# Conjectures on the Kodaira dimension

Mihnea Popa  Department of Mathematics, Harvard University, 1 Oxford Street,
Cambridge, MA 02138, USA<EMAIL_ADDRESS>
MP was partially supported by NSF grant DMS-2040378.

###  1\. Introduction

In this note I propose a few conjectures on the behavior of the Kodaira
dimension under morphisms of smooth complex varieties. The statements are of a
rather different flavor than the well-known subadditivity conjectured by
Iitaka; in some sense they complement it with ideas inspired by the study of
the hyperbolicity of parameter spaces. Even though there is one statement that
implies them all in §3, I will first discuss in §2 an intermediate conjecture
that is already of substantial interest and provides a “superadditivity”
counterpart to subaddivity. Concretely, here is essentially the main case of
Conjecture 2.1 in the text: if $X$ and $Y$ are smooth projective varieties,
and $f\colon X\to Y$ is an algebraic fiber space with general fiber $F$, which
is smooth away from a simple normal crossing divisor $D\subset Y$, then we
have

$\kappa(F)+\kappa\big{(}\omega_{Y}(D)\big{)}\geq\kappa(X).$

In §4 I will also state a weaker conjecture on the behavior of Kodaira _co_
dimension, which is potentially more accessible.

I was inspired to look for such statements while thinking about possible
extensions of results for families over abelian varieties in [PS1]. Various
cases follow from existing results and techniques, while others have been
established since the first announcement and are listed here. Moreover, it now
looks to be the case that, just as with Iitaka’s conjecture, they are implied
by the main conjectures of the minimal model program.

All the varieties in this note are defined over ${\mathbf{C}}$.

Acknowledgements. I thank C. Hacon, J. Kollár, F. Meng, S. Mori , S. G. Park,
C. Schnell and Z. Zhuang for comments and for answering my questions.

###  2\. A superadditivity conjecture for morphisms between smooth projective
varieties

Recall that Iitaka’s $C_{n,m}$ conjecture predicts that for an algebraic fiber
space $f\colon X\to Y$ of smooth projective varieties (meaning $f$ is a
surjective morphism, with connected fibers), we have subadditivity for the
Kodaira dimension, i.e.

$\kappa(X)\geq\kappa(F)+\kappa(Y).$

For surveys on this conjecture, see [Mori] and [Fujino].

Here I start by proposing a complementary “superadditivity” statement in this
same, most common, setting.

######  Conjecture 2.1.

Let $f:X\rightarrow Y$ be an algebraic fiber space between smooth projective
varieties, and let $V\subseteq Y$ be the open subset over which $f$ is smooth.
Then

$\kappa(F)+\kappa(V)\geq\kappa(X).$

Recall that the _log Kodaira dimension_ $\kappa(V)$ can be defined as follows:
after a birational base change we can assume that the complement
$D=Y\smallsetminus V$ is a simple normal crossing (SNC) divisor. One then
defines

$\kappa(V):=\kappa\big{(}Y,\omega_{Y}(D)\big{)},$

the Iitaka dimension of $\omega_{Y}(D)$, which is easily checked to be
independent of the choice of compactification of $V$ with simple normal
crossing boundary.

Conjecture 2.1 will be strengthened later on to one about smooth morphisms
between quasi-projective varieties, which provides the correction turning the
inequality into an equality; see Conjecture 3.1 below.

######  Remark 2.2 (Obvious cases).

The conjecture clearly holds when $\kappa(X)=-\infty$ (so in particular when
$\kappa(F)=-\infty$) and when $\kappa(V)=\dim Y$, i.e. $V$ is of log general
type. Recall that the Easy Addition lemma, see e.g. [Mori, Corollary
2.3(iii)], says that for any algebraic fiber space we have

$\kappa(F)+\dim Y\geq\kappa(X).$

######  Remark 2.3.

It is worth noting that an important class of fiber spaces for which $V$ is of
log general type is that of “moduli” families; more generally, by [PS2,
Theorem A], relying also on important work on Viehweg’s hyperbolicity
conjecture in [VZ2], [CP], this holds for every family with maximal variation
such that $F$ admits a good minimal model. The methods used in these works
have also been crucial for solving other cases of Conjecture 2.1.

