# On a numerical criterion for Fano fourfolds

Haidong Liu  Sun Yat-sen University, Department of mathematics, Guangzhou,
510275, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: , version 0.01)

###### Abstract.

In this paper, we prove a special case of Campana–Peternell’s conjecture in
dimension 4. Specifically, we show that a smooth projective fourfold $X$ with
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and strictly nef anti-canonical divisor
$-K_{X}$ is a Fano fourfold.

###### Key words and phrases:

strictly nef anti-canonical divisors, Fano fourfolds, Campana–Peternell’s
conjecture, Serrano’s conjecture

###### 2010 Mathematics Subject Classification:

Primary 14J45; Secondary 14J35, 14E30

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 On the weak Campana–Peternell’s conjecture
  4. 4 Non-vanishing for strictly nef anti-canonical divisors

##  1\. Introduction

In [2], Campana and Peternell proposed the following conjecture, which
provides a numerical criterion for Fano manifolds.

######  Conjecture 1.1 ([2]*Problem 11.4).

Let $X$ be a projective manifold such that the anti-canonical divisor $-K_{X}$
is strictly nef, i.e., $-K_{X}\cdot C>0$ for any curve $C$ on $X$. Then, $X$
is a Fano manifold, i.e., $-K_{X}$ is ample.

By [16]*Theorem 1.2, a projective manifold with strictly nef anti-canonical
divisor is rationally connected. Therefore, Conjecture 1.1 is equivalent to

######  Conjecture 1.2.

Let $X$ be a projective rationally connected manifold such that the anti-
canonical divisor $-K_{X}$ is strictly nef. Then, $-K_{X}$ is ample.

Then, Campana–Peternell’s conjecture stands for Conjecture 1.1 or 1.2,
depending on whether the rationally connectedness should be emphasized. This
conjecture has been confirmed in dimension 2 by Maeda [20] and in dimension 3
by Serrano [24]. In Serrano’s work [24], Campana–Peternell’s conjecture was
put into a more general framework by viewing as a special case of the so-
called Serrano’s conjecture, and then could be treated by induction on the
dimension. In this paper, we follow Serrano’s idea (as [1, 18] did) and give
an affirmative answer to Campana–Peternell’s conjecture in dimension 4 under
the assumption that $c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$.

######  Theorem 1.3 ($=$ Corollary 3.2 $+$ Theorem 4.1).

Let $X$ be a smooth projective (rationally connected) fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. Then, $X$ is a Fano fourfold.

The organization of this paper follows very closely to [18], by the same
reason that the unknown case of Serrano’s conjecture for Calabi–Yau threefolds
is far from reaching. Hence, we have to get around this difficulty by
carefully using the strict nefness; however, we can not solve the problem
completely as in [18]. In this paper, the case $c^{2}_{1}(X)\cdot c_{2}(X)=0$
is still remaining.

In [10]*Remark 3.6, Kollár expected to give a structure theory for minimal
threefolds with $c_{1}(X)\cdot c_{2}(X)=0$, and find a universal lower bound
$-c_{1}(X)\cdot c_{2}(X)\geq\epsilon>0$ for the rest. This expectation can be
generalized to any dimensions. In [9], Hao and Schreieder give a satisfactory
structure theory for minimal models of dimension $n$ with $\kappa(X)=n-1$ and
$c^{n-2}_{1}(X)\cdot c_{2}(X)=0$ (see [9]*Theorem 1.1). These works begin with
the Bogomolov–Miyaoka–Yau inequality

$(-1)^{n}c^{n-2}_{1}(X)\cdot c_{2}(X)\geq 0$

for non-general type. On the other side of the mirror, for projective terminal
varieties with nef anti-canonical divisors, Ou [23] shows that $c_{2}(X)$ also
has the positivity; specifically, [23]*Corollary 1.5 shows that

$c^{n-2}_{1}(X)\cdot c_{2}(X)\geq 0.$

Therefore, it is also natural to expect a structure theory for projective
terminal varieties with $c^{n-2}_{1}(X)\cdot c_{2}(X)=0$ and nef anti-
canonical divisors, and that there is a universal lower bound
$c^{n-2}_{1}(X)\cdot c_{2}(X)\geq\epsilon>0$ for the rest. This question is
simple in the surface case; in dimension 3, the author and Chen Jiang give a
relatively clear answer to this question in a forthcoming note [17]. We hope
that a satisfactory structure theory for $c^{2}_{1}(X)\cdot c_{2}(X)=0$ can
provide the last piece of the puzzle on Campana–Peternell’s conjecture in
dimension 4.

######  Acknowledgments.

The author is supported by the NSFC (grant no. 12001018). He would like to
thank Chen Jiang, Jie Liu, Wenhao Ou and Guolei Zhong for useful discussions
and suggestions.

Throughout this paper, we work over the complex number field $\mathbb{C}$. A
scheme is always assumed to be separated and of finite type over $\mathbb{C}$,
and a variety is a reduced and irreducible algebraic scheme. We will freely
use the basic notation in [5, 7, 12, 13].

##  2\. Preliminaries

In this section, we present some preliminary results.

###  2.1. Hirzebruch–Riemann–Roch formula

Let $X$ be a smooth projective fourfold. Then, since $c_{1}(X)=-K_{X}$, we
have that

$\displaystyle\chi(X,-mK_{X})$
$\displaystyle=\frac{K_{X}^{4}}{24}m^{4}+\frac{c_{1}\cdot(-K_{X})^{3}}{12}m^{3}+\frac{(c^{2}_{1}+c_{2})\cdot
K_{X}^{2}}{24}m^{2}+\frac{c_{1}c_{2}\cdot(-K_{X})}{24}m+\chi(\mathcal{O}_{X})$
$\displaystyle=\frac{K_{X}^{4}}{24}m^{4}+\frac{K_{X}^{4}}{12}m^{3}+\frac{(K_{X}^{4}+c_{2}\cdot
K_{X}^{2})}{24}m^{2}+\frac{c_{2}\cdot K_{X}^{2}}{24}m+\chi(\mathcal{O}_{X})$

from the Hirzebruch–Riemann–Roch formula. If $-K_{X}$ is nef but not big, then
$K_{X}^{4}=0$ and

(2.1) $\begin{split}\chi(X,-mK_{X})&=\frac{c_{2}\cdot
K_{X}^{2}}{24}m^{2}+\frac{c_{2}\cdot K_{X}^{2}}{24}m+\chi(\mathcal{O}_{X})\\\
&=\frac{c^{2}_{1}(X)\cdot
c_{2}(X)}{24}(m^{2}+m)+\chi(\mathcal{O}_{X}).\end{split}$

In [23], Ou proved a dual version of Miyaoka’s theorem [21]*Corollary 6.4,
concerning about the positivity of the second Chern class.

######  Theorem 2.1 ([23]*Corollary 1.5).

Let $X$ be a projective manifold of dimension $n$ with nef anti-canonical
divisor $-K_{X}$. Then for any nef divisors $H_{1},\cdots,H_{n-2}$, we have

$c_{2}(X)\cdot H_{1}\cdots H_{n-2}\geq 0.$

In particular, $c^{n-2}_{1}(X)\cdot c_{2}(X)=c_{2}(X)\cdot(-K_{X})^{n-2}\geq
0$ since $c_{1}(X)=-K_{X}$ is nef.

