Stability conditions on product threefolds of projective spaces and Abelian varieties

TL;DR

This paper proves a BG-type inequality conjecture for certain product threefolds, establishing the existence of Bridgeland stability conditions on these complex algebraic varieties.

Contribution

It demonstrates the validity of a key inequality conjecture, enabling the construction of Bridgeland stability conditions on specific product threefolds.

Findings

BG-type inequality conjecture proven for the specified threefolds

Existence of Bridgeland stability conditions established on these threefolds

Advancement in understanding stability conditions on complex threefolds

Abstract

In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds.

Equations

$$Z(\mathcal{A}\setminus \{0\})\subset \mathcal{H}.$$

$${\varphi }_Z(F)\le (\text{mbox}resp.<){\varphi }_Z(E).$$

$$0=E_0\subset E_1\subset \cdots \subset E_{n-1}\subset E_n=E$$

$${\varphi }_Z(E_1)>\cdots >{\varphi }_Z(E_n).$$

$$Z_{\omega ,B}:=-{\int }_X{e}^{-i\omega }.{\mathrm{ch}}^B:K(X)\to \mathbb{C}$$

$${\mu }_{\omega ,B}:\mathrm{Coh}(X)\to (-\infty ,+\infty ],E\mapsto \frac{{\omega }^{n-1}.{\mathrm{ch}}_1^B(E)}{{\mathrm{ch}}_0^B(E)},$$

$${\mathcal{T}}_{\omega ,B}:=⟨T\in \mathrm{Coh}(X):T\text{mbox}is{\mu }_{\omega ,B}\text{mbox}-semistablewith{\mu }_{\omega ,B}(T)>0⟩,$$

$${\mathcal{F}}_{\omega ,B}:=⟨F\in \mathrm{Coh}(X):F\text{mbox}is{\mu }_{\omega ,B}\text{mbox}-semistablewith{\mu }_{\omega ,B}(F)\le 0⟩.$$

$$\operatorname{Coh}^{\omega,B}(X):=\left\langle\mathcal{F}_{\omega,B}[1],\mathcal{T}_{\omega,B}\right\rangle=\left\{E\in\operatorname{Coh}(X):\begin{array}[]{ll}\mathcal{H}^{-1}(E)\in\mathcal{F}_{\omega,B},\mathcal{H}^{0}(E)\in\mathcal{T}_{\omega,B},\\ \mathcal{H}^{i}(E)=0\mbox{ for }i\neq-1,0\end{array}\right\}.$$

$${\nu }_{\alpha ,\beta }={\nu }_{\omega ,B}:{\mathrm{Coh}}^{\beta }(X)\to (-\infty ,+\infty ],E\mapsto \frac{H.{\mathrm{ch}}_2^{\beta }(E)-\frac{1}{2}{\alpha }^2{H}^3.{\mathrm{ch}}_0^{\beta }(E)}{H^2.{\mathrm{ch}}_1^{\beta }(E)}.$$

$${\mathrm{ch}}_3^{\beta }(E)\le \frac{1}{6}{\alpha }^2{H}^2.{\mathrm{ch}}_1^{\beta }(E).$$

$${\mathcal{A}}_{\alpha ,\beta }:=⟨{\mathcal{F}}_{\alpha ,\beta }^{{}^{\prime }}[1],{\mathcal{T}}_{\alpha ,\beta }^{{}^{\prime }}⟩.$$

$$\bar{\beta }(E):=\{\begin{array}{ll}\frac{H^2.{\mathrm{ch}}_1(E)-\sqrt{{\overline{\Delta }}_H(E)}}{H^3.{\mathrm{ch}}_0(E)}& ({\mathrm{ch}}_0(E)\ne 0)\\ \frac{H.{\mathrm{ch}}_2(E)}{H^2.{\mathrm{ch}}_1(E)}& ({\mathrm{ch}}_0(E)=0),\end{array}$$

$${\overline{\Delta }}_H(E):=(H^2.{\mathrm{ch}}_1(E){)}^2-2(H^3.{\mathrm{ch}}_0(E))(H.{\mathrm{ch}}_2(E)).$$

$${\mathrm{ch}}_3^{\bar{\beta }(E)}(E)\le 0.$$

$${\underline{m}}_{\ast }\mathcal{O}\left(D\right)=\underset{j}{⨁}{L}_j^{\ast \oplus {\eta }_j},$$

