Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

TL;DR

This paper establishes new inequalities relating Chern classes for terminal threefolds with nef anti-canonical divisors, providing classifications, examples, and extending Miyaoka--Kawamata type results.

Contribution

It introduces sharp bounds for Chern class intersections on such threefolds and offers a partial classification along with numerous examples.

Findings

Proved that c_1(X)·c_2(X) ≥ 1/252 when non-zero.

Established that for non-rationally connected X, c_1(X)·c_2(X) ≥ 4/5, which is sharp.

Extended Miyaoka--Kawamata inequalities to terminal weak Fano 3-folds.

Abstract

In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds.

Tables

Table 1. Table 1 . ℛ X subscript ℛ 𝑋 \mathcal{R}_{X} for χ = 1 𝜒 1 \chi=1 and c 1 ​ c 2 = 0 subscript 𝑐 1 subscript 𝑐 2 0 c_{1}c_{2}=0

No.	${ℛ}_X$	$r_X$
$1$	$\left\{5^5\right\}$	$5$
$2$	$\{2^3,4,8^2\}$	$8$
$3$	$\{2^3,5^2,10\}$	$10$
$4$	$\{2^2,3^2,4,12\}$	$12$
$5$	$\{2,4^6\}$	$4$
$6$	$\{3^4,4^2,6\}$	$12$
$7$	$\left\{3^9\right\}$	$3$
$8$	$\{2^6,4^4\}$	$4$
$9$	$\{2^5,3^4,6\}$	$6$
$10$	$\{2^{11},4^2\}$	$4$
$11$	$\left\{2^{16}\right\}$	$2$

Table 2. Table 2 . ℛ X subscript ℛ 𝑋 \mathcal{R}_{X} for χ = 1 𝜒 1 \chi=1 and − K 𝐾 -K not big

No.	${ℛ}_X$	$r_X$	$c_1{c}_2$
$1$	$\left\{5^2\right\}$	$5$	$72/5$
$2$	$\{2,3,6\}$	$6$	$14$
$3$	$\{2,4^2\}$	$4$	$15$
$4$	$\left\{3^3\right\}$	$3$	$16$
$5$	$\left\{7^3\right\}$	$7$	$24/7$
$6$	$\left\{2^4\right\}$	$2$	$18$
$7$	$\{2^2,{10}^2\}$	$10$	$6/5$
$8$	$\{2,4,8^2\}$	$8$	$3$
$9$	$\left\{5^4\right\}$	$5$	$24/5$
$10$	$\{2,3,5^2,6\}$	$30$	$22/5$
$11$	$\{2,4^2,5^2\}$	$20$	$27/5$
$12$	$\{3^3,5^2\}$	$15$	$32/5$
$13$	$\left\{5^5\right\}$	$5$	$0$
$14$	$\{2^4,5^2\}$	$10$	$42/5$
$15$	$\{2^3,4,8^2\}$	$8$	$0$
$16$	$\{2^3,5^2,10\}$	$10$	$0$
$17$	$\{2^3,6^3\}$	$6$	$2$
$18$	$\{2^2,3^2,4,12\}$	$12$	$0$
$19$	$\{2^2,3^2,6^2\}$	$6$	$4$
$20$	$\{2^2,3,4^2,6\}$	$12$	$5$
$21$	$\{2^2,4^4\}$	$4$	$6$
$22$	$\{2,3^4,6\}$	$6$	$6$
$23$	$\{2,3^3,4^2\}$	$12$	$7$
$24$	$\left\{3^6\right\}$	$3$	$8$
$25$	$\{2^5,3,6\}$	$6$	$8$
$26$	$\{2^5,4^2\}$	$4$	$9$
$27$	$\{2^4,3^3\}$	$6$	$10$
$28$	$\left\{2^8\right\}$	$2$	$12$
$29$	$\{2^4,3^3,5^2\}$	$30$	$2/5$
$30$	$\left\{3^9\right\}$	$3$	$0$
$31$	$\{2^8,5^2\}$	$10$	$12/5$
$32$	$\{2^6,4^4\}$	$4$	$0$
$33$	$\{2^5,3^4,6\}$	$6$	$0$
$34$	$\{2^5,3^3,4^2\}$	$12$	$1$
$35$	$\{2^4,3^6\}$	$6$	$2$
$36$	$\{2^9,3,6\}$	$6$	$2$
$37$	$\{2^9,4^2\}$	$4$	$3$
$38$	$\{2^8,3^3\}$	$6$	$4$
$39$	$\left\{2^{12}\right\}$	$2$	$6$
$40$	$\left\{2^{16}\right\}$	$2$	$0$

Table 3. Table 3 . Invariants of X 𝑋 X

No.	$c_2\left(X\right)$	$q\left(X\right)$	$h^2\left({𝒪}_X\right)$	$h^3\left({𝒪}_X\right)$	$\chi \left({𝒪}_X\right)$	$c_1{c}_2$
$(\text{<a><span>1</span></a>})$	$0$	$2$	$1$	$0$	$0$	$0$
$(\text{<a><span>2</span></a>})$	$0$	$1$	$0$	$0$	$0$	$0$
$(\text{<a><span>3</span></a>})$	$\ne 0$	$1$	$0$	$0$	$0$	$0$
$(\text{<a><span>4</span></a>})$	$\ne 0$	$0$	$1$	$0$	$2$	$48$
$(\text{<a><span>5</span></a>})$	$\ne 0$	$0$	$0$	$0$	$1$	$24$
$(\text{<a><span>6</span></a>})$	$\ne 0$	$0$	$0$	$0$	$1$	$24$

Table 4. Table 4 . G 𝐺 G for K3

$\#$	$G$	$\left|G\right|$	$\text{Sing}\left(S\right)$	${ℛ}_X$	$c_1{c}_2$
$1$	$C_2$	$2$	$8A_1$	$\left\{2^{16}\right\}$	$24$
$2$	$C_3$	$3$	$6A_2$	$\left\{3^{12}\right\}$	$16$
$3$	$D_4$	$4$	$12A_1$	$\left\{2^{24}\right\}$	$12$
$4$	$C_4$	$4$	$4A_3,2A_1$	$\{2^4,4^8\}$	$12$
$5$	$C_5$	$5$	$4A_4$	$\left\{5^8\right\}$	$48/5$
$6$	$D_6$	$6$	$3A_2,8A_1$	$\{2^{16},3^6\}$	$8$
$7$	$C_6$	$6$	$2A_5,2A_2,2A_1$	$\{2^4,3^4,6^4\}$	$8$
$8$	$C_7$	$7$	$3A_6$	$\left\{7^6\right\}$	$48/7$
$10$	$D_8$	$8$	$2A_3,9A_1$	$\{2^{18},4^4\}$	$6$
$14$	$C_8$	$8$	$2A_7,A_3,A_1$	$\{2^2,4^2,8^4\}$	$6$
$16$	$D_{10}$	$10$	$2A_4,8A_1$	$\{2^{16},5^4\}$	$24/5$
$17$	${𝔄}_4$	$12$	$6A_2,4A_1$	$\{2^8,3^{12}\}$	$4$
$18$	$D_{12}$	$12$	$A_5,A_2,9A_1$	$\{2^{18},3^2,6^2\}$	$4$
$34$	${𝔖}_4$	$24$	$2A_3,3A_2,5A_1$	$\{2^{10},3^6,4^4\}$	$2$
$55$	${𝔄}_5$	$60$	$2A_4,3A_2,4A_1$	$\{2^8,3^6,5^4\}$	$4/5$

Table 5. Table 5 . G s subscript 𝐺 𝑠 G_{s} for Enriques

$\#$	$G_s$	$\left|G\right|$	$\text{Sing}\left(S\right)$	${ℛ}_X$	$c_1{c}_2$
$1^{\prime }$	$C_2$	$4$	$4A_1$	$\left\{2^8\right\}$	$12$
$2^{\prime }$	$C_3$	$6$	$3A_2$	$\left\{3^6\right\}$	$8$
$3^{\prime }$	$D_4$	$8$	$6A_1$	$\left\{2^{12}\right\}$	$6$
$4^{\prime }$	$C_4$	$8$	$2A_3,A_1$	$\{2^2,4^4\}$	$6$
$5^{\prime }$	$C_5$	$10$	$2A_4$	$\left\{5^4\right\}$	$24/5$
$7^{\prime }$	$C_6$	$12$	$A_5,A_2,A_1$	$\{2^2,3^2,6^2\}$	$4$
${16}^{\prime }$	$D_{10}$	$20$	$A_4,4A_1$	$\{2^8,5^2\}$	$12/5$
${17}^{\prime }$	${𝔄}_4$	$24$	$3A_2,2A_1$	$\{2^4,3^6\}$	$2$