######  Corollary 2.2.

Let $X$ be a smooth projective rationally connected fourfold such that the
anti-canonical divisor $-K_{X}$ is nef but not big. Then, $\chi(X,-mK_{X})\geq
1$ for any $m\geq 0$.

###### Proof.

Since $X$ is rationally connected, we have that $h^{i}(X,\mathcal{O}_{X})=0$
for $i\geq 1$ by [11]*Chapter IV.3, Corollary 3.8. Then, it follows from (2.1)
and Theorem 2.1 that
$\chi(X,-mK_{X})\geq\chi(\mathcal{O}_{X})=h^{0}(X,\mathcal{O}_{X})=1$ for any
$m\geq 0$. ∎

Theorem 2.1 is a corollary of the following theorem, which will be crucial in
the proof of our non-vanishing result for the case $\nu(-K_{X})\leq 2$. This
theorem is a little bit more general than [23]*Theorem 1.4, but essentially it
can be reduced to [23]*Theorem 1.4, since reflexive product preserves
semistability on manifolds (see [3]*Section 5).

######  Theorem 2.3 ([23]*Theorem 1.4).

Let $X$ be a projective manifold and $(\mathcal{T}_{X})^{\otimes
m}\to\mathcal{Q}$ be a torsion free coherent quotient for some $m\geq 1$. If
$-K_{X}$ is nef, then $c_{1}(\mathcal{Q})$ is pseudoeffective.

###  2.2. Weak Campana–Peternell’s conjecture

As stated at the beginning of this paper, Campana–Peternell’s conjecture can
be viewed as a special case of Serrano’s conjecture, which is a weak analogue
of Fujita’s conjecture for strictly nef divisors. Then, as in Fujita’s
conjecture, it is natural to generalize Serrano’s conjecture to log pairs (see
[8]*Question 3.5 and [18]*Conjecture 1.5). For manifolds with strictly nef
anti-canonical divisors, the log version of Serrano’s conjecture can be stated
as follows, which is called the weak Campana–Peternell’s conjecture in this
paper:

######  Conjecture 2.4 (Weak Campana–Peternell’s conjecture).

Let $X$ be a projective (rationally connected) manifold such that the anti-
canonical divisor $-K_{X}$ is strictly nef. Let $V$ be a nonzero prime
divisor. Then, $V-mK_{X}$ is ample for $m\gg 1$.

It is obvious that Campana–Peternell’s conjecture is equivalent to the weak
Campana–Peternell’s conjecture plus the following non-vanishing conjecture for
strictly nef anti-canonical divisors (Conjecture 1.1 or 1.2 $\Leftrightarrow$
Conjecture 2.4 $+$ Conjecture 2.5).

######  Conjecture 2.5 (Non-vanishing conjecture for strictly nef anti-
canonical divisors).

Let $X$ be a projective (rationally connected) manifold such that the anti-
canonical divisor $-K_{X}$ is strictly nef. Then, we have that
$\kappa(-K_{X})\geq 0$.

The following lemma is a first step towards the weak Campana–Peternell’s
conjecture.

######  Lemma 2.6.

Let $X$ be a projective manifold such that the anti-canonical divisor $-K_{X}$
is strictly nef. Let $V$ be a nonzero prime divisor. Then, there exists a
positive integer $m_{0}$ (dependent on $V$) such that the Cartier divisor
$V-mK_{X}$ is strictly nef for any integer $m>m_{0}$. Moreover, if $V-mK_{X}$
is big for some $m>m_{0}+1$, then $V-mK_{X}$ is ample.

###### Proof.

Let $n:=\dim X$. Let $r\in\mathbb{Q}_{>0}$ be a sufficiently small number such
that $(X,rV)$ is a klt pair. Then, by [18]*Lemma 2.4, the $\mathbb{Q}$-divisor
$K_{X}+rV+s(-K_{X})$ is strictly nef for $s>2n$. In particular, $V-mK_{X}$ is
strictly nef for $m>m_{0}:=\lceil\frac{2n-1}{r}\rceil$.

If $V-mK_{X}$ is big for some $m>m_{0}+1$, then
$(K_{X}+V-mK_{X})-K_{X}=V-mK_{X}$ is strictly nef and big. It follows from the
basepoint-free theorem that $K_{X}+V-mK_{X}$ is semiample. That is,
$V-(m-1)K_{X}=K_{X}+V-mK_{X}$ is ample by the strict nefness. In particular,
$V-mK_{X}=V-(m-1)K_{X}+(-K_{X})$ is ample. ∎

The following calculations are much the same as those in [24]*Lemma 1.3 and
[18]*Theorems 3.3 and 3.5.

######  Corollary 2.7.

Let $X$ be a projective manifold of dimension $n$ such that the anti-canonical
divisor $-K_{X}$ is strictly nef. Let $V$ be a nonzero prime divisor. If
$V-mK_{X}$ is not ample for any $m\gg 1$, then we have that

$0=V^{n}=V^{n-1}\cdot K_{X}=\cdots=V\cdot K_{X}^{n-1}=K_{X}^{n}.$

###### Proof.

By Lemma 2.6, $V-mK_{X}$ and $V-mK_{X}-K_{X}$ are both (strictly) nef but not
big for $m\gg 1$. In particular, we have $(V-mK_{X}-K_{X})^{n}=0$ for some
$m$. This yields that

$0=(V-mK_{X}-K_{X})^{n}=\sum_{k=0}^{n}\big{(}_{k}^{n}\big{)}(V-mK_{X})^{k}\cdot(-K_{X})^{n-k}.$

Then, all the terms on the right-hand side should be zero since $-K_{X}$ is
also nef. From $(V-mK_{X})\cdot(-K_{X})^{n-1}=(-K_{X})^{n}=0$, we have that
$V\cdot(-K_{X})^{n-1}=0$; combining with
$(V-mK_{X})^{2}\cdot(-K_{X})^{n-2}=0$, we can conclude that
$V^{2}\cdot(-K_{X})^{n-2}=0$, and so forth. Finally, we can deduce the desired
equation. ∎

###  2.3. Almost strictly nef divisors

We recall from [18]*Subsection 2.1 the definition and some useful properties
of almost strictly nef divisors, which are the birational analogues of
strictly nef divisors.

######  Definition 2.8 ([1]*Definition 1.1 or [18]*Definition 2.1).

A $\mathbb{Q}$-Cartier divisor $L$ on a normal variety $X$ is called almost
strictly nef if there exists a surjective birational morphism $\mu\colon X\to
X^{*}$ and a strictly nef $\mathbb{Q}$-Cartier divisor $L^{*}$ on $X^{*}$ such
that $L=\mu^{*}L^{*}$. When a birational morphism $\mu\colon X\to X^{*}$ is
specified, we say that $L$ is almost strictly nef with respect to $\mu$.

######  Lemma 2.9 ([18]*Lemma 2.4).

Let $(X,\Delta)$ be a log canonical pair of dimension $n$ and $L$ be an almost
strictly nef Cartier divisor with respect to $\mu\colon X\to X^{*}$. If
$K_{X}+\Delta$ is $\mu$-nef, then the $\mathbb{Q}$-Cartier divisor
$K_{X}+\Delta+tL$ is nef for $t\geq 2n$.