$$L_j:=\mathcal{O}\left(\frac{1}{m}\left(-D+\sum _{\rho }{a}_{\rho }{D}_{\rho }\right)\right).$$

$$-K_Y-\frac{{\sum }_{\rho }{a}_{\rho }{D}_{\rho }}{m}=\sum _{\rho }{D}_{\rho }-\frac{{\sum }_{\rho }{a}_{\rho }{D}_{\rho }}{m}\ge \frac{1}{m}\sum _{\rho }{D}_{\rho }$$

$$\mathrm{hom}\left(\mathcal{O}(H^{\prime }),f^{(m^2,m)\ast }E\right)=0,$$

$$\mathrm{hom}\left(\mathcal{O},f^{(m^2,m)\ast }E\right)=O(m^4).$$

$${\mathrm{ext}}^2\left(\mathcal{O}(-H^{\prime }),f^{(m^2,m)\ast }E\right)=0,$$

$${\mathrm{ext}}^2\left(\mathcal{O},f^{(m^2,m)\ast }E\right)=O(m^4).$$

$$\underset{i=0}{\overset{3}{⨁}}{H}^{2i}(X)\ni (x,y,z,w)\mapsto (x,m^{2}y,m^{4}z,m^{6}w)\in \underset{i=0}{\overset{3}{⨁}}{H}^{2i}(X).$$

$$\mathrm{Hom}\left({\omega }_X^{\ast }\otimes f_{\ast }^{(a,b)}\left(E\otimes {\omega }_X\right),F\right)\cong \mathrm{Hom}\left(E,f^{(a,b)\ast }F\right).$$

$$\mathrm{hom}\left(L,E\right)=0.$$

$$\mathrm{hom}\left(E,L[1]\right)=0.$$

$$H^2.{\mathrm{ch}}_1^{\bar{\beta }}(L)>0,H.{\mathrm{ch}}_1^{\bar{\beta }}(L{)}^2>0.$$

$$\begin{array}{rl}\underset{\alpha \to +0}{\lim }{\nu }_{\alpha ,\bar{\beta }}(L)& =\frac{H.{\mathrm{ch}}_2^{\bar{\beta }}(L)}{H^2.{\mathrm{ch}}_1^{\bar{\beta }}(L)}\\ & =\frac{\frac{1}{2}H.{\mathrm{ch}}_1^{\bar{\beta }}(L{)}^2}{H^2.{\mathrm{ch}}_1^{\bar{\beta }}(L)}\\ & >0=\underset{\alpha \to 0}{\lim }{\nu }_{\alpha ,\bar{\beta }}(E).\end{array}$$

$$\mathrm{hom}\left(L,E\right)=0.$$

$$\chi \left(\mathcal{O},f^{(m^2,m)\ast }E\right)=m^6{\mathrm{ch}}_3(E)+O(m^4).$$

$$\chi \left(\mathcal{O},f^{(m^2,m)\ast }E\right)\le \mathrm{hom}\left(\mathcal{O},f^{(m^2,m)\ast }E\right)+{\mathrm{ext}}^2\left(\mathcal{O},f^{(m^2,m)\ast }E\right)$$

Full text

1. 1 Introduction
2. 2 Preliminaries
3. 3 Preparation for the Main Theorem
4. 4 Proof of the Main Theorem
5. A Counter-example for Conjecture 2.4

1. Introduction

1.1. Motivation and results

The notion of stability conditions on a triangulated category was introduced by Bridgeland in his paper [10]. Bridgeland stability condition is a mathematical subject realizing Douglas’ $\Pi$-stability in string theory [12], [13], [14]. It gives us new points of view in various scenes, such as birational geometry, counting invariants, Mirror symmetry, and so on (cf. [2], [3], [4], [5], [6], [8], [25], [26], [27], [28]).

Constructing stability conditions on the derived category of coherent sheaves of a given smooth projective variety $X$ is a starting problem for such applications. When $\dim X\leq 2$, the standard construction of stability conditions on $D^{b}(X)$ was given in [11] and [1]. In the case when $\dim X=3$, the construction problem of stability conditions on $D^{b}(X)$ is still open in general. In the paper [8], Bayer, Macrì and Toda proposed a conjectural approach for this problem. The problem was reduced to the conjectural Bogomolov-Gieseker (BG) type inequality for Chern characters (involving the third part of the Chern character) of certain semistable objects (called tilt-semistable objects) in the derived category. It is known that the original BG inequality conjecture holds for Abelian threefolds (cf. [17], [18], [7]), Fano threefolds of Picard rank one (cf. [8], [19], [22], [16]), some toric threefolds (cf. [9]), and their étale quotients (cf. [20]).