Equations

$$c_1(X{)}^{n-2}\cdot c_2(X)\ge 0.$$

$$c_1(X{)}^n\le b\cdot c_1(X{)}^{n-2}\cdot c_2(X)$$

$$\rho :{\pi }_1(Y)\to \mathrm{Aut}(F),$$

$$B_X=\{(b_i,r_i)\mid i=1,\cdots ,s;0<b_i\le \frac{r_i}{2};b_i\text{ is coprime to }{r}_i\}$$

$${\mathcal{R}}_X=\{3,3,7,11\}=\{3^2,7,11\}.$$

$$\chi (-n{K}_X)$$

$$\chi ({\mathcal{O}}_X)=\frac{1}{24}{c}_1(X)\cdot c_2(X)+\frac{1}{24}\sum _{r_i\in {\mathcal{R}}_X}(r_i-\frac{1}{r_i}).$$

$$h^2(X,{\mathcal{O}}_X)\le h^2(X^{\prime },{\mathcal{O}}_{X^{\prime }})=h^2(Y,{\mathcal{O}}_Y)\le 1$$

$$\chi ({\mathcal{O}}_X)=1+h^2(X,{\mathcal{O}}_X)\in \{1,2\}.$$

$$\sum _{r_i\in {\mathcal{R}}_X}(r_i-\frac{1}{r_i})\le 24\chi ({\mathcal{O}}_X)\le 48,$$

$$\sum _i(r_i-\frac{1}{r_i})\le 24\chi ({\mathcal{O}}_X)=24$$

$$c_1(X)\cdot c_2(X)=24-\sum _i(r_i-\frac{1}{r_i}).$$

$$l(2)=\sum _i\frac{b_i(r_i-b_i)}{2r_i}.$$

$$Q=\{[x_1:x_2:x_3:x_4]\in {\mathbb{P}}^3\mid x_1^2+x_2^2+x_3^2+x_4^2=0\}$$

$$(e;[x_1:x_2:x_3:x_4])\to (-e;[-x_1:-x_2:x_3:x_4]).$$

$$0\to {\mathcal{T}}_{X/Y}\to {\mathcal{T}}_X\to f^{\ast }{\mathcal{T}}_Y\to 0,$$

$$c_2(X)=c_2({\mathcal{T}}_{X/Y})+c_1({\mathcal{T}}_{X/Y})\cdot f^{\ast }{c}_1(Y)+f^{\ast }{c}_2(Y)=0.$$

$$0\to {\mathcal{T}}_F\to {\mathcal{T}}_X{|}_F\to {\mathcal{N}}_{F/X}\simeq {\mathcal{O}}_F(F)\to 0,$$

$$c_2(X)\cdot F=c_2(F)=e(F)=2+h^{1,1}(F)>0,$$

$$X^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1\to G_0\to G\to H\to 1$$

$$X^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1\to G_s\to G\to N\to 1$$

$$X^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0=:{\mathcal{E}}_0\subset {\mathcal{E}}_1\subset \cdots \subset {\mathcal{E}}_l:={\mathcal{T}}_X$$

$$\begin{array}{rl}2c_2({\mathcal{T}}_X)H_{\epsilon }^{n-2}& =\left(c_1({\mathcal{T}}_X{)}^2+\sum _{1\le i\le l}(2c_2({\mathcal{G}}_i)-c_1({\mathcal{G}}_i{)}^2)\right)H_{\epsilon }^{n-2}\\ & \ge \left(c_1({\mathcal{T}}_X{)}^2-\sum _{1\le i\le l}\frac{c_1({\mathcal{G}}_i{)}^2}{r_i}\right)H_{\epsilon }^{n-2}.\end{array}$$

$$\sum _{1\le i\le l}{a}_i=c_1({\mathcal{T}}_X{)}^n>0\text{and}{\mu }_{H_{\epsilon }}({\mathcal{G}}_i)=\frac{c_1({\mathcal{G}}_i)H_{\epsilon }^{n-1}}{r_i}=\frac{a_i}{r_i}+O(\epsilon ).$$

$$2c_2({\mathcal{T}}_X)H_{\epsilon }^{n-2}\ge \frac{(r_1-1)a_1}{r_1}+\sum _{2\le j\le l}{a}_j+O(\epsilon ).$$

Full text

1. 1 Introduction
2. 2 A structure theorem of Cao–Höring and Matsumura–Wang
3. 3 Reid’s formula and holomorphic Euler characteristics
4. 4 The case $\chi=1$
5. 5 Smooth cases
6. 6 Non-rationally connected singular cases
7. 7 The case when $-K$ is big
8. 8 Proof of main theorems

1. Introduction

In [26]*Question 4.8, the third author proposed the following question, which is a mirror version of Kollár’s expectation [22]*Remark 3.6 for minimal varieties.

Question 1.1**.**

Let $X$ be a terminal projective variety of dimension $n$ such that $-K_{X}$ is nef.

1. (1)

If $c_{1}(X)^{n-2}\cdot c_{2}(X)=0$, is there a classification of $X$? 2. (2)

If $c_{1}(X)^{n-2}\cdot c_{2}(X)>0$, is there a constant $\epsilon>0$ depending only on $n$ such that $c_{1}(X)^{n-2}\cdot c_{2}(X)\geq\epsilon$?

This expectation heavily depends on Ou’s result [37]*Corollary 1.5 on the pseudo-effectivity of the second Chern classes, which implies that for a terminal projective variety $X$ of dimension $n$ such that $-K_{X}$ is nef,

[TABLE]

It can be viewed as a mirror version of Miyaoka’s inequality [30]*Corollary 6.4 on the second Chern classes.

When $\dim X=2$, that is, when $X$ is a smooth projective surface such that $-K_{X}$ is nef, the answer to Question 1.1 is clear and well-known. If $c_{2}(X)=0$, then it is well-known that $X$ admits a finite étale cover by an abelian surface or a $\mathbb{P}^{1}$-bundle over an elliptic curve (see [37]*Theorem 1.8 and the references therein). If $c_{2}(X)\neq 0$, then the Noether’s formula $c_{2}(X)=12\chi(\mathcal{O}_{X})-c_{1}(X)^{2}$ implies that $\chi(\mathcal{O}_{X})\geq 1$ and hence $c_{2}(X)\geq 3$ as $c_{1}(X)^{2}\leq 9$; furthermore, if $X$ is not rational (or $-K_{X}$ is not big), then $c_{1}(X)^{2}=0$ and $c_{2}(X)\geq 12$.

The goal of this paper is to study Question 1.1 in dimension 3, which is also motivated by the classification theory of $3$-folds. For a terminal projective $3$-fold $X$ with $K_{X}\equiv 0$, $c_{1}(X)\cdot c_{2}(X)=0$ holds trivially. Such $3$-folds are known as “Calabi–Yau $3$-folds" and have already been studied in many references (for example, [19, 32, 35, 36, 16]), so we will always assume that $K_{X}\not\equiv 0$. In this case, there are also many interesting results, see [1, 7, 43, 4, 29, 25, 44] and the references therein.

Our main theorem is the following.

Theorem 1.2**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef. Suppose that $c_{1}(X)\cdot c_{2}(X)\neq 0$.

1. (1)

Then $c_{1}(X)\cdot c_{2}(X)\geq\frac{1}{252}$. 2. (2)

If $-K_{X}$ is not big, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{2}{5}$. 3. (3)

If $X$ is not rationally connected, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{4}{5}$.

Theorem 1.2 is obtained by a partial classification of terminal projective $3$-folds with nef anti-canonical divisors. We briefly summarize the results in the following.

Theorem 1.3**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$. Then $X$ is one of the following:

1. (1)

(Theorem 5.1, cf. **[1]**) $X$ is smooth and

1. (a)

$X$* is a $\mathbb{P}^{1}$-bundle over an abelian surface or a bi-elliptic surface;* 2. (b)

$X$* admits a locally trivial fibration over an elliptic curve, whose fibers are rationally connected;* 3. (c)

$X$* is a product of $\mathbb{P}^{1}$ with a K3 surface;* 4. (d)

$X$* is a $\mathbb{P}^{1}$-bundle over an Enriques surface, which is trivialized by the universal double étale cover;* 2. (2)

$X$* is rationally connected;* 3. (3)

(Theorem 6.2) $X$ is singular and not rationally connected, and $X=X^{\prime}/G$ where $X^{\prime}$ is either a $\mathbb{P}^{1}$-bundle over an abelian surface or a product of $\mathbb{P}^{1}$ with a K3 surface, and $G$ is a group acting on $X^{\prime}$ which can be classified.

From this classification, we can describe the structure of $X$ when $c_{1}(X)\cdot c_{2}(X)=0$.