For almost strictly nef divisors, we have a birational version of (the log
version of) Serrano’s conjecture (see [1]*Conjecture 2.2 or [18]*Conjecture
2.6). For the inductive work, we state the conjecture in the following weaken
form:

######  Conjecture 2.10.

Let $X$ be a projective canonical variety of dimension $n$ and $L$ be an
almost strictly nef Cartier divisor on $X$. Then, the divisor $K_{X}+tL$ is
big for $t>2n$.

######  Proposition 2.11.

Conjecture 2.4 holds in dimension $n$ if Conjecture 2.10 holds in dimension
$n-1$.

###### Proof.

By Corollary 2.7, we have that

(2.2) $0=V^{n}=V^{n-1}\cdot K_{X}=\cdots=V\cdot K_{X}^{n-1}=K_{X}^{n}.$

Let $\mu\colon T\to V$ be the composition of the normalization of $V$ and the
relative canonical model (see [18]*Subsection 2.3 for the definition) of its
normalization. As in [18]*Theorem 3.3, there exists an effective divisor $B$
such that $K_{T}+B=\mu^{*}(K_{X}+V)|_{T}$. Set

$L=\mu^{*}(-K_{X})|_{T},$

which is almost strictly nef with respect to $\mu$. By the assumption,
$K_{T}+tL$ is big for $t>2n-2$. It follows that $K_{T}+B+tL$ is also big for
$t>2n-2$. Moreover,

$K_{T}+B+tL=\mu^{*}(K_{X}+V-tK_{X})|_{T}$

is nef since $K_{X}+V-tK_{X}$ is nef for $t\gg 1$ by Lemma 2.6. In conclusion,
we have that

$0<(K_{T}+B+tL)^{n-1}=(\mu^{*}(K_{X}+V-tK_{X})|_{T})^{n-1}=(K_{X}+V-tK_{X})^{n-1}\cdot
V,$

contradicting the equation (2.2). ∎

######  Remark 2.12.

Similar results as Lemma 2.6, Corollary 2.7 and Proposition 2.11 also hold
when $K_{X}$ is strictly nef. Combining with [18]*Theorem 3.5, we have seen
that in all the three important special cases of Serrano’s conjecture ($K_{X}$
is strictly nef, $-K_{X}$ is strictly nef, and $K_{X}$ is trivial), we can
perform the induction on the dimension and reduce the problem onto the prime
divisor $V$.

###  2.4. Prime Calabi–Yau divisors

As in [18], we also have to consider the so-called prime Calabi–Yau divisors
in this paper, since the remaining case of Serrano’s conjecture for Calabi–Yau
threefolds is far from reaching. Precisely, a prime divisor $D$ on a normal
variety $X$ is said to be a _prime Calabi–Yau divisor_ if $D$ is a normal
variety with at worst canonical singularities satisfying that $K_{D}\sim 0$
and $H^{1}(D,\mathcal{O}_{D})=0$.

######  Lemma 2.13.

Let $X$ be a projective manifold such that the anti-canonical divisor $-K_{X}$
is nef. Let $D$ be a prime Calabi–Yau divisor on $X$. Then, $D$ is nef.

###### Proof.

By adjunction formula [12]*Proposition 5.73, we have that
$(K_{X}+D)|_{D}=K_{D}\sim 0$. Let $C$ be a curve on $X$. If $C\not\subset D$,
then we have $D\cdot C\geq 0$. If $C\subset D$, then we have $D\cdot
C=D|_{D}\cdot C=(K_{D}-K_{X}|_{D})\cdot C=-K_{X}\cdot C\geq 0$. Therefore, $D$
is nef. ∎

Finally, we assume that $X$ is a projective manifold of dimension $n$ such
that $c^{n-2}_{1}(X)\cdot c_{2}(X)=0$ and the anti-canonical divisor $-K_{X}$
is nef. We also assume that $D\sim-K_{X}$ is a smooth prime Calabi–Yau
divisor. Then, from the exact sequence

$0\to\mathcal{T}_{D}\to\mathcal{T}_{X}|_{D}\to\mathcal{O}_{D}(D)\to 0,$

we have that $c_{2}(X)|_{D}=c_{2}(D)+c_{1}(D)\cdot D|_{D}=c_{2}(D)$. It
follows that

$(-K_{X}|_{D})^{n-3}\cdot c_{2}(D)=(-K_{X}|_{D})^{n-3}\cdot
c_{2}(X)|_{D}=(-K_{X})^{n-2}\cdot c_{2}(X)=c^{n-2}_{1}(X)\cdot c_{2}(X)=0.$

In particular, we have that $(-K_{X}|_{D})\cdot c_{2}(D)=0$ in dimension
$n=4$. If moreover $-K_{X}$ is strictly nef, then these conditions turn out to
be the remaining case of Serrano’s conjecture in dimension 3. This is the only
reason that we have to treat Campana–Peternell’s conjecture in dimension 4
under the assumption that $c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$.

##  3\. On the weak Campana–Peternell’s conjecture

In this section, we prove the weak Campana–Peternell’s conjecture (Conjecture
2.4) in dimension 4 under the assumption that $c^{2}_{1}(X)\cdot c_{2}(X)\neq
0$.

######  Theorem 3.1.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. Let $V$ be a nonzero prime divisor. Then, $V-mK_{X}$ is ample
for $m\gg 1$.

###### Proof.

The first two steps of this proof follow exactly the proofs of [18]*Theorems
3.3 and 3.5. We sketch the proof here for the reader’s convenience.

######  Step 1.

By Corollary 2.7, we have that

(3.1) $0=V^{4}=V^{3}\cdot K_{X}=V^{2}\cdot K_{X}^{2}=V\cdot
K_{X}^{3}=K_{X}^{4}.$

Let $T$ be a relative canonical model of the normalization of $V$ and
$\mu\colon T\to V$ be the corresponding morphism. As in [18]*Theorem 3.3,
there exists an effective divisor $B$ such that
$K_{T}+B=\mu^{*}(K_{X}+V)|_{T}$. Set $L:=\mu^{*}(-K_{X})|_{T}$, which is
almost strictly nef with respect to $\mu$. Then for $t\gg 1$, $K_{T}+tL$ is
nef by Lemma 2.9, but not big by Proposition 2.11. Similarly, set

$D_{m}:=\mu^{*}(K_{X}+V-mK_{X})|_{T},$

which is almost strictly nef with respect to $\mu$ for any $m\gg 1$ since
$K_{X}+V-mK_{X}$ is strictly nef for any $m\gg 1$ by Lemma 2.6. Then,
$K_{T}+6D_{m}$ is nef for $m\gg 1$ by Lemma 2.9. For $m\gg 1$, we can see that

$K_{T}+B+6D_{m}=\mu^{*}(7(K_{X}+V)-6mK_{X})|_{T}=\mu^{*}(7V-(6m-7)K_{X})|_{T}$

is nef by Lemma 2.6, but not big by (3.1). It follows that $K_{T}+6D_{m}$ is
nef but not big for $m\gg 1$. Then using similar calculations for $K_{T}+tL$
and $K_{T}+6D_{m}$ as in Corollary 2.7, we have that

