedness
statement is a straightforward consequence of the definition of
$\mathrm{Bir}(Y)$ and the fact that $Y$ is $\dim(Y)$ algebraically bounded. ∎

Note that the special case of Theorem 3.1 with $p=\dim(X)$ combined with
Theorem 4.5 and Proposition 4.7 gives an alternative proof of (a slightly
weaker form of) Kobayachi-Ochiai finiteness theorems:

######  Corollary 4.8 (Kobayachi-Ochiai).

Let $X$ be a smooth projective variety of general type. Then for any
projective variety $Y$, the set of surjective morphisms
$\operatorname{Sur}(Y,X)$ is finite. Furthermore, the set of bimeromorphisms
$\operatorname{Bir}(X)$ is also finite.

###  4.3. Finiteness results for intermediate pseudo bounded and pseudo
hyperbolic varieties

Using Proposition 2.2, Theorem 4.5 and Proposition 4.7, we obtain immediately
the following:

######  Theorem 4.9.

Assume that $k$ is uncountable. Let $X$ be a projective
pseudo-$p$-algebraically bounded variety, with $1\leq p\leq\dim(X)$. Then the
set of bimeromorphisms $\operatorname{Bir}(X)$ is finite, and for any
projective variety $Y$, the set of surjective morphisms
$\operatorname{Sur}(Y,X)$ is finite. ∎

In particular, this gives Theorem 0.8 as $\mathbb{C}$ is uncountable. Using
this time Proposition 2.2 and Proposition 2.3, with Theorem 4.5 and
Proposition 4.7, we obtain immediately the following, where the uncountability
hypothesis has been dropped:

######  Theorem 4.10.

Let $X$ be a projective pseudo-$p$-algebraically hyperbolic variety, with
$1\leq p\leq\dim(X)$. Then the set of bimeromorphisms $\operatorname{Bir}(X)$
is finite, and for any projective variety $Y$, the set of surjective morphisms
$\operatorname{Sur}(Y,X)$ is finite. ∎

## References

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latexText page 8 contains only floats

11institutetext: Chonnam National University

# An Ensemble Approach for Multiple Emotion Descriptors Estimation Using
Multi-task Learning

Irfan Haider  Minh-Trieu Tran  Soo-Hyung Kim  Hyung-Jeong Yang  Guee-Sang Lee

###### Abstract

This paper illustrates our submission method to the fourth Affective Behavior
Analysis in-the-Wild (ABAW) Competition. The method is used for the Multi-Task
Learning Challenge. Instead of using only face information, we employ full
information from a provided dataset containing face and the context around the
face. We utilized the InceptionNet V3 model to extract deep features then we
applied the attention mechanism to refine the features. After that, we put
those features into the transformer block and multi-layer perceptron networks
to get the final multiple kinds of emotion. Our model predicts arousal and
valence, classifies the emotional expression and estimates the action units
simultaneously. The proposed system achieves the performance of 0.917 on the
MTL Challenge validation dataset.

###### Keywords:

Multi-Task Learning, Emotion Recognition, Deep Learning, Context Information,
Spatial Attention

##  1 Introduction

Emotion recognition plays an important role in many researches because of its
multiple application in different fields such as autonomous driving, or
medicine. Emotion Recognition is also significant in the human effective
behaviour research and it has been studied many times with the development of
big data and deep learning technologies. Emotion can be detected by several
ways such as images, videos, speech and by using bio- signals. Human facial
expressions are not enough to recognise the emotion by images or videos, scene
context also effective to rather than only using the facial features[9].
Meanwhile, people recognize human emotions not only from facial features but
also from their surroundings, such as movement, communication with others, and
location[2][1].

Traditional recognition studies only focused on facial features and ignore the
other information i.e., contextual information that is very effective in
recognising the human emotion accurately. Previously, due to a lack of vast
amounts of data collected in actual scenarios, research focused on controlled
scenarios[12]. However, digital networks and social media have previously
become broadly used and a huge amount of data has become available.

Human emotions have recently been the focus of psychology. The most common
type of emotion representation is deterministic, which includes the seven
basic categories of anger, disgust, fear, happiness, sadness, surprise, and
neutrality[6].

Emotion recognition research is booming in computer vision. Continuous
research in this field and development in deep learning it is gaining more and
more attention. The proof is available as large number of datasets Aff-
Wild[29][15], s-Aff-Wild2[21][14][20][19][17][13], Aff-Wild2[18].

