Kawamata–Miyaoka type inequality for $\mathbb{Q}$-Fano varieties with canonical singularities

We thank the referees for their detailed reports, which help us to significantly improve the exposition of this paper. We would like to thank Wenhao Ou and Qizheng Yin for very helpful discussions and Yuri Prokhorov for useful communications. J. Liu is supported by the National Key Research and Development Program of China (No. 2021YFA1002300), the NSFC grants (No. 12001521 and No. 12288201), the CAS Project for Young Scientists in Basic Research (No. YSBR-033) and the Youth Innovation Promotion Association CAS.

Let $X$ be an $n$-dimensional normal projective variety and let $\scr{F}$ be a torsion free coherent sheaf with rank $r\geq 1$ on $X$. The reflexive hull of $\scr{F}$ is defined to be its double dual $\scr{F}^{**}$. We say that $\scr{F}$ is reflexive if $\scr{F}=\scr{F}^{**}$. For any positive integer $m$, we denote by $\wedge^{[m]}\scr{F}$ the reflexive hull of $\wedge^{m}\scr{F}$ and the determinant $\det(\scr{F})$ of $\scr{F}$ is defined as $\wedge^{[r]}\scr{F}$. Let $f\colon Y\rightarrow X$ be a surjective morphism. We denote by $f^{[*]}\scr{F}$ the reflexive pull-back of $\scr{F}$; in other words, $f^{[*]}\scr{F}$ is the reflexive hull of $f^{*}\scr{F}$. If $X$ is smooth in codimension $k$, then the Chern classes $c_{i}(\scr{F})$ are defined as the elements $j_{*}c_{i}(\scr{F}|_{X_{\operatorname{reg}}})$ sitting in the Chow groups $A^{i}(X)$ for $i\leq k$, where $X_{\operatorname{reg}}$ is the smooth locus of $X$ and $j\colon A^{i}(X_{\operatorname{reg}})\hookrightarrow A^{i}(X)$ is the natural inclusion by taking closure [BS58, § 6]. In particular, if $X$ is smooth in codimension two, then $c_{1}(\scr{F})^{2}$ and $c_{2}(\scr{F})$ are well-defined in $A^{2}(X)$ and the discriminant $\Delta(\scr{F})$ of $\scr{F}$ is defined as

Let $D_{1},\dots,D_{n-1}$ be a collection of nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors on $X$. Denote by $\alpha$ the one cycle $D_{1}\cdots D_{n-1}$. The slope $\mu_{\alpha}(\scr{F})$ of $\scr{F}$ (with respect to $\alpha$) is defined as

We note that $c_{1}(\scr{F})\cdot\alpha$ is well-defined because the $D_{i}$’s are $\mathbb{Q}$-Cartier. One can then define (semi-)stability of $\scr{F}$ (with respect to $\alpha$) as usual. Moreover, there exists the so-called Harder–Narasimhan filtration of $\scr{F}$, i.e., a filtration as follows

such that the graded pieces, $\scr{G}_{i}\coloneqq\scr{F}_{i}/\scr{F}_{i-1}$, are semi-stable torsion free sheaves with strictly decreasing slopes [Miy87, Theorem 2.1]. We call $\scr{F}_{1}$ the maximal destabilising subsheaf of $\scr{F}$. We can also define

The following easy result is well-known to experts and we include a complete proof for the lack of explicit references.

Let $X$ be a normal projective variety such that its canonical divisor $K_{X}$ is $\mathbb{Q}$-Cartier. Then the Gorenstein index $r_{X}$ of $X$ is defined as the smallest positive integer $m$ such that $mK_{X}$ is Cartier. For a $\mathbb{Q}$-Fano variety $X$, we can define the Fano index of $X$ in two different ways:

Let $X$ be a $\mathbb{Q}$-Fano variety with klt singularities. Then the Picard group $\operatorname{Pic}(X)$ is a finitely generated torsion free $\mathbb{Z}$-module and the numerical equivalence coincides with the $\mathbb{Q}$-linear equivalence [IP99, Proposition 2.1.2]. In particular, we have

where “$\sim_{\mathbb{Q}}$” means $\mathbb{Q}$-linear equivalence and “$\equiv$” means numerical equivalence. This implies directly that the Fano indices $\iota_{X}$ and $\hat{\iota}_{X}$ are well-defined positive integers such that $\iota_{X}|\hat{\iota}_{X}$.

The tangent sheaf $\scr{T}_{X}$ of a normal projective variety $X$ is defined to be the dual sheaf $(\Omega^{1}_{X})^{*}$ and the Chern classes $c_{i}(X)$ are defined to be the Chern classes of $\scr{T}_{X}$ whenever they are well-defined in the sense of the definition in § 2.1.