(3.2) $\begin{split}&K_{T}^{3}=K_{T}^{2}\cdot L=K_{T}\cdot L^{2}=L^{3}=0,\\\
&K_{T}^{3}=K_{T}^{2}\cdot D_{m}=K_{T}\cdot D_{m}^{2}=D_{m}^{3}=0,\\\
&K_{T}^{3}=K_{T}^{2}\cdot D_{m+1}=K_{T}\cdot D_{m+1}^{2}=D_{m+1}^{3}=0,\\\
&2K_{T}\cdot D_{m}\cdot
L=K_{T}\cdot(D_{m+1}^{2}-D_{m}^{2}-L^{2})=0\end{split}$

for any $m\gg 1$. Directly from (3.1) or from $(D_{m}+L)^{3}=D_{m+1}^{3}=0$ in
(3.2), we can see that $D_{m}^{2}\cdot L=D_{m}\cdot L^{2}=0$. It follows that

(3.3) $\begin{split}&L^{2}\cdot
B=L^{2}\cdot(K_{T}+B)=L^{2}\cdot(D_{m}-mL)=0,\\\ &D_{m}^{2}\cdot
B=D_{m}^{2}\cdot(K_{T}+B)=D_{m}^{2}\cdot(D_{m}-mL)=0,\\\ &L\cdot K_{T}\cdot
B=L\cdot K_{T}\cdot(K_{T}+B)=L\cdot K_{T}\cdot(D_{m}-mL)=0,\\\ &D_{m}\cdot
K_{T}\cdot B=D_{m}\cdot K_{T}\cdot(K_{T}+B)=D_{m}\cdot
K_{T}\cdot(D_{m}-mL)=0,\end{split}$

and

(3.4) $\begin{split}&K^{2}_{T}\cdot
B=K_{T}^{2}\cdot(K_{T}+B)=K_{T}^{2}\cdot(D_{m}-mL)=0,\\\ &K_{T}\cdot
B^{2}=K_{T}\cdot(K_{T}+B)^{2}=K_{T}\cdot(D_{m}-mL)^{2}=0,\\\
&B^{3}=(K_{T}+B)^{3}=(D_{m}-mL)^{3}=0\end{split}$

for any $m\gg 1$. Then, we can derive contradictions on $B$ using these
equations as in [18]*Theorems 3.3 and 3.5.

######  Step 2.

If some integral component $S$ of $B$ is not $\mu$-exceptional, then as $L$
and $K_{T}+tL$ are nef, we have that $0=L^{2}\cdot B\geq a_{S}L^{2}\cdot S\geq
0$ and

$0=(K_{T}+tL)\cdot L\cdot B\geq a_{S}(K_{T}+tL)\cdot L\cdot S\geq 0$

by (3.3), where $a_{S}>0$ is the coefficient of $S$ in $B$. Similar results
also hold after replacing $L$ by $D_{m}$ and $K_{T}+tL$ by $K_{T}+6D_{m}$ for
$m\gg 1$. These indicates that

(3.5) $\begin{split}&L\cdot K_{T}\cdot S=L^{2}\cdot S=0,\\\ &D_{m}\cdot
K_{T}\cdot S=D_{m}^{2}\cdot S=0,\\\ &2L\cdot D_{m}\cdot
S=(D_{m+1}^{2}-D_{m}^{2}-L^{2})\cdot S=0\end{split}$

for $m\gg 1$. By exactly the same arguments as in Case 1 of the proof of
[18]*Theorems 3.3, there exists a curve $C\in|(K_{T}+S+rL)|_{S}|$ which is not
contained in the $\mu|_{S}$-exceptional locus for $r>3$. Then, from (3.5), we
can conclude that

$0<L|_{S}\cdot C=L\cdot(K_{T}+S+rL)\cdot S=L\cdot K_{T}\cdot S+L\cdot
S^{2}+rL^{2}\cdot S=L\cdot S^{2}.$

On the other hand, from (3.5), we have that

$\begin{split}0&=L\cdot(D_{m}-mL)\cdot S=L\cdot(K_{T}+B)\cdot S=L\cdot
K_{T}\cdot S+L\cdot B\cdot S\\\ &=L\cdot(B-a_{S}S)\cdot S+a_{S}L\cdot
S^{2}\geq a_{S}L\cdot S^{2}.\end{split}$

This is a contradiction.

Therefore, we assume that $\dim\mu(B)\leq 1$. In this case, $B$ is
$\mu$-exceptional and $V$ is normal. Assume that $B\neq 0$. Then,
$\dim\mu(B)=1$ by $B^{3}=0$ in (3.4). As in [18]*Theorem 3.3, we have that
$(K_{T}+\mu^{*}H)\cdot B\cdot L>0$ for a very ample divisor $H$ on $V$. It
follows that $H\cdot\mu(B)\cdot(-K_{X})|_{V}>0$, which is impossible.
Therefore, we have that $B=0$, $T=V$ is a Gorenstein variety with at worst
canonical singularities, $L=-K_{X}|_{V}$ is strictly nef, and $K_{V}+tL$ is
not ample for any $t\gg 1$ (see Proposition 2.11). As showed in the last
paragraph of the proof of [18]*Theorem 3.5, we can further assume that $V$ is
a prime Calabi–Yau divisor.

######  Step 3.

In the remaining case, $V$ is a prime Calabi–Yau divisor such that
$(K_{X}+V)|_{V}=K_{V}\sim 0$. Then, $V$ is nef on $X$ by Lemma 2.13. We
consider the long cohomology sequence induced by the exact sequence

(3.6)
$0\to\mathcal{O}_{X}(K_{X})\to\mathcal{O}_{X}(K_{X}+V)\to\mathcal{O}_{V}(K_{X}+V)\simeq\mathcal{O}_{V}\to
0.$

Since $X$ is rationally connected, we have that
$H^{0}(X,\mathcal{O}_{X}(K_{X}))=H^{4}(X,\mathcal{O}_{X})=0$ and
$H^{1}(X,\mathcal{O}_{X}(K_{X}))=H^{3}(X,\mathcal{O}_{X})=0$ by Serre’s
duality. It follows that

$H^{0}(X,\mathcal{O}_{X}(K_{X}+V))\cong
H^{0}(V,\mathcal{O}_{V})\cong\mathbb{C}.$

That is, there exists an effective divisor $V_{1}\sim K_{X}+V$ and all the
coefficients of irreducible components of $V_{1}$ are positive integers. If
$V_{1}\neq 0$, then we replace $V$ by $V_{1}$ and consider
$V_{1}-mK_{X}=V-(m-1)K_{X}$ from the beginning instead. If some reduced and
irreducible component $E$ of $V_{1}$ is not a prime Calabi–Yau divisor, then
$E-mK_{X}$ is ample for $m\gg 1$ as above; it follows that
$V_{1}-2mK_{X}=E-mK_{X}+(V_{1}-E)-mK_{X}$ is also ample since
$(V_{1}-E)-mK_{X}$ is strictly nef for $m\gg 1$ by Lemma 2.6. Therefore, we
finish the proof unless any reduced and irreducible component of $V_{1}$ is a
prime Calabi–Yau divisor. Note that in this case, $V_{1}$ is nef by Lemma 2.13
again.