Some problems related to emotion recognition are discussed in 3rd Affective
Behavior Analysis in-the wild (ABAW) Competition is held in conjunction with
IEEE International Conference on Computer Vision and Pattern Recognition
(CVPR), 2022. 3rd ABAW includes Valence-Arousal (VA) Estimation Challenge,
Expression (Expr) Classification Challenge, Action Unit (AU) Detection
Challenge, Multi-Task-Learning (MTL) Challenge. Many teams participate every
time in ABAW[11] challenge to improve the performance of human emotion models
in the real world.

##  2 Related Work

For many years, researchers have researched about human emotion recognition
utilizing various forms of human emotion, such as fundamental facial
expressions, action units, and valence arousal. Mostly researchers focused on
the facial features to recognize the human emotion[9][24]. Some methods are
based on the facial action system[7][8], which uses a set of local facial
postures to encrypt facial expression. The majority of studies have depended
on analyzing human faces, which restricts their capability to utilise semantic
features for emotion recognition in the wild[16].

Many researchers have researched about how to improve recognising the emotion
and to improve the performance, they used different inputs. Some used only
images[4][27][28] and some used audio data[5][23] with images. Different
researches used different approaches[10] for classifying and recognising the
human emotions. In emotion recognition different approaches has been used for
classifying the emotions. Deep Convolutional network(CNN) is used in different
fields of research[22][25]. Especially CNN have big contribution in the
computer vision field. After having good performance[30] in computer vision,
CNN is investigated to classifying the facial emotions[26].

In our study we used CNN due to its better performance and we used
Inception-V3[31] model that is a deep CNN architecture. This model is proposed
by Szegedy et al. in the Large-Scale ImageNet Visual Identification Challenge
2014. It’s goal was to improve computational performance and reduce the
parameters in applications. Inception-V3 model is faster than VGG. Basically
it is the modification of AlexNet. Last year top 1st ranked author of ABAW
challenge also used the same model in their work and achieved good results[3].

Figure 1: Illustration of our proposed architecture.

##  3 Proposed Method

Our proposed architecture has been shown in figure1. We employ two kinds of
information from the provided dataset: cropped and cropped aligned images.
Cropped images include the facial, while the cropped aligned images contain
the facial and also the context around. Based on the results of the first
ranking last year[3], we utilize the model and add some modifications. Instead
of using only the cropped aligned images, we explored full information from
the provided dataset. Basically, our approach is based on three main parts:
feature extraction, attention mechanism, and classifiers.

For the first part, we used Inception Net V3 to extract the features from the
input image. Inception v3 is a deep network for the image recognition field.
The model itself contains symmetric and asymmetric building blocks with
integrated convolutions and pooling layers inside. This model shows a superb
result with higher than 78 % accuracy on the ImageNet dataset. The feature
extractor gives the feature output with the sizes of 768 $\times$ 17 $\times$
17.

In the second part, we will mention the mechanism modules that help the model
to focus better on meaningful regions. After having the feature maps, we put
them through the attention module to learn the spatial attention for each
region. Based on the combination of attention maps and features, we have the
embedding feature vector of each region.

The final part included transformer block and multi-layer perception networks.
We apply the transformer to learn the correlation between different regions
related to different action units. After having the output features from the
transformer block, we put them into action unit classifiers. Note that each
action unit has a separate classifier. Besides the action unit classifiers, we
proposed the valence/arousal predictors and expression classifier. For the
valence/arousal predictor, we use two fully connected layers. We employ a
multi-layer perception with three fully connected layers for the expression
classifier.

During training, we try multiple types of loss functions. Particularly, in the
action unit prediction and emotional expression classification, we use the
cross-entropy loss function. With the arousal/valence prediction task, a
negative Concordance Correlation Coefficient (CCC) is employed as a loss
function. Finally, the total loss is the sum of three elemental losses.

We use several kinds of evaluation metrics in this paper. Firstly, the F1
score metric is used in the evaluation step to compute the performance of
action unit prediction and emotional expression classification tasks.
Secondly, the performance of valence/arousal prediction is calculated by CCC
metric. The total metric for multi-task prediction is based on the combination
of the F1 score and CCC metrics.

##  4 Experimental Results

The proposed method is trained on the system with only one Nvidia GTX 3090 GPU
card, 24GB memory, and with SGD optimizer. We trained the model for 50 epochs,
batch size 24, and the learning rate of 0.001. Note that we train in the same
number of epochs for a fair comparison between the proposed method and last
year’s first rank method.

Table 1 shows comparison results between our proposed method with the top-
ranking ABAW3 method in the EXPR classification task. The results prove two
things.