Corollary 1.4**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$. Then $c_{1}(X)\cdot c_{2}(X)=0$ if and only if $X$ is one of the following:

1. (1)

$X$* admits a quasi-étale cover by a $\mathbb{P}^{1}$-bundle over an abelian surface;* 2. (2)

$X$* admits a locally trivial fibration over an elliptic curve, whose fibers are rationally connected;* 3. (3)

$X$* is rationally connected, $K_{X}^{3}=0$, and the set $\mathcal{R}_{X}$ of local indices of the virtue singularities (see §3 for the definition) belongs to Table LABEL:tab1 of Theorem 4.1 except types $(5),(6)$ and $(10)$.*

Remark 1.5**.**

The lower bound of Theorem 1.2(3) is sharp by Example 6.4, while the lower bounds of Theorem 1.2(1)(2) might not be sharp. The main difficulty is the case when $X$ is non-Gorenstein and rationally connected, in which case there are no good methods to classify $X$ in a more explicit way. For example, one natural strategy in birational geometry is to run a minimal model program, and then describe each step of the minimal model program and classify the outcomes (which are Mori fiber spaces in this case). But so far neither of these can be done in an explicit way as singularities are involved. Also, we will lose the nefness of $-K_{X}$ when running the minimal model program.

Finally, we focus on the case when $-K_{X}$ is nef and big. Such $X$ is usually called a weak Fano variety; and $X$ is called a Fano variety if $-K_{X}$ is ample. Terminal weak Fano $3$-folds have been also studied in many references (see [8, 17] and the references therein). In fact, $c_{1}c_{2}>2$ for the majority of known examples of terminal Fano $3$-folds (see Example 7.1). So it is interesting to improve the lower bound of $c_{1}c_{2}$ for terminal weak Fano $3$-folds or to find new examples with small $c_{1}c_{2}$.

Another interesting problem is whether there exist terminal weak Fano $3$-folds with $c_{1}c_{2}=0$. In this direction, we prove that this can not happen.

Theorem 1.6** (Corollary 7.3).**

Let $X$ be a terminal weak Fano variety of dimension $n$. Then $c_{1}(X)^{n-2}\cdot c_{2}(X)>0$.

As a corollary, we can show a Miyaoka–Kawamata type inequality (see [20, 5] for the case $\rho(X)=1$).

Corollary 1.7** (Corollaries 7.4 and 7.6).**

There exists a positive rational number $b$ depending only on $n$ such that the inequality

[TABLE]

holds for any terminal weak Fano variety $X$ of dimension $n$. In particular, we can choose $b=2^{4}\cdot 3^{6}\cdot 7$ when $n=3$.

Remark 1.8**.**

The universal upper bound $b=2^{4}\cdot 3^{6}\cdot 7$ in Corollary 1.7 is far from being sharp. We would conjecture that $b$ should be 3, the same as the constant in [30]*Theorem 1.1 (see Remark 7.7).

Remark 1.9**.**

Recently, the third author and Jie Liu proved in [27] that for terminal $\mathbb{Q}$-Fano varieties, i.e., $\mathbb{Q}$-factorial terminal Fano varieties with Picard number one, the bound $b$ in Corollary 1.7 can be improved to be $4$ (and to be $25/8$ in dimension 3).

Acknowledgments**.**

The authors would like to thank Professors Vladimir Lazić, Shin-ichi Matsumura, Thomas Peternell and Shilin Yu for useful discussions and suggestions. The first author would like to thank Kento Fujita for useful comments. The third author would like to thank Xiaobin Li for his computer support. We thank the referees for useful comments and suggestions.

This work was supported by National Natural Science Foundation of China for Innovative Research Groups (Grant No. 12121001) and National Key Research and Development Program of China (Grant No. 2020YFA0713200). The first author was supported by Grant-in-Aid for Early Career Scientists $\sharp$ 22K13907. The second author is a member of LMNS, Fudan University.

Throughout this paper, we work over the complex number field $\mathbb{C}$. We will freely use the basic notation in [24].

2. A structure theorem of Cao–Höring and Matsumura–Wang

Let $X$ be a terminal projective variety such that $-K_{X}$ is nef. There exists a structure theorem of $X$ by [7]*Theorem 1.3 for smooth cases and [29]*Theorem 1.1 for klt cases. We recall these results in the setting of terminal singularities as follows.

Definition 2.1**.**

Let $f\colon X\to Y$ be a proper surjective morphism between normal varieties with connected fibers. Then $f$ is called a locally constant fibration if $f$ is a locally trivial fiber bundle with a representation

[TABLE]

where $\pi_{1}(Y)$ is the fundamental group of $Y$ and $F$ is the fiber of $f$, such that $X$ is isomorphic to the quotient of $\widetilde{Y}\times F$ by the action of $\pi_{1}(Y)$ given by $\delta\cdot(\widetilde{y},z)=(\delta\cdot\widetilde{y},\rho(\delta)(z))$ where $\widetilde{Y}$ is the universal cover of $Y$.

Theorem 2.2** (cf. [7, 40, 29]).**

Let $X$ be a terminal projective variety such that $-K_{X}$ is nef. Then, there exists a finite cover $\pi\colon X^{\prime}\to X$ étale in codimension $2$ such that $X^{\prime}$ admits a holomorphic MRC fibration $f\colon X^{\prime}\to Y$ satisfying the following properties:

1. (1)

$f$* is locally constant;* 2. (2)

any fiber $F$ of $f$ is a rationally connected variety with terminal singularities; 3. (3)

$Y$* is a terminal projective variety with $K_{Y}\sim_{\mathbb{Q}}0$.*

Moreover, if $X$ is smooth, then we may choose $\pi$ to be the identity map and $Y$ is smooth.

Proof.

The existence of $\pi$ and $f$ is by [29]*Theorem 1.1 and the last statement is by [7]*Theorem 1.3.

Here note that by [29]*Theorem 1.1, $\pi$ is étale in codimension $1$ and $Y,F$ are klt. So we shall explain why $\pi$ is étale in codimension $2$ and $Y,F$ are terminal in our setting.

Since $X$ is terminal, the singular locus $\operatorname{Sing}X$ is of codimension at least 3 by [24]*Corollary 5.18. Since $\pi$ is étale in codimension $1$, $\pi$ is actually étale over the smooth locus $X\setminus\operatorname{Sing}X$ by the purity of ramification locus (Zariski’s purity theorem). Therefore, $\pi\colon X^{\prime}\to X$ is étale in codimension $2$.

Since $X$ is terminal, $X^{\prime}$ is also terminal by [24]*Proposition 5.20; then $Y,F$ are also terminal since $f$ is locally constant. ∎

Lemma 2.3**.**

Let $F$ be a rationally connected projective variety with rational singularities. Then $H^{j}(F,\mathcal{O}_{F})=0$ for $j\geq 1$. In particular, $\chi(\mathcal{O}_{F})=1$.

Proof.

As $F$ has only rational singularities, it suffices to show that for a resolution $F^{\prime}$ of $F$, $H^{j}(F^{\prime},\mathcal{O}_{F^{\prime}})=0$ for $j\geq 1$. Note that $F^{\prime}$ is again rationally connected, so the conclusion follows from [23]*Corollary IV.3.8. ∎

Proposition 2.4**.**

Let $f\colon X^{\prime}\to Y$ be the fibration in Theorem 2.2. Then $h^{i}(X^{\prime},\mathcal{O}_{X^{\prime}})=h^{i}(Y,\mathcal{O}_{Y})$ for each $i\geq 0$. In particular, $\chi(\mathcal{O}_{X^{\prime}})=\chi(\mathcal{O}_{Y})$.

Proof.

Since any fiber $F$ of $f$ is rationally connected and terminal, $H^{j}(F,\mathcal{O}_{F})=0$ for $j\geq 1$ by Lemma 2.3. In particular, $R^{j}f_{*}\mathcal{O}_{X^{\prime}}=0$ for $j\geq 1$ as the fibration is locally constant. Then $h^{i}(X^{\prime},\mathcal{O}_{X^{\prime}})=h^{i}(Y,\mathcal{O}_{Y})$ by the Leray spectral sequence. ∎

Lemma 2.5**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef. If the irregularity $q(X):=h^{1}(X,\mathcal{O}_{X})>0$, then $X$ is smooth.

Proof.

By [40]*Theorem A or [29]*Corollary 4.9, the Albanese map of $X$ is locally constant and surjective. Any fiber of the Albanese map of $X$ has dimension at most $2$ and has terminal singularities, so it is smooth, which implies that $X$ is smooth. ∎

If a normal variety $X$ is smooth in codimension $k$, then by extending from the smooth locus, the Chern classes $c_{i}(X)$ can be defined as elements sitting in the Chow groups $A^{i}(X)$ for $i\leq k$. In particular, $c_{1}(X)$ and $c_{2}(X)$ are well-defined for terminal varieties. The following lemma for Chern classes is easy but useful.