Repeatedly, we will get a sequence of $V_{i}$ which is a union of prime
Calabi–Yau divisors (using (3.6) for each reduced and irreducible component of
$V_{i}$), such that $V_{i}\sim r_{i}K_{X}+V_{i-1}$ for some integer $r_{i}\geq
1$ and $V-(\sum r_{i})(-K_{X})=V_{i}$ is nef by Lemma 2.13. If $V_{i}\neq 0$
for infinitely many $i$, then for a fixed curve $C$ on $X$, we will have

$V_{i}\cdot C=V\cdot C-(\sum r_{i})(-K_{X})\cdot C<0$

for $i\gg 1$ since $-K_{X}$ is strictly nef and $\sum r_{i}\to+\infty$ for
$i\to+\infty$. This contradicts that $V_{i}$ is nef. Hence, we must stop after
finitely many steps of replacements. That is, $V\sim-(\sum r_{i})K_{X}$ for
some positive integer $i$. Since $K_{V}=(K_{X}+V)|_{V}\sim 0$ and $-K_{X}$ is
strictly nef, we must have that $\sum r_{i}=1$. That is, $V_{1}=0$ and $V\sim-
K_{X}$.

Finally, we consider the exact sequence:

$0\to\mathcal{O}_{X}(-mK_{X}+K_{X})\to\mathcal{O}_{X}(-mK_{X})\to\mathcal{O}_{V}(-mK_{X})\simeq\mathcal{O}_{V}(mL)\to
0.$

By (2.1) and the assumption, we have that

$\chi(V,mL)=\chi(X,-mK_{X})-\chi(X,-mK_{X}+K_{X})=\frac{c^{2}_{1}(X)\cdot
c_{2}(X)}{12}m>0$

for $m\geq 1$. In this case, it is well-known that the strictly nef divisor
$L=-K_{X}|_{V}$ is ample (see [22] for example). In particular,
$(-K_{X})^{4}=(-K_{X})^{3}\cdot V=(-K_{X}|_{V})^{3}>0$, which implies that
$-K_{X}$ is ample on $X$. ∎

The ampleness part of Theorem 1.3 follows immediately:

######  Corollary 3.2.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. If $\kappa(-K_{X})\geq 0$, then $-K_{X}$ is ample.

The next corollary of Theorem 3.1 is very similar to [18]*Lemma 4.5. It plays
the same role in Section 4 as [18]*Lemma 4.5 in [18]*Section 4.

######  Corollary 3.3.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. If $\kappa(-K_{X})=-\infty$, then the divisor $-K_{X}-D$ is not
pseudoeffective for any nonzero effective $\mathbb{R}$-divisor $D$ on $X$.

###### Proof.

Assume that there exists a nonzero effective $\mathbb{R}$-divisor $D$ such
that $E:=-K_{X}-D$ is pseudoeffective. By Theorem 3.1, the divisor
$H:=D-mK_{X}$ is ample for $m\gg 1$. Since $E$ is pseudoeffective and $H$ is
ample, we have that

$\begin{split}&(-K_{X})^{3}\cdot E\geq 0,\quad(-K_{X})^{2}\cdot H\cdot E\geq
0,\quad(-K_{X})\cdot H^{2}\cdot E\geq 0,\\\ &(-K_{X})^{3}\cdot H\geq
0,\quad(-K_{X})^{2}\cdot H^{2}\geq 0,\quad(-K_{X})\cdot H^{3}\geq
0.\end{split}$

Since $\kappa(-K_{X})=-\infty$, we have that $(-K_{X})^{4}=0$. From
$0=(m+1)(-K_{X})^{4}=(-K_{X})^{3}\cdot(E+H)\geq 0$, we obtain
$(-K_{X})^{3}\cdot E=(-K_{X})^{3}\cdot H=0$. Then, we have
$0=(m+1)(-K_{X})^{3}\cdot H=(-K_{X})^{2}\cdot(E+H)\cdot H\geq 0$, which
implies that $(-K_{X})^{2}\cdot H\cdot E=(-K_{X})^{2}\cdot H^{2}=0$.
Similarly, from

$0=(m+1)(-K_{X})^{2}\cdot H^{2}=(-K_{X})\cdot(E+H)\cdot H^{2}=(-K_{X})\cdot
H^{2}\cdot E+(-K_{X})\cdot H^{3}\geq 0,$

we finally obtain $(-K_{X})\cdot H^{2}\cdot E=(-K_{X})\cdot H^{3}=0$. However,
a multiple of the numerical class $H^{3}$ can be represented by an effective
curve since $H$ is ample, and thus $(-K_{X})\cdot H^{3}=0$ contradicts the
strict nefness of $-K_{X}$. ∎

######  Remark 3.4.

Actually, we prove in Theorem 3.1 that for any smooth projective fourfold $X$
with strictly nef anti-canonical divisor $-K_{X}$, the divisor $V-mK_{X}$ is
ample for $m\gg 1$ unless the prime divisor $V$ is linearly equivalent to
$-K_{X}$ (i.e., unless $V\sim-K_{X}$). In this case, we can improve the
situations a little bit further by adding assumptions as in [19]*Theorem 1.6.
We will also see in the next section that, throughout this paper, the
assumption $c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ is only used in the last
paragraph of the proof of Theorem 3.1 to avoid the remaining case of Serrano’s
conjecture for Calabi–Yau threefolds. Therefore, if the log version of
Serrano’s conjecture is totally solved in dimension 3, then Conjecture 1.1
holds true in dimension 4.

##  4\. Non-vanishing for strictly nef anti-canonical divisors

In this section, we prove the non-vanishing part of Theorem 1.3.

######  Theorem 4.1 (Theorems 4.2 and 4.5).

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. Then, we have that $\kappa(-K_{X})\geq 0$.

If $\nu(-K_{X})=4$, then the anti-canonical divisor $-K_{X}$ is big. By the
basepoint-free theorem, $-K_{X}$ is semiample, hence ample from the strict
nefness. If $\nu(-K_{X})=0$, then $-K_{X}$ is trivial, contradicting that it
is strictly nef. Therefore, it suffices to consider $\nu(-K_{X})=1,2,3$.

###  4.1. The case $\nu(-K_{X})=3$

In this case, we have the non-vanishing result by the standard arguments
without assuming $c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$.

######  Theorem 4.2.

Let $X$ be a smooth projective rationally connected fourfold such that the
anti-canonical divisor $-K_{X}$ is nef. If $\nu(-K_{X})=3$, then
$\kappa(-K_{X})\geq 0$.

###### Proof.

Let $H$ be a very ample smooth hypersurface on $X$. Then, $-K_{X}|_{H}$ is nef
and big on $H$ (see [19]*Lemma 2.1). Let us consider the long cohomology
sequence induced by the exact sequence:

$0\to\mathcal{O}_{X}(-mK_{X})\to\mathcal{O}_{X}(H-mK_{X})\to\mathcal{O}_{H}(H-mK_{X})\to
0.$

By adjunction, we have that

$(H-mK_{X})|_{H}=(K_{X}+H-(m+1)K_{X})|_{H}=K_{H}+(m+1)(-K_{X})|_{H}.$

Therefore, it follows from the Kawamata–Viehweg vanishing theorem that

$H^{i}(H,\mathcal{O}_{H}(H-mK_{X}))=H^{i}(H,\mathcal{O}_{H}(K_{H}+(m+1)(-K_{X})|_{H}))=0$

for $i\geq 1$ and $m\geq 0$. On the other hand, since $H+m(-K_{X})$ is ample
for any $m\geq 0$, we have

$H^{i}(X,\mathcal{O}_{X}(H-mK_{X}))=H^{i}(X,\mathcal{O}_{X}(K_{X}+H+(m+1)(-K_{X})))=0$

for any $i\geq 1$ and $m\geq 0$ by Kodaira’s vanishing theorem. Therefore, we
have that $H^{i}(X,\mathcal{O}_{X}(-mK_{X}))=0$ holds for $i\geq 2$ and $m\geq
0$. Note that $\chi(X,-mK_{X})\geq 1$ by Corollary 2.2. Hence,

$h^{0}(X,\mathcal{O}_{X}(-mK_{X}))=h^{1}(X,\mathcal{O}_{X}(-mK_{X}))+\chi(X,-mK_{X})\geq
1,$

which implies that $\kappa(-K_{X})\geq 0$. ∎

###  4.2. The case $\nu(-K_{X})\leq 2$

In this case, we follow closely the arguments in [18]*Subsection 4.2, which is
based on the strategy of [14, 15] and the references therein.