######  Remark 2.4 (Smooth case).

In the smooth case, when combined with Iitaka’s subadditivity, Conjecture 2.1
leads to the following additivity formula, also generalized later from a
different point of view by Conjecture 3.1:

######  Conjecture 2.5.

If $f:X\rightarrow Y$ is smooth algbraic fiber space between smooth projective
varieties, with general fiber $F$, then

$\kappa(X)=\kappa(F)+\kappa(Y).$

######  Remark 2.6 (Variation and the Kebekus-Kovács conjecture).

Recall that Viehweg’s $C_{n,m}^{+}$ conjecture states that, when
$\kappa(Y)\geq 0$, one has

$\kappa(X)\geq\kappa(F)+{\rm max}\\{{\rm Var}(f),\kappa(Y)\\}.$

(See [Mori, §7] for a survey.) It is interesting to note that, when combined
with $C_{n,m}^{+}$, Conjecture 2.1 implies the Kebekus-Kovács conjecture [KK,
Conjecture 1.6] for such $Y$, i.e the inequality

$\kappa(V)\geq{\rm Var}(f).$

This last conjecture has been verified when the general fiber of $f$ is of
general type by Taji [Taji] and Wei-Wu [WW]; the paper [Taji] also addresses
the case when the general fiber of $f$ has a good minimal model.

######  Remark 2.7 (Domain of general type).

It is also amusing to spell out the special case when $X$ is of general type.
When $f$ is smooth (i.e. $V=Y$), or when $Y$ is not uniruled, this was proved
in [PS3]; the full statement is a consequence of a more general result by
Park, see [Park, Corollary 1.6].

######  Theorem 2.8.

With the notation in Conjecture 2.1, if $X$ is of general type, then $V$ is of
log general type.

The non-obvious cases of Conjecture 2.1 and Conjecture 2.5 are summarized in
the next two theorems.

######  Theorem 2.9.

Conjecture 2.1 holds when:

  1. (1)

$Y$ is an abelian variety, or more generally a variety of maximal Albanese
dimension.111This means that $Y$ admits a generically finite (not necessarily
surjective) morphism to an abelian variety.

  2. (2)

$Y$ is a curve.

  3. (3)

$f$ is smooth, and either the general fiber of the Iitaka fibration of $Y$
admits a good minimal model,222More precisely one needs to assume a conjecture
of Campana-Peternell, which is in turn a consequence of the existence of good
minimal models. or $Y$ is uniruled. In particular, it holds when $f$ is smooth
and $Y$ is a surface or a threefold.

  4. (4)

$f$ is smooth, and $\kappa(Y)\geq\dim Y-3\geq 0$.

  5. (5)

$F$ is of general type, so in particular when $X$ is of general type.

  6. (6)

$F$ has semiample canonical bundle.

###### Proof.

Part (1) is shown in [MP], using techniques from [LPS].

Part (2) is clear when $g(Y)\geq 2$. When $Y=\mathbf{P}^{1}$ it follows from
[VZ1, Theorem 0.2], while when $Y$ is elliptic it follows from (1).

Part (3) is established in [PS3]; see Theorem C and Corollaries E, F in _loc.
cit._

Part (4) follows by applying Lemma 2.11 below to a model $g\colon
Y^{\prime}\to Z$ of the Iitaka fibration of $Y$, with $Y^{\prime}$ and $Z$
smooth. Its general fiber $G$ has dimension at most $3$, hence (2) and (3)
apply.

Parts (5) and (6) follow from more general results in [PS3], [Park] for (5),
and [Campana] for both, explained in the next section. ∎

######  Theorem 2.10.

Conjecture 2.5 holds when

  1. (1)

$f$ is a fiber bundle.

  2. (2)

$Y$ is of general type.

  3. (3)

$Y$ is an abelian variety, or more generally a variety of maximal Albanese
dimension.

  4. (4)

$Y$ is a curve.

  5. (5)

$Y$ is a surface.

  6. (6)

$Y$ is uniruled.

  7. (7)

$Y$ is a good minimal model with $\kappa(Y)=0$.

  8. (8)

$X$ is a good minimal model with $\kappa(X)=0$.

  9. (9)

$F$ is of general type.

  10. (10)

$F$ has semiample canonical bundle.

  11. (11)

we assume the conjectures of the MMP.

###### Proof.

Part (1) is one of the original results on the Iitaka conjecture, obtained (in
a more general setting) in [NU].

For (2), (3), (4), and (5), since we know either from Remark 2.2 or from the
previous statement that Conjecture 2.1 is settled in these cases, Conjecture
2.5 follows as in Remark 2.4, as in all these cases we know that Iitaka’s
conjecture holds (see [Kawamata2], [Viehweg1], [Kawamata3], [CaP], [HPS]).

For part (6), see [PS3, Proposition G]; the main input is the case
$Y={\mathbf{P}}^{1}$, proved in [VZ1, Theorem 0.2].

Part (7) is [PS3, Theorem H(ii)]. On the other hand, if $X$ is a good minimal
model with $\kappa(X)=0$, then so is $Y$ by [TZ], hence (8) also follows.