First is the meaning of context in emotion recognition which contains faces
can bring more useful information for the EXPR classification task than only
the face information. It is similar to AU classification and Valence
prediction tasks, shown in Tables 2, 3, respectively. However, for the Arousal
prediction task results, shown in Table 4, information that comes from the
face only seems more meaningful than the combination of face and context
around.

The second thing is that the results show the ensemble approach brings better
results than using only a separate model.

The information of the training procedure are presented in Figures 2,3,4,5,6.
Figure 2 is the illustration of accuracy in action unit prediction during
training steps. Figure 3 is the illustration of the F1 score in expression
classification during training steps. Figures 4, 5 is the illustration of the
Concordance Correlation Coefficient in Arousal and Valence predictions during
training steps. Fig 6 presents loss reduction during training steps. From
those images, the orange line indicates the performance of training from
facial (cropped images). The blue line indicates the performance of training
from facial and context around (cropped aligned images). The grey line
indicates the performance of training from last year’s first ranking provided
source code from facial and context around.

Figure 2: Illustration of accuracy in action unit prediction during training steps. The orange line indicates the performance of training from facial (cropped images). The blue line indicates the performance of training from facial and context around (cropped aligned images). The grey line indicates the performance of training from last year’s first ranking provided source code from facial and context around. Figure 3: Illustration of F1 score in expression classification during training steps. The orange line indicates the performance of training from facial (cropped images). The blue line indicates the performance of training from facial and context around (cropped aligned images). The grey line indicates the performance of training from last year’s first ranking provided source code from facial and context around. Figure 4: Illustration of Concordance Correlation Coefficient in arousal prediction during training steps. The orange line indicates the performance of training from facial (cropped images). The blue line indicates the performance of training from facial and context around (cropped aligned images). The grey line indicates the performance of training from last year’s first ranking provided source code from facial and context around. Figure 5: Illustration of Concordance Correlation Coefficient in valence prediction during training steps. The orange line indicates the performance of training from facial (cropped images). The blue line indicates the performance of training from facial and context around (cropped aligned images). The grey line indicates the performance of training from last year’s first ranking provided source code from facial and context around. Figure 6: Illustration loss reduction during training steps. The orange line indicates the performance of training from facial (cropped images). The blue line indicates the performance of training from facial and context around (cropped aligned images). The grey line indicates the performance of training from last year’s first ranking provided source code from facial and context around. Table 1: The comparison results between our proposed method with the top-ranking ABAW3 method in the EXPR classification task. Method | F1 Score
---|---
Last year 1st rank | 0.1023
Ours (Cropped) | 0.1692
Ours (Cropped Aligned) | 0.2194
Ours (Ensemble) | 0.3055
Table 2: The comparison results between our proposed method with the top-ranking ABAW3 method in AU classification task. Method | F1 Score
---|---
Last year 1st rank | 0.4714
Ours (Cropped) | 0.4856
Ours (Cropped Aligned) | 0.4860
Ours (Ensemble) | 0.4939
Table 3: The comparison results between our proposed method with the top-ranking ABAW3 method in the Valence Prediction task. Method | CCC
---|---
Last year 1st rank | 0.0534
Ours (Cropped) | 0.0520
Ours (Cropped Aligned) | 0.1091
Ours (Ensemble) | 0.1597
Table 4: The comparison results between our proposed method with the top-ranking ABAW3 method in the Arousal Prediction task. Method | CCC
---|---
Last year 1st rank | 0.0504
Ours (Cropped) | 0.0735
Ours (Cropped Aligned) | 0.0446
Ours (Ensemble) | 0.0754
Table 5: The comparison results between our proposed method with baseline and the top-ranking ABAW3 method in the Multi-Task Prediction. Method | Total Metric
---|---
Baseline | 0.3000
Last year 1st rank | 0.6256
Ours | 0.9170

##  5 Conclusion

This paper presents an ensemble approach for multiple emotion descriptors
estimation using multi-task learning. The context is one of the important
factors in improving the performance of emotion recognition tasks.
Additionally, Instead of using a separate framework, the ensemble approach
still can bring more benefit to recognition issues. Our code implementation is
available at https://github.com/tmtvaa/abaw4.

##  6 Acknowledgement

This research was supported by the Bio & Medical Technology Development
Program of the National Research Foundation (NRF) & funded by the Korean
government (MSIT) (NRF-2019M3E5D1A02067961) and by Basic Science Research
Program through the National Research Foundation of Korea(NRF) funded by the
Ministry of Education(NRF-2018R1D1A3B05049058 & NRF-2020R1A4A1019191).

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