######  Lemma 4.3.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. If $\kappa(-K_{X})=-\infty$, then
$H^{0}(X,\Omega^{q}_{X}(-mK_{X}))=0$ for $0<q<n$ and $m\gg 1$.

###### Proof.

The proof is very similar to [14]*Proposition 3.4. If there exists an integer
$q$ such that $H^{0}(X,\Omega^{q}_{X}(-mK_{X}))\neq 0$ for infinitely many
positive integers $m$, then as in [14]*Proposition 3.4 or [15]*Lemma 4.1,
there exists a positive integer $r$ and a Cartier divisor $F$ such that
$\mathcal{O}_{X}(-F)$ is a subsheaf saturated in $\bigwedge^{r}\Omega^{q}_{X}$
and

$H^{0}(X,\mathcal{O}_{X}(-F-m^{\prime}K_{X}))\neq 0$

for infinitely many positive integers $m^{\prime}$. By the duality of Theorem
2.3, we have that $F$ is pseudoeffective. Therefore, there exists an effective
divisor $E_{m^{\prime}}$ such that

$-m^{\prime}K_{X}\sim E_{m^{\prime}}+F$

for infinitely many positive integers $m^{\prime}$. Then, Corollary 3.3
implies that $E_{m^{\prime}}=0$ for all $m^{\prime}$. In particular,
$-m^{\prime}_{1}K_{X}\sim F\sim-m^{\prime}_{2}K_{X}$ for two different
integers $m^{\prime}_{1}$ and $m^{\prime}_{2}$, which implies that $K_{X}$ is
trivial. This contradicts both that $\kappa(-K_{X})=-\infty$ and that $-K_{X}$
is strictly nef. ∎

######  Lemma 4.4.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. Assume that $-K_{X}$ admits a singular Hermitian metric $h$ with
positive curvature current such that the closed subschemes $V_{m}$ defined by
multiplier ideal sheaves $\mathcal{I}(h^{\otimes m})$ are of dimension $\leq
1$ for any $m\gg 1$. Then, we have $\kappa(-K_{X})\geq 0$.

###### Proof.

For a contradiction, we assume that $\kappa(-K_{X})=-\infty$. By the hard
Lefschetz theorem [6]*Theorem 0.1, the morphism

$H^{0}(X,\Omega^{2}_{X}(-mK_{X})\otimes\mathcal{I}(h^{\otimes
m}))\twoheadrightarrow
H^{2}(X,\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X})\otimes\mathcal{I}(h^{\otimes
m}))$

is surjective. By Lemma 4.3, we have that $H^{0}(X,\Omega^{2}_{X}(-mK_{X}))=0$
for any $m\gg 1$. It follows that
$H^{0}(X,\Omega^{2}_{X}(-mK_{X})\otimes\mathcal{I}(h^{\otimes
m}))=H^{2}(X,\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X})\otimes\mathcal{I}(h^{\otimes
m}))=0$ for any $m\gg 1$. On the other hand, we have that
$H^{2}(V_{m},K_{X}-mK_{X})=0$ for any $m\gg 1$ by the dimensional reason.
Then, from the long cohomology sequence induced by the exact sequence

$0\to\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X})\otimes\mathcal{I}(h^{\otimes
m})\to\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X})\to\operatorname{\mathcal{O}}_{V_{m}}(K_{X}-mK_{X})\to
0,$

we can see that $H^{2}(X,\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X}))=0$ for
any $m\gg 1$. Note that by Serre’s duality, we have that
$H^{4}(X,\operatorname{\mathcal{O}}_{X}(K_{X}-mK_{X}))=H^{0}(X,\operatorname{\mathcal{O}}_{X}(mK_{X}))=0$.
Note also that $\chi(X,K_{X}-mK_{X})\geq 1$ for $m\geq 1$ by Corollary 2.2.
This indicates that

$h^{0}(X,\mathcal{O}_{X}(-(m-1)K_{X}))\geq\chi(X,K_{X}-mK_{X})\geq 1$

for any $m\gg 1$, contradicting the assumption $\kappa(-K_{X})=-\infty$. ∎

Then, the proof of [18]*Theorem 4.7 works almost verbatim after replacing
[18]*Lemma 4.5 by Corollary 3.3 and [18]*Lemma 4.6 by Lemma 4.4. We sketch the
proof here for the reader’s convenience.

######  Theorem 4.5.

Let $X$ be a smooth projective rationally connected fourfold such that
$c^{2}_{1}(X)\cdot c_{2}(X)\neq 0$ and the anti-canonical divisor $-K_{X}$ is
strictly nef. If $\nu(-K_{X})\leq 2$, then $\kappa(-K_{X})\geq 0$.

###### Proof.

For a contradiction, we assume that $\kappa(-K_{X})=-\infty$. Let $h$ be a
singular Hermitian metric on $\operatorname{\mathcal{O}}_{X}(-K_{X})$ such
that the curvature current $\sqrt{-1}\Theta_{h}(-K_{X})$ is positive. Consider
the Siu decomposition of $\sqrt{-1}\Theta_{h}(-K_{X})$:

$\sqrt{-1}\Theta_{h}(-K_{X})=R+\sum_{i}\lambda_{i}[D_{i}],$

where $[D_{i}]$ is a current of integration over a prime divisor $D_{i}$ and
the positive real number $\lambda_{i}$ is the generic Lelong number along
$D_{i}$. If $[D_{i}]$ is nonzero for some $i$, then $-mK_{X}-D_{i}$ is
pseudoeffective for $m\gg 1$, contradicting to Corollary 3.3.