Parts (9) and (10) follow from (5) and (6) in Theorem 2.9 and the fact that
Iitaka’s conjecture holds when $F$ is of general type by [Kollar], and when
$F$ has semiample canonical bundle by [Kawamata3].

Part (11) is [PS3, Corollary D]; a more careful explanation of what exactly is
needed is given in _loc. cit._ ∎

We note that via standard methods one can perform a reduction on the base,
which in particular generates further examples (and was already used in
Theorem 2.9(4)). We restrict the discussion to smooth morphisms for
simplicity.

######  Lemma 2.11.

Any smooth algebraic fiber space over $Y$ satisfies Conjecture 2.1 if there is
a birational morphism $Y^{\prime}\to Y$, and a nontrivial fibration $g\colon
Y^{\prime}\to Z$ of smooth projective varieties with general fiber $G$, such
that $\kappa(Y)=\dim Z+\kappa(G)$333This happens for instance if $g$ is a
model of the Iitaka fibration of $Y$, or if $Z$ is of general type. and any
smooth fiber space over $G$ satisfies Conjecture 2.1.

###### Proof.

Let $f\colon X\to Y$ be a smooth algebraic fiber space. After a birational
base change on $Y$, we may assume that there is a fiber space $g\colon Y\to Z$
to a smooth projective variety, such that its fiber $G$ satisfies Conjecture
2.1. (Doing the corresponding birational base change on $X$ is ok, since the
fiber product is still a smooth fibration.) Let’s denote by $F$ the general
fiber of $f$, and by $H$ the general fiber of $h=g\circ f$. We then have that
$H\to G$ is a smooth fibration with fiber $F$, so by assumption

$\kappa(F)+\kappa(G)\geq\kappa(H).$

Adding $\dim Z$ to both sides of the inequality and using the hypothesis, we
obtain

$\kappa(F)+\kappa(Y)\geq\dim Z+\kappa(H)\geq\kappa(X),$

where the last inequality is the Easy Addition formula applied to $h$. ∎

######  Remark 2.12.

One consequence is that in order to establish Conjecture 2.1 for smooth
morphisms, it would be enough to show it over all bases $Y$ with
$\kappa(Y)=-\infty$ or $\kappa(Y)=0$. Indeed, when $\kappa(Y)>0$, we can
simply consider a smooth model $g\colon Y^{\prime}\to Z$ of the Iitaka
fibration of $Y$. Then either $Y$ is of general type and the statement is
clear, or the fiber $G$ of $g$ is positive dimensional with $\kappa(G)=0$, and
the lemma applies.

###  3\. The most general conjecture

The strongest proposal about projective morphisms that I would like to make,
easily seen to imply all the other conjectures in this note, is the following:

######  Conjecture 3.1.

If $f:U\rightarrow V$ is smooth projective algebraic fiber space between
smooth quasi-projective varieties, with general fiber $F$, then

$\kappa(U)=\kappa(F)+\kappa(V).$

In other words, in the presence of smooth morphisms, subadditivity in the log
version of Iitaka’s conjecture should become _additivity_. As noted in the
previous section, this is quite open even when $U$ and $V$ are projective.

There is a mounting body of evidence in favor of this conjecture. To begin
with, it is known to hold when $V$ is of log general type, i.e.
$\kappa(V)=\dim V$, without any smoothness hypothesis on $f$; see Remark 3.7
below. Here are some sample recent results on other cases. The first regards
base spaces that compactify to abelian varieties:

######  Theorem 3.2 ([MP, Theorem A]).

Let $f\colon X\to A$ be an algebraic fiber space, with $X$ a smooth projective
variety and $A$ an abelian variety. Assume that $f$ is smooth over an open set
$V\subseteq A$, and denote $U=f^{-1}(V)$ and the general fiber of $f$ by $F$.
Then $\kappa(U)=\kappa(V)+\kappa(F)$.

Another is that the conjecture holds when $U$ is of log general type; this is
a recent theorem of Park. Extending a result shown in [PS3] in the projective
case, he proves the following more general fact:

######  Theorem 3.3 ([Park, Theorem 1.5]).

In the situation of Conjecture 3.1, assume that $\kappa(F)\geq 0$. Then

$V~{}{\rm is~{}of~{}log~{}general~{}type}\iff\kappa(U)=\kappa(F)+\dim V.$

In particular, if $U$ is of log general type, then $V$ is of log general type.

In [Park, Theorem 1.10], Park also completes the proof of the conjecture when
$V$ is a curve; due to results in [VZ1] and [MP], the cases that he needs to
establish are when $V$ is ${\mathbf{P}}^{1}$ minus one or two points.