Lemma 2.6**.**

Let $X$ be a normal projective $\mathbb{Q}$-Gorenstein variety of dimension $n$ which is smooth in codimension $2$. Let $\pi\colon X^{\prime}\to X$ be a finite morphism which is étale in codimension $2$, then $c_{i}(X^{\prime})=\pi^{*}c_{i}(X)$ for $i=1,2$. In particular, $c_{1}(X^{\prime})^{n-2}\cdot c_{2}(X^{\prime})=\deg\pi\cdot c_{1}(X)^{n-2}\cdot c_{2}(X)$.

3. Reid’s formula and holomorphic Euler characteristics

A basket $B$ is a collection of pairs of coprime integers (permitting weights). Let $X$ be a terminal projective $3$-fold. According to Reid [39], there is a basket of virtual orbifold points

[TABLE]

associated to $X$, where a pair $\left(b_{i},r_{i}\right)$ corresponds to an orbifold point $Q_{i}$ of type $\frac{1}{r_{i}}\left(1,-1,b_{i}\right)$. Here recall that every singularity of $X$, as a $3$-dimensional terminal singularity, can be locally deformed into a set of cyclic quotient singularities, and these cyclic quotient singularities are called virtual orbifold points of $X$. Denote by $\mathcal{R}_{X}$ the collection of $r_{i}$ (permitting weights) appearing in $B_{X}$ or as a set of integers with weights where weights appear in superscripts, say for example,

[TABLE]

Note that the Cartier index $r_{X}$ of $K_{X}$ is just ${\text{l.c.m.}}\{r_{i}\mid r_{i}\in\mathcal{R}_{X}\}$.

According to Reid [39]*10.3, for any positive integer $n$,

[TABLE]

where $l(n+1)=\sum_{i}\sum_{j=1}^{n}\frac{\overline{jb_{i}}\left(r_{i}-\overline{jb_{i}}\right)}{2r_{i}}$ and the first sum runs over $B_{X}$. Here $\overline{jb_{i}}$ means the smallest non-negative residue of $jb_{i}\bmod r_{i}$.

In this case, $\chi(\mathcal{O}_{X})$ is related to $c_{1}(X)\cdot c_{2}(X)$ and the singularities as follows:

Theorem 3.1** ([19, 39]).**

Let X be a terminal projective $3$-fold. Then

[TABLE]

As a corollary, we get the finiteness of singularities of terminal $3$-folds with nef anti-canonical divisors.

Corollary 3.2**.**

Let $X$ be a terminal projective $3$-fold $X$ such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$. Then the following hold:

1. (1)

$\chi(\mathcal{O}_{X})\in\{0,1,2\}$. Moreover, $\chi(\mathcal{O}_{X})\neq 0$ if $X$ is singular. 2. (2)

*There are only finitely many possibilities for $\mathcal{R}_{X}$, $B_{X}$, and $c_{1}(X)\cdot c_{2}(X)$. *

Proof.

(1) We may assume that $X$ is not rationally connected by Lemma 2.3.

Let $f\colon X^{\prime}\to Y$ be the fibration in Theorem 2.2. Then $\dim Y\in\{1,2\}$ as $K_{X}\not\equiv 0$ and $X$ is not rationally connected. Moreover, $Y$ is smooth and $K_{Y}\sim_{\mathbb{Q}}0$. Hence $\chi(\mathcal{O}_{X^{\prime}})=\chi(\mathcal{O}_{Y})\in\{0,1,2\}$ by Proposition 2.4 and the classical classification of curves and surfaces.

Let $q(X):=h^{1}(X,\mathcal{O}_{X})$ be the irregularity of $X$. If $q(X)>0$, then $X$ is smooth by Lemma 2.5. Hence $\chi(\mathcal{O}_{X})=\chi(\mathcal{O}_{X^{\prime}})\in\{0,1,2\}$ by Theorem 2.2.

If $q(X)=0$ (which contains the case that $X$ is singular), then as $X^{\prime}\to X$ is finite,

[TABLE]

by Proposition 2.4 and the classical classification of curves and surfaces. Also $h^{3}(X,\mathcal{O}_{X})=0$ as $K_{X}\not\equiv 0$. Hence

[TABLE]

(2) By [37]*Corollary 1.5, $c_{1}(X)\cdot c_{2}(X)\geq 0$. Hence by Theorem 3.1, as

[TABLE]

there are only finitely many possibilities for $\mathcal{R}_{X}$ and hence also for $B_{X}$. Therefore, by Theorem 3.1 again, there are only finitely many possibilities for $c_{1}(X)\cdot c_{2}(X)$. ∎

If $-K_{X}$ is nef but not big, then there is a strong restriction on $B_{X}$.

Lemma 3.3**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef but not big. Then $l(n+1)$ is an integer for any positive integer $n$. In particular, $l(2)=\sum_{i}\frac{b_{i}(r_{i}-b_{i})}{2r_{i}}$ is an integer.

Proof.

Since $(-K_{X})^{3}=0$, by (3.1), $l(n+1)=(2n+1)\chi(\mathcal{O}_{X})-\chi(-nK_{X})$ is an integer. ∎

4. The case $\chi=1$

In this section, we study the case when $X$ is a terminal projective $3$-fold such that $-K_{X}$ is nef and $\chi(\mathcal{O}_{X})=1$. In particular, it contains the case that $X$ is rationally connected by Lemma 2.3.

Theorem 4.1**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $\chi(\mathcal{O}_{X})=1$. Then the following statements hold.

1. (1)

Either $c_{1}(X)\cdot c_{2}(X)\geq\frac{1}{252}$ or $c_{1}(X)\cdot c_{2}(X)=0$, where in the latter case, $\mathcal{R}_{X}$ is one of the following types:

2. (2)

If moreover $-K_{X}$ is not big, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{2}{5}$ or $c_{1}(X)\cdot c_{2}(X)=0$; and $\mathcal{R}_{X}$ is one of the following types:

Proof.

(1) Note that

[TABLE]

by (3.3) and

[TABLE]

With the aid of a computer program, we can list all possible $\mathcal{R}_{X}$ satisfying (4.1). Among them, there are $11$ possibilities of $\mathcal{R}_{X}$ as listed in Table LABEL:tab1 with $c_{1}(X)\cdot c_{2}(X)=0$, while $c_{1}(X)\cdot c_{2}(X)\geq\frac{1}{4\cdot 7\cdot 9}=\frac{1}{252}$ for the remaining ones. Here the minimal value is obtained by $\mathcal{R}_{X}=\{2^{3},4,7,9\}$.

(2) As $-K_{X}$ is not big, $l(2)\in\mathbb{Z}$ by Lemma 3.3, where

[TABLE]

With the aid of a computer program, we can list all possible $\mathcal{R}_{X}$ satisfying (4.1) such that $l(2)\in\mathbb{Z}$ for some basket $B_{X}=\{(b_{i},r_{i})\}$, as in Table LABEL:tab2. Here the minimal value is obtained by $\mathcal{R}_{X}=\{2^{4},3^{3},5^{2}\}$ as No.29 of Table LABEL:tab2. ∎

The following example provides a rationally connected terminal projective $3$-fold such that $-K_{X}$ is nef but not big with $c_{1}(X)\cdot c_{2}(X)=0$.

Example 4.2**.**

Let $E$ be an elliptic curve and

[TABLE]

be a smooth quadric surface in $\mathbb{P}^{3}$. Let $X^{\prime}=E\times Q$. It is easy to see that $-K_{X^{\prime}}$ is nef with $(-K_{X^{\prime}})^{3}=0$, and $c_{1}(X^{\prime})\cdot c_{2}(X^{\prime})=0$. Consider the involution $\tau$ on $X^{\prime}$:

[TABLE]

Let $X\coloneqq X^{\prime}/\tau$ and $f\colon X\to E/-1\simeq\mathbb{P}^{1}$ be the natural projection. Then $\pi\colon X^{\prime}\to X$ is étale in codimension $2$ and $X$ is a $\mathbb{Q}$-factorial terminal $3$-fold with exactly 16 points of type $\frac{1}{2}(1,1,1)$. Note that $X$ is rationally connected by [12]*Corollary 1.3, $-K_{X}$ is nef but not big, and $c_{1}(X)\cdot c_{2}(X)=0$ by Lemma 2.6. In this case, $\mathcal{R}_{X}=\{2^{16}\}$ as No.11 of Table LABEL:tab1 (or No.40 of Table LABEL:tab2).

Remark 4.3**.**

In the case $\chi(\mathcal{O}_{X})=2$, we can still use computer to list all the possibilities of $\mathcal{R}_{X}$, which are numerous (over 1000 candidates). In §6, we will use a more geometric method to deal with this case.

5. Smooth cases

For a smooth projective $3$-fold $X$ such that $-K_{X}$ is nef, a detailed classification has been given by Bauer–Peternell in [1]. Combining with Cao–Höring’s recent result [7], we summarize the result as the following.