A foliation $\scr{F}$ on a normal projective variety $X$ is called algebraically integrable if the leaf of $\scr{F}$ through a general point of $X$ is an algebraic variety. For an algebraically integrable foliation $\scr{F}$ with rank $r>0$ such that $K_{\scr{F}}$ is $\mathbb{Q}$-Cartier, there exists a unique proper subvariety $T^{\prime}$ of the Chow variety of $X$ whose general point parametrises the closure of a general leaf of $\scr{F}$ (viewed as a reduced and irreducible cycle in $X$). Let $T$ be the normalisation of $T^{\prime}$ and let $U\rightarrow T^{\prime}\times X$ be the normalisation of the universal family, with induced morphisms:

Then $\nu\colon U\rightarrow X$ is birational and, for a general point $t\in T$, the image $\nu(\pi^{-1}(t))\subsetneq X$ is the closure of a leaf of $\scr{F}$. We shall call the diagram (2.2.1) the family of leaves of $\scr{F}$ [AD14, Lemma 3.9]. Thanks to [AD14, Lemma 3.7 and Remark 3.12], there exists a canonically defined effective $\mathbb{Q}$-divisor $\Delta$ on $U$ such that

where $\scr{F}_{U}$ is the algebraically integrable foliation on $U$ induced by $\pi$. In particular, the divisor $\Delta$ is $\nu$-exceptional as $\nu_{*}K_{\scr{F}_{U}}=K_{\scr{F}}$. For a general fibre $F$ of $\pi$, the pair $(F,\Delta_{F})$ is called the general log leaf of $\scr{F}$, where $\Delta_{F}=\Delta|_{F}$ [AD14, Definition 3.11 and Remark 3.12]. Here the restriction $\Delta|_{F}$ is well-defined because $U$ is smooth at codimension one points of $F$. The following proposition plays a key role in the proof of Proposition 3.8.

Let $X$ be a $\mathbb{Q}$-factorial threefold with terminal singularities. Then $X$ has only isolated singularities and it follows from [Mor85, Theorems 12, 23 and 25] that each singular point of $X$ can be deformed to a (unique) collection of a finite number of terminal quotient singularities $\{Q_{i}\}_{i\in I}$. Write the type of the orbifold point $Q_{i}$ as $\frac{1}{r_{i}}\left(1,-1,b_{i}\right)$ with $\gcd(b_{i},r_{i})=1$ and $0<b_{i}\leq r_{i}/2$. We can then associate to $Q_{i}$ the pair $(b_{i},r_{i})$ and the basket $B_{X}$ of $X$ is defined to be the collection of all such pairs (permitting weights) appeared in the deformations of singular points of $X$ [Rei87, § 6]. Denote by $\mathcal{R}_{X}$ the collection of $r_{i}$ (permitting weights) appearing in $B_{X}$. For simplicity, we write $\mathcal{R}_{X}$ as a set of positive integers whose weights appear in superscripts, say for example,

Note that $\operatorname{lcm}(\mathcal{R}_{X})$ coincides with the Gorenstein index $r_{X}$ of $X$ by [Suz04, Lemma 1.2]. Let $D$ be a Weil divisor on $X$. According to [Rei87, Theorem 10.2], we have

where the last sum runs over $B_{X}$. Recall that the local Weil class group of $X$ at a singular point $P$ is generated by $K_{X}$ [Kaw88, Corollary 5.2]; so $D\sim lK_{X}$ around $P$ for some $l\in\mathbb{Z}$. At an orbifold point $Q$ of type $\frac{1}{r_{Q}}(1,-1,b_{Q})$ associated to $P$, the local index of $D$ at $Q$ is defined to be the unique integer $0\leq i_{Q}<r_{Q}$ such that $lK_{X}\sim i_{Q}K_{X}$ around $Q$ by passing to a deformation if necessary and then the number $c_{Q}(D)$ is defined as

where the symbol $\overline{\bullet}$ means the smallest residue mod $r_{Q}$ and $\sum_{j=0}^{-1}\coloneqq 0$.

Let now $X$ be a $\mathbb{Q}$-Fano threefold with terminal singularities. In the case $D=K_{X}$, the formula (2.3.1) and Serre’s duality imply

In the case $D=-nK_{X}$ for some positive integer $n$, the formula (2.3.1) implies

where $l(n+1)=\sum_{Q}\sum_{j=0}^{n}\frac{\overline{jb_{Q}}\left(r_{Q}-\overline{jb_{Q}}\right)}{2r_{Q}}$ and the first sum runs over $B_{X}$. When $n=1$, together with the Kawamata–Viehweg vanishing theorem, we immediately get the following easy but useful lemma.

In the case $-K_{X}\sim\iota_{X}A$, where $A$ is an ample Weil divisor, by [Suz04, Lemma 1.2], the numbers $A^{3}$ and $r_{X}$ satisfy

If $\iota_{X}\geq 3$, combining (2.3.1) with the Kawamata–Viehweg vanishing yields

for $-\iota_{X}<t<0$. In particular, when $t=-1$, we obtain

We discuss an inequality proved by Langer in [Lan04] and its several variants in this subsection.

Recall that a Mehta–Ramanathan-general curve $C$ on an $n$-dimensional normal projective variety is the complete intersection of $n-1$ sufficiently ample divisors in general positions.

With the notions above, one can derive the following result.

We start with the following result, which is essentially proved in [Ou23]. See also [Pet12, Theorem 1.3] for the smooth case.

The following result is a slight generalisation of [IJL23, Corollary 7.5].

Given two $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor classes $\delta$ and $\delta^{\prime}$ on a projective variety $X$, we will denote by $\delta\leq\delta^{\prime}$ (resp. $\delta<\delta^{\prime}$) if $\delta^{\prime}-\delta$ is effective (resp. effective and non-zero). The following result is the key ingredient of the proof of Theorem 1.1.

The inequalities (3.8.1) and (3.8.2) in Proposition 3.8 above are both sharp as shown by the following example of the so-called normal generalised cones.