Moreover, the methods of [PS3] show that the conjecture holds when $F$ is of
general type, assuming that $\kappa(V)\geq 0$; see Remark 4 in _loc. cit._
However, more generally and quite importantly, Campana [Campana] has shown
that this last assumption can be removed, and furthermore:

######  Theorem 3.4 ([Campana, Theorem 1]).

Conjecture 3.1 holds when $F$ has semiample canonical bundle.

Using the most general result of [Taji], still under review, the same proof
shows that the conjecture actually holds when $F$ has a good minimal model.

Let’s finish by summarizing this discussion:

######  Theorem 3.5.

Conjecture 3.1 holds when:

  1. (1)

$V$ is of log general type [Kawamata1], [Maehara].

  2. (2)

$V$ compactifies to an abelian variety [MP].

  3. (3)

$V$ is a curve [VZ1],[MP], [Park].

  4. (4)

$U$ is of log general type [Park].

  5. (5)

$F$ is of general type [PS3], [Campana].

  6. (6)

$F$ has semiample canonical bundle (and possibly, more generally, $F$ has a
good minimal model) [Campana].

The log version. Even though the logarithmic Kodaira dimension necessarily
made an appearance, all the statements above should be seen as being about the
standard case of varieties. Iitaka’s conjecture however has more general log
analogues; see e.g. [Fujino] for a general overview. In this direction, a
generalization of Conjecture 3.1 is as follows:

######  Conjecture 3.6.

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective
varieties, and let $E$ be an SNC divisor on $X$ and $D$ an SNC divisor on $Y$
such that ${\rm Supp}(f^{*}D)\subseteq E$. Assume that $f$ is log-smooth over
$V=Y\smallsetminus D$,444The log-smooth condition means that each stratum of
the pair $(X,E)$, including $X$ itself, is smooth over $V$ via $f$. and let
$F$ be a general fiber over a point of $V$. Then

$\kappa(X,K_{X}+E)=\kappa(Y,K_{Y}+D)+\kappa(F,K_{F}+E_{F}).$

The reader can specialize this statement to various weaker versions, including
those discussed earlier in the standard case. For instance, the case of log
smooth morphisms to smooth projective varieties corresponds to $D=0$.

######  Remark 3.7 (Base of log general type).

Due to a result of Kawamata [Kawamata1, Theorem 30] for $\kappa(X,K_{X}+E)\geq
0$, and Maehara [Maehara, Corollary 2] in general, Conjecture 3.6 holds when
the base is of log general type, i.e. $\kappa(Y,K_{Y}+D)=\dim Y$. This is of
course a result about Iitaka’s conjecture (since the opposite inequality
follows from Easy Addition), so it does not require the log-smoothness
hypothesis.

###  4\. A conjecture on the Kodaira codimension

A weaker but quite interesting consequence of the conjectures in the previous
sections can be phrased in terms of the _Kodaira codimension_ of a smooth
quasi-projective variety $U$, defined in [Mori] as

$\kappa c(U):=\dim U-\kappa(U).$

Concretely, at least for smooth fiber spaces, the inequality

$\kappa(F)+\kappa(V)\geq\kappa(U)$

predicted as part of Conjecture 3.1 implies the following:

######  Conjecture 4.1.

If $f:U\rightarrow V$ is smooth projective morphism of smooth quasi-projective
varieties, then

$\kappa c(U)\geq\kappa c(V).$

I am including this statement because I like its symmetric form, but also
because, although weaker, it can still be very useful. For instance, this is
what was originally proved in [PS1] when $V$ is an abelian variety. It may
also possibly be more approachable in some instances. Note though that while
writing the first version of this note, in many significant cases Conjecture
4.1 was known to hold, while the stronger conjectures in the earlier sections
were not; however in view of recent work this is not the case anymore.

######  Remark 4.2.

(1) The conjecture holds when $f$ is étale, since then $\kappa(U)=\kappa(V)$.
When the dimension of the fibers is positive, one may also assume that $f$ is
an algebraic fiber space, as then it is not hard to see that in the Stein
factorization the finite morphism is in fact étale.

(2) The conjecture is not true when $f$ is not smooth. For instance, it is
well known that there exist smooth projective surfaces $S$ of general type
with $q(S)=1$, so with a surjective morphism to an elliptic curve; such a
morphism cannot be smooth. Or consider any base point free pencil on a variety
with $\kappa(X)\geq 0$. This induces a morphism $f\colon X\to\mathbf{P}^{1}$,
but such a morphism is never smooth by [VZ1, Theorem 0.2]. There are many
other examples.

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