Theorem 5.1**.**

Let $X$ be a smooth projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$. Let $q(X):=h^{1}(X,\mathcal{O}_{X})$ be the irregularity of $X$. Then $X$ is one of the following cases:

1. (1)

$X$* is a $\mathbb{P}^{1}$-bundle over an abelian surface;* 2. (2)

$X$* is a $\mathbb{P}^{1}$-bundle over a bi-elliptic surface;* 3. (3)

$X$* admits a locally trivial fibration over an elliptic curve whose fibers are rationally connected;* 4. (4)

$X=\mathbb{P}^{1}\times Y$, where $Y$ is a K3 surface; 5. (5)

$X$* is a $\mathbb{P}^{1}$-bundle over an Enriques surface, which is trivialized by the universal double étale cover;* 6. (6)

$X$* is rationally connected.*

Proof.

By Theorem 2.2, there exists a locally constant fibration $f\colon X\to Y$ such that any fiber $F$ of $f$ is rationally connected and $Y$ is a smooth projective variety with $K_{Y}\sim_{\mathbb{Q}}0$. Since $K_{X}\not\equiv 0$, $\dim Y\in\{0,1,2\}$.

Case 1**.**

If $\dim Y=2$, then the fiber $F$ is $\mathbb{P}^{1}$. Since $K_{Y}\sim_{\mathbb{Q}}0$, $Y$ belongs to one of the $4$ classes of minimal surfaces with Kodaira dimension [math] by the Enriques–Kodaira classification.

If $Y$ is an abelian surface or a bi-elliptic surface, then we get Case (1) or (2). From the exact sequence

[TABLE]

we have

[TABLE]

Here we used the facts that $\mathcal{T}_{X/Y}$ is a line bundle as $\dim F=1$, $c_{1}(Y)=0$, and $c_{2}(Y)=0$.

If $Y$ is a K3 surface or an Enriques surface, then $q(X)=h^{1}(X,\mathcal{O}_{X})=h^{1}(Y,\mathcal{O}_{Y})=0$ by Proposition 2.4. Then by [1]*Corollary 4.4 and Corollary 3.2, we get Case (4) or (5).

Case 2**.**

If $\dim Y=1$, then $Y$ is an elliptic curve and we get Case (3). From the exact sequence

[TABLE]

we have

[TABLE]

which implies that $c_{2}(X)\neq 0$. Here we used the fact that $F$ is a rationally connected surface.

Case 3**.**

If $\dim Y=0$, then $X$ is rationally connected and we get Case (6).

By Proposition 2.4, $h^{i}(X,\mathcal{O}_{X})=h^{i}(Y,\mathcal{O}_{Y})$ for each $i\geq 0$. By the classical Hirzebruch–Riemann–Roch theorem, $24\chi(\mathcal{O}_{X})=c_{1}(X)\cdot c_{2}(X).$ So to complete Table LABEL:tab3, we only need to compute $c_{2}(X)$ and $h^{i}(Y,\mathcal{O}_{Y})$ for each $i\geq 0$, which are clear from the above discussions and the classification of $Y$. ∎

Remark 5.2**.**

In Cases (1) and (2) of Theorem 5.1, the second Chern class $c_{2}(X)$ is trivial. In Case (1), after a finite étale cover, we can write $X=\mathbb{P}(E)$ where $E$ is a numerically flat bundle of rank 2 (see [1]*Remark 4.3); for Case (2), after an étale base change, it can be reduced to Case (1).

Remark 5.3**.**

While the equality $c_{1}(X)\cdot c_{2}(X)=0$ holds trivially in Cases (1) and (2) of Theorem 5.1, we have shown that neither $c_{1}(X)$ nor $c_{2}(X)$ is trivial in Case (3). Since the fiber $F$ is a smooth rationally connected surface such that $-K_{F}=-K_{X}|_{F}$ is nef, we have a complete classification of these surfaces (see [1]*Propositions 1.5 and 1.6 and the last paragraph of [1]*Preliminaries).

On the other hand, there are also classification results for rationally connected $3$-folds with nef anti-canonical divisors, see for example [1, 43, 44].

6. Non-rationally connected singular cases

In this section, we give a classification of non-rationally connected singular terminal projective $3$-folds with nef and non-trivial anti-canonical divisors.

Lemma 6.1**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$. Assume that $X$ is singular and not rationally connected. Let $f\colon X^{\prime}\to Y$ be the fibration in Theorem 2.2. Then $\dim Y=2$.

Proof.

If $\dim Y=0$, then $X^{\prime}$ is rationally connected, which contradicts to the assumption.

If $\dim Y=1$, then consider the MRC fibration $X\dashrightarrow Z$ of $X$ (which is an almost holomorphic map whose general fibers are rationally connected such that $Z$ is not uniruled). Since $X$ is not rationally connected, $\dim Z\geq 1$. On the other hand, as $X^{\prime}$ is fibered by rationally connected surfaces and $Z$ is not uniruled, we have $\dim Z\leq 1$. But then $h^{1}(X,\mathcal{O}_{X})\geq h^{1}(Z,\mathcal{O}_{Z})\geq 1$, which implies that $X$ is smooth by Lemma 2.5, a contradiction.

If $\dim Y=3$, then $X^{\prime}\simeq Y$ and $K_{X^{\prime}}\equiv 0$. It follows that $K_{X}\equiv 0$, a contradiction. ∎

Theorem 6.2**.**

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $K\not\equiv 0$. Assume that $X$ is singular and not rationally connected.

*Then there exists a finite cover $\pi\colon X^{\prime}\to X$ étale in codimension $2$ which is Galois (in the sense of [29]Definition 2.11) and a morphism $f\colon X^{\prime}\to Y$ satisfying the following properties:

1. (1)

One of the following holds:

1. (a)

$X^{\prime}$* is a $\mathbb{P}^{1}$-bundle over $Y$ where $Y$ is an abelian surface, or* 2. (b)

$X^{\prime}=\mathbb{P}^{1}\times Y$* where $Y$ is a K3 surface.* 2. (2)

Denote by $G$ the Galois group of $\pi$, then $G$ acts faithfully on $Y$ and there exists a commutative diagram

[TABLE]

such that $\mu$ is étale in codimension $1$ and $S$ is a surface with Du Val singularities of type A with $K_{S}\sim_{\mathbb{Q}}0$ and $h^{1}(S,\mathcal{O}_{S})=0$. In particular, the minimal resolution of $S$ is either a K3 surface or an Enriques surface. Furthermore, $\chi(\mathcal{O}_{X})=\chi(\mathcal{O}_{S})=2$ or $1$ respectively. 3. (3)

In Case (a), $c_{1}(X)\cdot c_{2}(X)=0$. If the minimal resolution of $S$ is a K3 surface, then $G$ is a cyclic group $C_{n}$ for some $n\in\{2,3,4,6\}$; if the minimal resolution of $S$ is an Enriques surface, then $G\in\{C_{2}\times C_{2},D_{8}\}$. 4. (4)

In Case (b), $G$ acts diagonally on $\mathbb{P}^{1}\times S$. If the minimal resolution of $S$ is a K3 surface, then $G$ is classified in Table LABEL:tab4, and in particular $c_{1}(X)\cdot c_{2}(X)\geq\frac{4}{5}$. If the minimal resolution of $S$ is an Enriques surface, then there exists a normal subgroup $G_{s}$ of $G$ of index $2$ which is classified in Table LABEL:tab5, and in particular $c_{1}(X)\cdot c_{2}(X)\geq 2$.

Proof.

(1) By Theorem 2.2, there exists a finite cover $\pi\colon X^{\prime}\to X$ étale in codimension $2$ such that $X^{\prime}$ admits a locally constant fibration $f\colon X^{\prime}\to Y$, where $Y$ is a terminal projective variety with $K_{Y}\sim_{\mathbb{Q}}0$ and any fiber $F$ of $f$ is rationally connected. By Lemma 6.1, $\dim Y=2$. In this case, $Y$ is smooth and so is $X^{\prime}$.

By Theorem 5.1, after replacing $Y$ by an étale cover if necessary, we may assume that either $X^{\prime}$ is a $\mathbb{P}^{1}$-bundle over an abelian surface or $X^{\prime}=\mathbb{P}^{1}\times Y$ where $Y$ is a K3 surface. Note that $X^{\prime}\to X$ might not be Galois as we required.

Let $\gamma\colon W\to X^{\prime}$ be the Galois closure of $\pi$ (see [29]*Definition 2.11 and [13]*Theorem 3.7). Since $\pi$ is étale in codimension $2$, $\gamma$ is also étale in codimension $2$, and hence étale as $X^{\prime}$ is smooth. In particular, $W$ is also smooth.

If $X^{\prime}$ is a $\mathbb{P}^{1}$-bundle over an abelian surface, then $-K_{W}=\gamma^{*}(-K_{X^{\prime}})$ is nef and we have $h^{1}(W,\mathcal{O}_{W})\geq h^{1}(X^{\prime},\mathcal{O}_{X^{\prime}})=2$. So we may apply Theorem 5.1 to $W$ to conclude that $W$ is again a $\mathbb{P}^{1}$-bundle over an abelian surface. We may replace $\pi\colon X^{\prime}\to X$ by $\pi\circ\gamma\colon W\to X$ to get the desired Galois finite cover.