Therefore, we assume that $\sum_{i}\lambda_{i}[D_{i}]=0$. In this case, as in
the proof of [18]*Theorem 4.7, we can use the Bedford–Taylor product [4]*Lemma
7.4 and the approximation theorem [4]*Main Theorem 1.1 to get a Siu
decomposition of $(-K_{X})^{2}$:

(4.1) $\displaystyle(-K_{X})^{2}\ni T+S\geq
T+\sum_{S_{i}}\nu(R,S_{i})^{2}[S_{i}],$

where $T,S$ are positive $(2,2)$-currents, $\nu(R,S_{i})$ is the Lelong number
along any subvariety $S_{i}\subset X$ of codimension 2, and the inequality in
(4.1) follows from [4]*(7.5) and the properties of the Bedford–Taylor product.
Taking the wedge products of a Kähler form $\\{\omega\\}$ and the numerical
class $-K_{X}$ yields that

$0=(-K_{X})^{3}\cdot\\{\omega\\}\geq\\{T\\}\cdot(-K_{X})\cdot\\{\omega\\}+\\{\sum_{S_{i}}\nu(R,S_{i})^{2}[S_{i}]\\}\cdot(-K_{X})\cdot\\{\omega\\},$

where $\\{\bullet\\}$ denotes the the numerical class of $(1,1)$-currents. The
equality on the left-hand side follows from the assumption $\nu(-K_{X})\leq
2$. Then, since $-K_{X}$ is nef and $T$ is a positive $(2,2)$-current, each
term on the right-hand side is zero. In particular, we have that
$(-K_{X})|_{S_{i}}\equiv 0$ for any subvariety $S_{i}$ with $\nu(R,S_{i})>0$.
On the other hand, we can choose a curve $C$ on $S_{i}$ such that $-K_{X}\cdot
C>0$ since $-K_{X}$ is strictly nef. This is a contradiction, and so the
upper-level set of Lelong numbers

$E_{c}(\sqrt{-1}\Theta_{h}(-K_{X})):=\\{x\in
X\,|\,\nu(\sqrt{-1}\Theta_{h}(-K_{X}),x)\geq c\\}$

is a (possibly) countable union of Zariski closed subsets of codimension $\geq
3$. Then, by Skoda’s lemma, we can see that $V_{m}$ in Lemma 4.4 is of
codimension $\geq 3$, and thus we obtain the conclusion $\kappa(-K_{X})\geq
0$. ∎

## References

  * [1] F. Campana, J. A. Chen, Th. Peternell, _Strictly nef divisors_ , Math. Ann. 342 (2008), 565–585.
  * [2] F. Campana, Th. Peternell, _Projective manifolds whose tangent bundles are numerically effective_ , Math. Ann. 289 (1991), 169–187.
  * [3] F. Campana, Th. Peternell, _Geometric stability of the cotangent bundle and the universal cover of a projective manifold_ , Bull. Soc. Math. France 139 (2011), no. 1, 41–74, with an appendix by Matei Toma.
  * [4] J.-P. Demailly, _Regularization of closed positive currents and intersection theory_ , J. Algebraic Geom. 1 (1992), no. 3, 361–409.
  * [5] J.-P. Demailly, _Analytic Methods in Algebraic Geometry_ , Surveys of Modern Mathematics 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
  * [6] J.-P. Demailly, Th. Peternell, M. Schneider, _Pseudo-effective line bundles on compact kähler manifolds_ , Internat. J. Math. 12 (2001), no. 6, 689–741.
  * [7] O. Fujino, _Foundations of the minimal model program_ , MSJ Memoirs, 35. Mathematical Society of Japan, Tokyo, 2017.
  * [8] J. Han, W. Liu, _On numerical nonvanishing for generalized log canonical pairs_ , Doc. Math. 25 (2020), 93–123.
  * [9] F. Hao, S. Schreieder, _Equality in the Bogomolov–Miyaoka–Yau inequality in the non-general type case_ , J. Reine Angew. Math. 775 (2021), 87–115.
  * [10] J. Kollár, _Log surfaces of general type; some conjectures_ , in: Classification of algebraic varieties, Contemp. Math. 162, American Mathematical Society, Providence (1994), 261–275.
  * [11] J. Kollár, _Rational Curves on Algebraic Varieties_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32, Springer-Verlag Berlin Heidelberg, 1996.
  * [12] J. Kollár, S. Mori, _Birational geometry of algebraic varieties_ , Cambridge tracts in mathematics, vol. 134, Cambridge University Press, 1998.
  * [13] R. Lazarsfeld, _Positivity in algebraic geometry I, II_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, 49, Springer-Verlag, Berlin, 2004.
  * [14] V. Lazić, K. Oguiso, Th. Peternell, _Nef line bundles on Calabi–Yau threefolds I_ , Int. Math. Res. Not. IMRN 2020, no. 19, 6070–6119.
  * [15] V. Lazić, Th. Peternell, _Abundance for varieties with many differential forms_ , Épijournal Géom. Algébrique, 2 (2018), Art. 1, 35.
  * [16] D. Li, W. Ou, X. Yang, _On projective varieties with strictly nef tangent bundles_ , J. Math. Pures Appl. (9) 128 (2019), 140–151.
  * [17] H. Liu, C. Jiang, _On Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors_ , preprint.
  * [18] H. Liu, S. Matsumura, _Strictly nef divisors on K-trivial fourfolds_ , arXiv:2105.07259v2.
  * [19] H. Liu, R. Svaldi, _Rational curves and strictly nef divisors on Calabi–Yau threefolds_ , arXiv:2010.12233.
  * [20] H. Maeda, _A criterion for a smooth surface to be Del Pezzo_ , Math. Proc. Cambridge Phil. Soc. 113 (1993), 1–3.
  * [21] Y. Miyaoka, _The Chern Classes and Kodaira Dimension of a Minimal Variety_ , Adv. Stud. Pure Math. 10 (1987), 449–476.
  * [22] K. Oguiso, _On algebraic fiber space structures on a Calabi–Yau 3-fold_ , Internat. J. Math. 4 (1993), no. 3, 439–465, with an appendix by Noboru Nakayama.
  * [23] W. Ou, _On generic nefness of tangent sheaves_ , arXiv:1703.03175v4.
  * [24] F. Serrano, _Strictly nef divisors and Fano threefolds_ , J. Reine Angew. Math. 464 (1995), 187–206.


11institutetext: Faculty of Computer and Information Science, University of
Ljubljana, Slovenia 22institutetext: Department of Applied Informatics,
Comenius University Bratislava, Slovakia 22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Safe Reinforcement Learning in a Simulated Robotic Arm

Luka Kovač 11 Igor Farkaš 22

###### Abstract

Reinforcement learning (RL) agents need to explore their environments in order
to learn optimal policies. In many environments and tasks, safety is of
critical importance. The widespread use of simulators offers a number of
advantages, including safe exploration which will be inevitable in cases when
RL systems need to be trained directly in the physical environment (e.g. in
human-robot interaction). The popular Safety Gym library offers three mobile
agent types that can learn goal-directed tasks while considering various
safety constraints. In this paper, we extend the applicability of safe RL
algorithms by creating a customized environment with Panda robotic arm where
Safety Gym algorithms can be tested. We performed pilot experiments with the
popular PPO algorithm comparing the baseline with the constrained version and
show that the constrained version is able to learn the equally good policy
while better complying with safety constraints and taking longer training time
as expected.

###### Keywords:

safe exploration reinforcement learning robotic arm

##  1 Introduction

Reinforcement learning (RL) L agents need to explore their environments to
learn optimal behaviours. Sometimes an agent might perform a dangerous action,
therefore exploration is risky. Safe RL can be defined as the process of
learning to maximize the reward and at the same time to ensure respecting
safety constraints during learning [2]. It is usually possible to train the
agent in a simulated environment, and then after learning to transfer the
learned policy to a physical agent in the real world. However, because of
difficulties in simulating certain behaviours (e.g. human interaction, real-
world scenarios in traffic, etc.) agent’s learning is transferred to the real
world, where safety concerns are of great importance.