However, counter-examples for the original BG inequality conjecture were constructed in the case when $X$ is the blow-up of a smooth projective threefold at a point (cf. [21], [23]). Furthermore, by using the argument of [21], we can show that the BG inequality conjecture does not hold even when $X$ is a Calabi-Yau threefold containing a plane. See Appendix A of this paper. Hence we need to modify the inequality in general. In this direction, it was shown that some modified versions of the BG inequality conjecture hold for every Fano threefolds (cf. [9]). On the other hand, it seems still important to study for which variety the original BG inequality conjecture holds. In this paper, we give three new examples which satisfy the original BG inequality conjecture:

Theorem 1.1**.**

Let $X$ be $\mathbb{P}^{1}\times S$, $\mathbb{P}^{2}\times C$, or $\mathbb{P}^{1}\times\mathbb{P}^{1}\times C$, where $S$ is an Abelian surface and $C$ is an elliptic curve. Then the original BG inequality conjecture holds for $X$.

See Theorem 2.6 for the precise statement. In particular, the above theorem implies:

Theorem 1.2**.**

Let $X$ be as above. Then there exist Bridgeland stability conditions on $X$.

1.2. Strategy of the proof of the main theorem

The idea of proof is borrowed from that of [7] and [9]. Roughly speaking, they considered the Euler characteristic $\chi(\mathcal{O},\underline{m}^{*}E)$ of the pull back of a given tilt-semistable object $E$ by the multiplication map (resp. toric Frobenius morphism) $\underline{m}\colon X\to X$ on an Abelian threefold (resp. a toric threefold) $X$. Then by the Riemann-Roch theorem, we know that $\chi(\mathcal{O},\underline{m}^{*}E)$ is a polynomial of degree $6$ (resp. $3$) with respect to $m$ and its leading coefficient is $\operatorname{ch}_{3}(E)$.

On the other hand, they showed that $\operatorname{ext}^{i}(\mathcal{O},\underline{m}^{*}E)=O(m^{4})$ (resp. $O(m^{2})$) for even $i$. In this way, they got an inequality for the third part of the Chern character, i,e, $\operatorname{ch}_{3}(E)\leq 0$. To approximate $\operatorname{ext}^{i}(\mathcal{O},\underline{m}^{*}E)$, it was important that $\underline{m}$ is étale in the case when $X$ is an Abelian threefold, while the toric Frobenius splitting (Theorem 3.1) was essential when $X$ is a toric threefold.

In this paper, we consider the product of the multiplication map on an Abelian variety and the toric Frobenius morphisms on the projective spaces. Then we approximate $\operatorname{ext}^{i}(\mathcal{O},\underline{m}^{*}E)$ combining the methods in [7] and [9]. Note that our approach cannot apply to the product threefolds of an elliptic curve and other toric surfaces for a technical reason (see Remark 3.7).

1.3. Plan of the paper

The paper is organized as follows. In Section 2, we recall the notion of stability conditions. After that, we recall the work of [8] and state our main theorem. In Section 3, we collect key results which we will use in the proof of our main theorem. In Section 4, we prove our main theorem. In Appendix A, we will show that the original BG inequality conjecture for a Calabi-Yau threefold containing a plane does not hold.

Acknowledgement**.**

I would like to thank my supervisor Professor Yukinobu Toda. He suggested this problem to me and gave various comments and advices. I would also like to thank Genki Ouchi for useful discussions.

Notation and Convention**.**

In this paper we always work over $\mathbb{C}$. We use the following notations:

• •

$\operatorname{ch}^{B}=(\operatorname{ch}_{0}^{B},\cdots,\operatorname{ch}_{n}^{B}):=e^{-B}.\operatorname{ch}$, where $\operatorname{ch}$ denotes the Chern character and $B\in H^{2}(X;\mathbb{R})$.

• •

$\operatorname{ch}^{\beta}:=\operatorname{ch}^{\beta H}$, where $H$ is an ample divisor and $\beta\in\mathbb{R}$.

• •

$H.\operatorname{ch}^{B}:=(H^{n}.\operatorname{ch}^{B}_{0},\cdots,H.\operatorname{ch}^{B}_{n-1},\operatorname{ch}^{B}_{n})$.