If $X^{\prime}=\mathbb{P}^{1}\times Y$ where $Y$ is a K3 surface, then $W=X^{\prime}$ as $X^{\prime}$ is simply connected. So $\pi$ is the desired Galois finite cover.

(2) Let $g\in G$ be an automorphism of $X^{\prime}$. Note that $f$ is exactly the MRC fibration of $X^{\prime}$. By the (birational) functoriality of MRC fibrations (see [23]*5.5 Theorem), $g$ induces a birational map $g_{Y}\colon Y\dasharrow Y$. Since $Y$ is a minimal surface, $g_{Y}$ is an automorphism. So $G$ naturally acts on $Y$. Denote $S\coloneqq Y/G$ and $\mu\colon Y\to S$ the quotient map. There is a commutative diagram

[TABLE]

As $\pi$ is étale in codimension $2$, for any general fiber $F\simeq\mathbb{P}^{1}$ of $f$, $g|_{F}\colon F\to g(F)\simeq\mathbb{P}^{1}$ is étale, hence is an identity by the simple connectedness of $\mathbb{P}^{1}$. Therefore, $\deg\mu=\deg\pi=|G|$ and the action of $G$ on $Y$ is faithful.

By construction, $-K_{X}$ is ample over $S$ as $-K_{X^{\prime}}$ is ample over $Y$, so $h\colon X\to S$ is a $\mathbb{Q}$-conic bundle in the sense of [31](1.1) Definition, hence $S$ has only Du Val singularities of type $A$ by [31](1.2.7) Theorem. Moreover, $R^{i}h_{*}\mathcal{O}_{X}=0$ for $i>0$ by [31]*(2.3) Theorem and hence $\chi(\mathcal{O}_{X})=\chi(\mathcal{O}_{S})$.

By Lemma 2.5, $h^{1}(X,\mathcal{O}_{X})=0$, which implies that $h^{1}(S,\mathcal{O}_{S})=0$. On the other hand, $0\sim K_{Y}=\mu^{*}(K_{S}+R)$ where $R$ is the ramification divisor. If $K_{S}$ is not pseudo-effective, then $S$ is birational to a ruled surface over $\mathbb{P}^{1}$ (as $h^{1}(S,\mathcal{O}_{S})=0$) by the minimal model program, which implies that $S$ is rational. But this implies that $X$ is rationally connected by [12]*Corollary 1.3, a contradiction. So $K_{S}$ is pseudo-effective, which implies that $K_{S}\equiv 0$ and $R=0$. The minimal resolution $S^{\prime}$ of $S$ is a smooth projective surface with $K_{S^{\prime}}\equiv 0$ and $h^{1}(S^{\prime},\mathcal{O}_{S^{\prime}})=0$, so it is either a K3 surface or an Enriques surface. Moreover, we can also conclude that $\mu$ is étale in codimension $1$ by $R=0$.

(3) Suppose that $X^{\prime}$ is a $\mathbb{P}^{1}$-bundle over $Y$ where $Y$ is an abelian surface. Then $c_{1}(X)\cdot c_{2}(X)=0$ by Theorem 5.1 and Lemma 2.6.

We recall the well-known results of quotients of abelian surfaces from [10, 41, 45] and the references therein. There exists an exact sequence of groups

[TABLE]

where $G_{0}$ is the maximal translation subgroup. Then there is a commutative diagram

[TABLE]

Here $Y/G_{0}$ is an abelian surface. Since $X^{\prime}\to X$ is étale in codimension $2$, $X^{\prime}\to X^{\prime}/G_{0}$ is also étale in codimension $2$. This implies that $-K_{X^{\prime}/G_{0}}$ is nef and $X^{\prime}/G_{0}$ is terminal by [24]*Proposition 5.20. On the other hand, $h^{1}(X^{\prime}/G_{0},\mathcal{O}_{X^{\prime}/G_{0}})\geq h^{1}(Y/G_{0},\mathcal{O}_{Y/G_{0}})=2$, so $X^{\prime}/G_{0}$ is smooth by Lemma 2.5. By Theorem 5.1, $X^{\prime}/G_{0}\to Y/G_{0}$ is a $\mathbb{P}^{1}$-bundle. So we may replace $X^{\prime}\to Y$ by $X^{\prime}/G_{0}\to Y/G_{0}$ and assume further that $H=G$, namely, $G$ does not contain translations.

If the minimal resolution of $S$ is a K3 surface, then $G=C_{n}$ for some $n\in\{2,3,4,6\}$ by [41]*(2,2) or [11]*Proposition 4.3, as $S$ has only Du Val singularities of type A.

If the minimal resolution of $S$ is an Enriques surface, then $G\in\{C_{2}\times C_{2},D_{8}\}$ by [45]*Theorem 2.1.

(4) Suppose that $X^{\prime}=\mathbb{P}^{1}\times Y$ where $Y$ is a K3 surface.

Since $\mathbb{P}^{1}$ is the anti-canonical model of $X^{\prime}$, the action of $G$ on $X^{\prime}$ naturally induces an action on $\mathbb{P}^{1}$. There is a commutative diagram

[TABLE]

It is clear that $G$ acts diagonally on $\mathbb{P}^{1}\times Y$ (but the action might not be faithful on $\mathbb{P}^{1}$).

Case 1**.**

Suppose that the minimal resolution of $S$ is a K3 surface.

Take a general fiber $T$ of $p$ and take $T^{\prime}\simeq Y$ to be a fiber of $p^{\prime}$ mapping to $T$, then $X^{\prime}\to X\to S$ induces finite maps $T^{\prime}\to T\to S$. By (2), $T^{\prime}\to S$ (which is just $\mu$) is étale in codimension $1$. Then $T^{\prime}\to T$ is étale as $T$ is smooth. Hence $T$ can only be a K3 surface or an Enriques surface. Since the minimal resolution of $S$ is a K3 surface, $K_{S}\sim 0$ and hence $T$ is a K3 surface and $T^{\prime}\to T$ is an isomorphism. This implies that $|G|=\deg\pi=\deg\tau$ and $G$ acts on $\mathbb{P}^{1}$ faithfully. In particular, $G$ is a finite subgroup of $\text{PGL}_{2}(\mathbb{C})$, which can only be one of the following groups:

• •

a cyclic group $C_{n}$,

• •

a dihedral group $D_{2n}$,

• •

the tetrahedral group $\mathfrak{A}_{4}$,

• •

the octahedral group $\mathfrak{S}_{4}$, or

• •

the icosahedral group $\mathfrak{A}_{5}$.

On the other hand, Xiao [42]*Theorem 3 lists all possible groups $G$ acting on a K3 surface $Y$ such that the minimal resolution of $Y/G$ is a K3 surface. So we get all possible $G$ with corresponding $\text{\rm Sing}(S)$ as listed in Table LABEL:tab4. Then $c_{1}(X)\cdot c_{2}(X)=\frac{c_{1}(X^{\prime})\cdot c_{2}(X^{\prime})}{|G|}=\frac{48}{|G|}$ by Lemma 2.6. The computation of $\mathcal{R}_{X}$ will be explained in Example 6.4.

Case 2**.**

Suppose that the minimal resolution of $S$ is an Enriques surface.

Then there exists an exact sequence of groups induced by the action on $\mathbb{C}\simeq H^{0}(K_{Y})$:

[TABLE]

where $G_{s}$ consists of symplectic automorphisms and $N$ is a cyclic group. Then there is a commutative diagram

[TABLE]

As $K_{Y/G_{s}}\sim 0$ and $2K_{S}\sim 0$, we have that $N=C_{2}$ and $Y/G_{s}\to S$ is an étale double cover induced by the $2$-torsion Cartier divisor $K_{S}$. Note that the minimal resolution of $Y/G_{s}$ is a K3 surface, hence $X^{\prime}/G_{s}$ satisfies Case (b) of the theorem. Therefore, $G_{s}$ and $\text{\rm Sing}(Y/G_{s})$ are listed in Table LABEL:tab4. As $Y/G_{s}\to S$ is an étale double cover, we may rule out the cases in which singularities of $Y/G_{s}$ appear not in couples, and $\text{\rm Sing}(S)$ (resp. $\mathcal{R}_{X}$) is the half of $\text{\rm Sing}(Y/G_{s})$ (resp. $\mathcal{R}_{X^{\prime}/G_{s}}$).∎

Remark 6.3**.**

Conversely, if $X$ admits a non-trivial finite cover $\pi\colon X^{\prime}\to X$ étale in codimension 2 where $X^{\prime}$ is either a $\mathbb{P}^{1}$-bundle over an abelian surface $Y$ or a product of $\mathbb{P}^{1}$ with a K3 surface $Y$, then similar arguments as in Theorem 6.2 show that $X$ is not rationally connected. Indeed, as in the proof of Theorem 6.2(1)(2), after replacing $\pi$ by its Galois closure, we can assume that $X=X^{\prime}/G$ for some Galois group $G$; by the functoriality of MRC fibrations, there exists an induced fibration $X\to S:=Y/G$, where $Y$ is the abelian surface or the K3 surface. If $X$ is rationally connected, then $S$ is also rationally connected. It follows that the ramified locus $R$ of $\mu\colon Y\to S$ on $Y$ has dimension $1$. However, as $\pi$ is étale in codimension 2 and $\mathbb{P}^{1}$ is simple connected, the induced action of $G$ is faithful on generic points of irreducible components of $R$ as the first paragraph of the proof of Theorem 6.2(2), which contradicts to the fact that $R$ is the ramified locus.