To address these problems, OpenAI created Safety Gym, a suite of environments
and tools for measuring progress toward RL agents that respect safety
constraints while learning [3], not only in testing. Safety Gym offers three
different agent types (point, car, quadruped), different tasks (goal, button,
push) and different safety constraints (hazards, vases, etc.). With those
tools, one can create different layouts for trying out novel RL algorithms and
having a common ground for benchmarking and evaluating them.

Our work integrates a new model of an agent (a robotic arm) into the Safety
Gym environment. In a simulated environment, we are able to evaluate the
agent’s behaviour regarding the safety concerns. Research in this direction
can produce significant contributions into human-robot interaction in the
future.

An optimal policy in constrained RL is given by:

$\pi^{*}=\arg\max_{\pi\in{\rm\Pi}_{C}}J_{r}(\pi)\quad\quad\quad{\rm\Pi}_{C}=\\{\pi:J_{c_{i}}(\pi)\leq
d_{i},\ i=1,...,k\\}$ (1)

where $J_{r}(\pi)$ is a reward-based objective function and each $J_{c_{i}}$
is a cost-based constraint function, involving thresholds $d_{i}$ (a human-
selected hyperparameters). These constraint functions form a feasible set (of
allowable policies) ${\rm\Pi}_{C}$ that has been defined in the framework of
constrained Markov Decision Processes [1]. In our case, $d_{i}=1$, if the arm
collides with the obstacle, otherwise it is 0. Hence, Lagrangian method uses a
two-component loss function (reward-based and cost-based). In eq. 1, the cost-
based component is included within the space of acceptable policies
${\rm\Pi}_{C}$. The optimization problem can also be expressed as

$\max_{\theta}\min_{\lambda\geq 0}L(\phi,\theta)\doteq f(\theta)-\lambda
g(\theta)$

where the two terms of the loss function correspond to the reward and the
cost, involving policy network parameters $\theta$ and Lagrangian
hyperparameter $\lambda$ [3].

##  2 Finding a technical solution

On one hand, it is positive that there exist various Python libraries and
robotic simulators built on a variety of physics simulation engines. On the
other hand, combining them or making extensions may often not be easy. Our
primary motivation was to integrate safe RL algorithms with a robotic arm (not
included in the Safety Gym library) that can be used in human-robot
interaction. Finding a solution was not straightforward, though. The
integration could be achieved in two ways: (1) Bringing a robotic arm model
into Safety Gym framework, or (2) using a different or a customized
environment with a robotic arm and integrate just the safety algorithms into
it. This led us to the exploration of feasible options.

Safety Gym is built on the MuJoCo physics engine [5], so we first tried to
import a Reacher model (a simplified robotic arm) from OpenAI Gym to Safety
Gym. This should be compatible, since both are based on MuJoCo. But various
technical problems (a lot of dependencies, the need to use older versions of
Python and Tensorflow) discouraged us from pursuing this line of
investigation.

Within the second option, we tried to connect Safety Gym with commonly used
robotic simulator CoppeliaSim using PyRep library built on top of it – but
this did not work due to incompatibility issues.

Finally, we used a PyBullet physics simulation engine that is built with
python and is an open source project, so it is well documented and with a lot
of examples already on the web. That helped a lot to set up the environment in
the desirable way. Because there are already a lot of examples, we found the
environment with a robotic arm that is implemented with PyBullet and is
compatible with OpenAI Gym - panda-gym. Our source code with the installation
guide and instructions of how to run the environment can be found
here.111https://github.com/lukakovac99/robotic-arm-safeRL We also implemented
two aditional arms to the environment – xarm and kuka – that can be used for
training with safety algorithms.

##  3 Experiments

We used the Proximal Policy Optimization (PPO), a well-known efficient policy
gradient method for RL [4] in our pilot experiments. We compared the basic PPO
with its constrained version (cPPO) using the panda-gym robotics arm (with 7
DoF).

Figure 1: Panda arm learned to reach the target (yellow cube) without
colliding with an obstacle (red) in front of it.

Regarding the action representation, we considered two options: (a) in
PyBullet, the action representation is given by a vector [$dx,dy,dz$] which
means changes of the tip of the arm in 3D Cartesian space (we label it AR1).
Those values are used to calculate the new position of the tip and via inverse
kinematics to calculate how much the joints should change. (b) We also tested
a “classical” actor output representation computed directly in the joint space
as a 7-dim. vector of DoF angle changes in each step (AR2). These values are
then directly added to move the arm (forward kinematics). We used dense reward
hence simulating robotic vision enabling the robor to estimate the distance
between the tip and the target, which served as information for calculating
the (inversely proportional) reward. Last but not least, we added an obstacle
on the table in front of the target object (see Fig.1).

Table 1: Average cost (with std) per one run of the classical PPO algorithm and its constrained version in case of Panda arm reaching for a target, using two action representation formats. | PPO | cPPO
---|---|---
3D | 17.6$\pm$1.3 | 11.9$\pm$3.6
7DoF | 23.8$\pm$5.0 | 17.0$\pm$1.9

In our four experiments (AR1/2, c/PPO) we used separate feedforward MLP policy
networks with two hidden layers, each with 64 neurons, 1000 steps per epoch,
maximum 200 epochs of training, and maximum number of steps per episode = 500.
The experiments lead to two observations (see Table 1): (a) Regarding AR type,
the agent learns faster (roughly with speedup factor of 2) and easier when
using AR1 than AR2 (this is probably due to higher dimensionality of the state
vector in the latter case). (b) Regarding the algorithm, cPPO yield lower
average costs for both AR types. This makes Lagrangian PPO safer, with a
tradeoff for length of training. Performance of both algorithms in case of AR1
is illustrated in Fig. 2.

Figure 2: Comparison of PPO and cPPO using panda arm in terms of reward (left)
and cost (right). Constrained PPO is slower in learning and reaching the
reward. On the other hand, it is keeping the cost at lower values hence making
the arm behavior safer.

##  4 Conclusion

We presented pilot results with a robotic arm (panda gym) environment that is
compatible with OpenAI Safety Gym, and verified the correct functionality on a
selected algorithm (PPO). Constrained (Lagrangian) PPO algorithm was observed
to have a longer learning time, but eventually learned the policies at the
same level of efficiency while being all the way safer.

The available code provides opportunities for experimenting with the robotic
arm in various setups, trying also other algorithms available in Safety Gym
(TRPO, cTRPO and CPO), adding a proper obstacle representation, obstacle
generation methods, or developing different safe tasks for the agent to
perform.

Acknowledgement: L.K. was supported by Erasmus mobility stipend, and I.F. by
the Horizon Europe project TERAIS, no. 101079338 and by the national project
APVV-21-0105.

## References

  * [1] Altman, E.: Constrained Markov Decision Processes. Routledge, New York (1999)
  * [2] Garcıa, J., Fernández, F.: A comprehensive survey on safe reinforcement learning. Journal of Machine Learning Research 16(1), 1437–1480 (2015)
  * [3] Ray, A., Achiam, J., Amodei, D.: Benchmarking safe exploration in deep reinforcement learning. arXiv preprint arXiv:1910. 01708 7(1.), 01708 (2019)
  * [4] Schulman, J., Wolski, F., Dhariwal, P.: Proximal policy optimization algorithms. arXiv preprintv arXiv (2017)
  * [5] Todorov, E., Erez, T., Tassa, Y.: MuJoCo: A physics engine for model-based control. In: IEEE International Conference on Intelligent Robots and Systems (2012)