In the rest of this section, we exhibit concrete examples satisfying Theorem 6.2 with $\chi(\mathcal{O}_{X})=2$.

Example 6.4**.**

Let $G\subset\text{PGL}_{2}(\mathbb{C})$ be a finite group. Let $Y$ be an abelian surface or a K3 surface with a faithful $G$ action, such that $S\coloneqq Y/G$ has only Du Val singularities of type A and the minimal resolution of $S$ is a K3 surface. Then consider $X\coloneqq(\mathbb{P}^{1}\times Y)/G$, where $G$ acts diagonally on $\mathbb{P}^{1}\times Y$.

We will check that $X$ is a terminal projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$ satisfying Theorem 6.2 with $X^{\prime}=\mathbb{P}^{1}\times Y$ and $\chi(\mathcal{O}_{X})=2$.

First we check that $X$ has terminal singularities. It is clear that over the smooth locus of $S$, the quotient maps are étale and hence $X$ is smooth. On the other hand, let $P\in S$ be a singular point of type $A_{n}$. By Lemma 6.5, there exists an analytic open neighborhood $U\subset Y$ over a neighborhood of $P$ such that locally $X=(\mathbb{P}^{1}\times U)/C_{n+1}$, which implies that $X$ has $2$ terminal singularities of type $\frac{1}{n+1}(\pm 1,1,-1)$ locally over $P\in S$. In particular, each singular point of type $A_{n}$ on $S$ contributes two cyclic quotient singularities of index $n+1$ on $X$, so we can gather all those singularities to form $\mathcal{R}_{X}$. This local computation also shows that $\mathbb{P}^{1}\times Y\to X$ is étale in codimension $2$ and hence $-K_{X}$ is nef. Finally, $\chi(\mathcal{O}_{X})=\chi(\mathcal{O}_{S})=2$ by Theorem 6.2(2).

This example shows that all groups $G$ in Theorem 6.2(3)(4) can be realized when the minimal resolution of $S$ is a K3 surface. For the existence of surfaces $Y$ with certain group actions we refer to [10], [41]*(2.2), [11]*Proposition 4.3, [34]*Theorem 4.5, [42]Theorem 3, [33](0.4) Example. In particular, when $Y$ is a K3 surface and $G=\mathfrak{A}_{5}$, we get an example of a terminal projective $3$-fold $X$ with $c_{1}(X)\cdot c_{2}(X)=\frac{4}{5}$ attaining the minimal value in Theorem 6.2.

Lemma 6.5**.**

Let $Y$ be a smooth surface with a finite group $G$ acting freely in codimension $1$. Suppose that $P\in Y/G$ is a Du Val singularity of type $A_{n}$ and $Q\in Y$ is a point in its preimage. Then the stabilizer of $Q$ is a subgroup of $G$ isomorphic to $C_{n+1}$ and there exists an analytic open neighborhood $U$ of $Q$ such that $Y/G\simeq U/C_{n+1}$ in a neighborhood of $P$.

Proof.

As $P$ is of type $A_{n}$, the stabilizer of $Q$ is isomorphic to $C_{n+1}$ (see for example [42]*Page 74). Other parts are clear. ∎

7. The case when $-K$ is big

In this section, we consider the case when $X$ is a weak Fano $3$-fold.

Example 7.1**.**

There are many known examples of Fano $3$-folds, for example, Iano-Fletcher’s list [14]§16.6, §16.7 of $\mathbb{Q}$-factorial terminal Fano $3$-folds of Picard number $1$ and Kasprzyk’s classification [18] of terminal toric Fano $3$-folds (which can also be found in http://www.grdb.co.uk/search/toricf3t). The baskets of all these examples are known. By (3.2), we can compute that for all these examples, $c_{1}c_{2}>2$, where the minimal one is obtained by [14]§16.7 No.85 with $c_{1}c_{2}=\frac{431}{180}$.

Theorem 7.2**.**

Let $X$ be a $\mathbb{Q}$-factorial klt weak Fano variety of dimension $n$. Assume that $X$ is smooth in codimension $2$. Then $c_{1}(X)^{n-2}\cdot c_{2}(X)>0$.

Proof.

Let $\mathcal{T}_{X}$ be the tangent sheaf of $X$ and $H$ be an ample Cartier divisor on $X$. Set $H_{\varepsilon}:=-K_{X}+\varepsilon H$ for any $\varepsilon>0$. By [28]*Proposition 6.4 (see also [6]*Proposition 2.3), if $\varepsilon>0$ is small enough, then the $(H_{\varepsilon}^{n-1})$-Harder–Narasimhan filtration

[TABLE]

is independent of $\varepsilon$. Set $\mathcal{G}_{i}:=\mathcal{E}_{i}/\mathcal{E}_{i-1}$ and $r_{i}:=\operatorname{rank}\mathcal{G}_{i}$. By [37]*Theorem 1.3, $\mathcal{T}_{X}$ is $H_{\varepsilon}^{n-1}$-generically nef, in particular, $\mu_{H_{\varepsilon}}(\mathcal{G}_{1})>\cdots>\mu_{H_{\varepsilon}}(\mathcal{G}_{l})\geq 0$. Since the sheaf $\mathcal{G}_{i}$ is an $H_{\varepsilon}$-semistable sheaf, the Bogomolov–Gieseker inequality yields

[TABLE]

Set $a_{i}:=c_{1}(\mathcal{G}_{i})c_{1}(\mathcal{T}_{X})^{n-1}$. Then, we obtain that

[TABLE]

From $\mu_{H_{\varepsilon}}(\mathcal{G}_{1})>\cdots>\mu_{H_{\varepsilon}}(\mathcal{G}_{l})\geq 0$, we obtain $a_{1}/r_{1}\geq a_{2}/r_{2}\geq\cdots\geq a_{l}/r_{l}\geq 0$ for sufficiently small $\varepsilon>0$.

Claim 1**.**

The following estimate holds:

[TABLE]

In particular, if $a_{2}>0$ or $r_{1}\neq 1$, then $c_{2}(\mathcal{T}_{X})c_{1}(\mathcal{T}_{X})^{n-2}>0$.

Proof of Claim 1.

This argument is the same as that in [15]*Appendix. By the Hodge index theorem, we obtain

[TABLE]

We also obtain the following equalities

[TABLE]

From (7.1)-(7.3), we obtain

[TABLE]

If $a_{2}>0$ or $r_{1}\neq 1$, then by taking $\varepsilon\rightarrow 0$, we obtain

[TABLE]

Therefore, it suffices to show that $a_{2}>0$ or $r_{1}\neq 1$ holds. For a contradiction, we assume that $a_{2}=0$ and $r_{1}=1$. Note that $l>1$. Set $\mathcal{F}:=\mathcal{E}_{1}\subset\mathcal{T}_{X}$. From $a_{2}=0$,

[TABLE]

Since then $c_{1}(\mathcal{F})H_{\varepsilon}^{n-1}>0$, taking a general curve $C$ representing $(mH_{\varepsilon})^{n-1}$ for $m\gg 1$, we have that $\mathcal{F}|_{C}$ is ample, hence $\mathcal{F}$ is an algebraically integrable foliation with rationally connected leaves by [21]*Theorem 1. Let $f\colon X\dashrightarrow Y$ be a dominant rational map induced by $\mathcal{F}$. Notice that the Zariski closure of a general fiber of $f$ is a rational curve since $\operatorname{rank}\mathcal{F}=1$ and thus $\dim Y=\dim X-1\geq 1$. Take a resolution $\pi\colon\tilde{X}\to X$ of the indeterminacy locus of $f$. Here we have the following commutative diagram:

[TABLE]

We may assume that both $\tilde{X}$ and $Y$ are smooth. Since $X$ is klt, there exist $\pi$-exceptional $\mathbb{Q}$-effective divisors $D$ and $G$ without common component on $\tilde{X}$ such that

[TABLE]

and $(\tilde{X},D)$ is a klt pair.

Claim 2**.**

Let $A_{Y}$ be an ample Cartier divisor on $Y$ and $\tilde{\mathcal{F}}$ be the foliation induced by $\tilde{f}$. Then, there exists $\delta>0$ such that

[TABLE]

is pseudo-effective.

Proof of Claim 2.

The proof is almost the same as that in [37]*Theorem 1.10. Since $-\pi^{*}K_{X}$ is nef and big, there exists an ample $\mathbb{Q}$-divisor $A_{\tilde{X}}$ and an effective $\mathbb{Q}$-divisor $D^{\prime}$ on $\tilde{X}$ such that

[TABLE]

and $(\tilde{X},D+D^{\prime})$ is a klt pair. By taking $\delta>0$ small enough, $A_{\tilde{X}}-\delta\tilde{f}^{*}A_{Y}$ is ample, so we may assume that $A_{\tilde{X}}-\delta\tilde{f}^{*}A_{Y}$ is effective and sufficiently general. Set $\Delta:=D+D^{\prime}+(A_{\tilde{X}}-\delta\tilde{f}^{*}A_{Y})$. Hence, $(\tilde{X},\Delta)$ is klt and

[TABLE]

For any general fiber $\tilde{F}$ of $\tilde{f}$,

[TABLE]

In particular, $\kappa(\tilde{F},(K_{\tilde{X}}+\Delta)|_{\tilde{F}})\geq 0$. Hence, by [9]*Proposition 4.1, $K_{\tilde{\mathcal{F}}}+\Delta$ is pseudo-effective. ∎

Since both $K_{\tilde{\mathcal{F}}}-\pi^{*}K_{\mathcal{F}}$ and $D$ are $\pi$-exceptional, by Claim 2, we obtain

[TABLE]

From (7.4), we have $0\geq\tilde{f}^{*}A_{Y}\cdot\pi^{*}(-K_{X})^{n-1}$. On the other hand, since $-\pi^{*}K_{X}$ is nef and big, $-\pi^{*}K_{X}\sim_{\mathbb{Q}}A+E$ holds for some ample $\mathbb{Q}$-divisor $A$ and some effective $\mathbb{Q}$-divisor $E$, hence we have

[TABLE]

This is a contradiction. ∎

We can remove the assumption on $\mathbb{Q}$-factoriality for terminal weak Fano varieties.

Corollary 7.3**.**

Let $X$ be a terminal weak Fano variety of dimension $n$. Then $c_{1}(X)^{n-2}\cdot c_{2}(X)>0$.

Proof.

Let $\rho\colon Z\to X$ be a $\mathbb{Q}$-factorialization of $X$ (see [3]*Corollary 1.4.3 for the existence of a $\mathbb{Q}$-factorialization). Note that $\rho$ is small. Hence $Z$ is again terminal, and smooth in codimension $2$.

By definition, $c_{1}(Z)=c_{1}(\rho^{-1}(X_{0}))=\rho^{*}c_{1}(X_{0})=\rho^{*}c_{1}(X)$ since $\rho$ is small, and $c_{2}(Z)=c_{2}(\rho^{-1}(X_{0}))+B$ where $B$ is an element of the second Chow group of $Z$, supporting on the exceptional locus of $\rho$. It follows that $\rho_{*}c_{2}(Z)=c_{2}(X_{0})=c_{2}(X)$ and $c_{1}(Z)^{n-2}\cdot c_{2}(Z)=\rho^{*}(c_{1}(X)^{n-2})\cdot c_{2}(Z)=c_{1}(X)^{n-2}\cdot\rho_{*}c_{2}(Z)=c_{1}(X)^{n-2}\cdot c_{2}(X)$ by the projection formula. Therefore, the conclusion follows by applying Theorem 7.2 to $Z$. ∎

Combining with BAB theorem proved by Birkar, we can obtain an inequality similar to [20]*Proposition 1:

Corollary 7.4**.**

There exists a positive rational number $b$ depending only on $n$ such that the inequality

[TABLE]

holds for any terminal weak Fano variety $X$ of dimension $n$.

Proof.

Fix a positive integer $n$. By [2], the set of terminal weak Fano varieties $X$ of dimension $n$ is bounded. On the other hand, Corollary 7.3 implies that $c_{1}(X)^{n-2}\cdot c_{2}(X)>0$. Hence the function $\frac{c_{1}(X)^{n}}{c_{1}(X)^{n-2}\cdot c_{2}(X)}$ is well-defined and locally constant in flat families. So it has a uniform upper bound for any terminal weak Fano variety $X$ of dimension $n$. ∎

We would like to obtain an effective bound $b$. We obtain a partial result by the same argument of Theorem 7.2.

Corollary 7.5**.**

Let $X$ be a $\mathbb{Q}$-factorial klt weak Fano variety of dimension $n$. Assume that $X$ is smooth in codimension $2$. Then, one of the following holds:

1. (1)

$c_{1}(X)^{n}\leq 4c_{1}(X)^{n-2}\cdot c_{2}(X)$. 2. (2)

There exists a dominant rational map $f\colon X\dashrightarrow Y$ such that the Zariski closure of a general fiber of $f$ is a rational curve.

Proof.

We use the same notation as in the proof of Theorem 7.2. If $r_{1}=1$, then by [21]*Theorem 1, we obtain (2). So we assume that $r_{1}\geq 2$. By the same calculations, we have

[TABLE]

Therefore, if $r_{1}>2$ or $a_{2}>0$, then by taking $\varepsilon\rightarrow 0$, we obtain

[TABLE]

If $r_{1}=2$ and $a_{2}=0$, then by the same argument after Claim 1 of Theorem 7.2, we have $n=r_{1}=2$.

Indeed, if $n>2$, then from the discussion after Claim 1 in Theorem 7.2, there exists a dominant rational map $f\colon X\dashrightarrow Y$ such that the Zariski closure of the general fiber is a rationally connected surface. From $n>2$, we have $\dim Y\geq 1$, which is a contradiction by using the same argument after Claim 2. Hence, we can conclude that $n=r_{1}=2$.

In this case, $X$ is a surface and $\mathcal{T}_{X}$ is $H_{\varepsilon}$-semistable, thus by the Bogomolov–Gieseker inequality, we obtain $4c_{2}(\mathcal{T}_{X})\geq c_{1}(\mathcal{T}_{X})^{2}$. ∎

For terminal weak Fano $3$-folds, the upper bound $b$ can be more explicit:

Corollary 7.6**.**

There exists a positive integer $b=2^{4}\cdot 3^{6}\cdot 7$ such that an inequality

[TABLE]

holds for any terminal weak Fano $3$-fold $X$.

Proof.

It follows directly by the facts that $c_{1}(X)^{3}=(-K_{X})^{3}\leq 324$ by [17]*Theorem 1.1 and $c_{1}(X)\cdot c_{2}(X)\geq\frac{1}{252}$ by Theorem 1.2 and Corollary 7.3. ∎

Remark 7.7**.**

For Gorenstein terminal weak Fano $3$-folds, there exists a sharp bound $b=3$. In fact, $c_{1}(X)\cdot c_{2}(X)=24$ by Theorem 3.1 and by the fact that $\chi(\mathcal{O}_{X})=1$. On the other hand, $c_{1}(X)^{3}\leq 72$ by Prokhorov [38].

8. Proof of main theorems

Proof of Theorem 1.2.

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $c_{1}(X)\cdot c_{2}(X)\neq 0$.

If $X$ is rationally connected, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{1}{252}$ by Theorem 4.1; moreover, if $-K_{X}$ is not big, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{2}{5}$.

If $X$ is smooth, then $c_{1}(X)\cdot c_{2}(X)\geq 24$ by Theorem 5.1.

If $X$ is singular and not rationally connected, then $c_{1}(X)\cdot c_{2}(X)\geq\frac{4}{5}$ by Theorem 6.2.

These three cases conclude Theorem 1.2. ∎

Proof of Theorem 1.3.

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $K_{X}\not\equiv 0$.

If $X$ is rationally connected, then we get (2).

If $X$ is smooth and not rationally connected, then $X$ is classified in Theorem 5.1, and we get (1).

If $X$ is singular and not rationally connected, then $X$ is classified in Theorem 6.2, and we get (3). ∎

Proof of Corollary 1.4.

Let $X$ be a terminal projective $3$-fold such that $-K_{X}$ is nef and $c_{1}(X)\cdot c_{2}(X)=0$.

If $X$ is not rationally connected, then we get (1)(2) from the structure of $X$ in Theorem 5.1 and Theorem 6.2.

If $X$ is rationally connected, then $-K_{X}$ is not big by Corollary 7.3, hence $K_{X}^{3}=0$. By Theorem 4.1, $\mathcal{R}_{X}$ is in Table LABEL:tab1. On the other hand, as $-K_{X}$ is not big, we can rule out (5), (6) and (10) in Table LABEL:tab1 by Table LABEL:tab2. ∎