• •

$K(\mathcal{A})$ : the Grothendieck group of an abelian category $\mathcal{A}$.

• •

$\hom(E,F):=\dim\operatorname{Hom}(E,F)$.

• •

$\operatorname{ext}^{i}(E,F):=\dim\operatorname{Ext}^{i}(E,F)$.

• •

$D^{b}(X):=D^{b}(\operatorname{Coh}(X))$ : the bounded derived category of coherent sheaves on a smooth projective variety $X$.

2. Preliminaries

2.1. Bridgeland stability condition

In this subsection, we recall the definition of stability conditions due to Bridgeland [10]. First we define the notion of stability functions:

Definition 2.1**.**

1. (1)

Let $\mathcal{A}$ be an abelian category. A stability function on $\mathcal{A}$ is a group homomorphism $Z\colon K(\mathcal{A})\to\mathbb{C}$ such that

[TABLE]

Here $\mathcal{H}:=\mathbb{H}\cup\mathbb{R}_{<0}$ is the union of upper half plane and negative real line. 2. (2)

Let $Z$ be a stability function on an abelian category $\mathcal{A}$. For $E\in\mathcal{A}$, define $\phi_{Z}(E):=\frac{1}{\pi}\arg Z(E)\in(0,1]$. Then $E$ is $Z$-semistable (resp. stable) if for every proper non-zero subobject $F\subset E$,

[TABLE] 3. (3)

$Z$ satisfies the Harder-Narashimhan property (HN property) if the following property holds: For every non-zero object $E\in\mathcal{A}$, there exists a finite filtration

[TABLE]

such that $F_{i}:=E_{i}/E_{i-1}$ is $Z$-semistable for every $i=1,\cdots,n$ and

[TABLE]

Now we can define the notion of stability conditions on a triangulated category.

Definition 2.2**.**

Let $\mathcal{D}$ be a triangulated category. A stability condition $\sigma=(Z,\mathcal{A})$ on $\mathcal{D}$ is a pair consisting of the heart of a bounded t-structure $\mathcal{A}$ on $\mathcal{D}$ and a stability function $Z$ on $\mathcal{A}$ (called central charge) satisfying the HN-property.

2.2. Stability conditions on smooth projective varieties.

In this subsection, we recall the works about the stability conditions on smooth projective varieties. Let $X$ be a smooth projective variety, $\omega$ an ample $\mathbb{R}$-divisor on $X$, and $B$ any $\mathbb{R}$-divisor on $X$. Conjecturally, a group homomorphism

[TABLE]

becomes the central charge of some stability condition on $D^{b}(X)$ (cf. [8], Conjecture 2.1.2).

When $\dim X=1$, the pair $(Z_{\omega,B},\operatorname{Coh}(X))$ is a stability condition on $X$ and this coincides with the Mumford’s slope stability.

However in $\dim X\geq 2$, we need a more complicated construction of the wanted heart as follows. Let us define the slope function on $\operatorname{Coh}(X)$ as

[TABLE]

where $n=\dim X$. Define subcategories of $\operatorname{Coh}(X)$ as follows:

[TABLE]

Here, we denote by $\langle S\rangle$ the extension closure of a set of objects $S\subset\operatorname{Coh}(X)$. Due to the HN-property of $\mu$-stability, the pair $(\mathcal{T}_{\omega,B},\mathcal{F}_{\omega,B})$ is a torsion pair on $\operatorname{Coh}(X)$ in the sence of [15]. Then we can construct a new heart, called the tilting heart of $\operatorname{Coh}(X)$ with respect to the torsion pair:

[TABLE]

In $\dim X=2$, $\operatorname{Coh}^{\omega,B}(X)$ is the required heart:

Theorem 2.3** (([11], [1])).**

Let $\dim X=2$. Then the pair $(Z_{\omega,B},\operatorname{Coh}^{\omega,B}(X))$ is a stability condition on $X$.

In $\dim X=3$, Bayer, Macrì and Toda provided the conjectural approach to construct the required heart ([8]). The idea is to tilt the heart $\operatorname{Coh}^{\omega,B}(X)$ once again by using a new slope function. Let us recall the work [8] of Bayer, Macrì and Toda. In the followings, assume that $\dim X=3$. Let $H$ be an ample divisor on $X$ and let $\omega:=\alpha\sqrt{3}H,B:=\beta H$  $(\alpha,\beta\in\mathbb{R},\alpha>0)$. Define a slope function on $\operatorname{Coh}^{\beta}(X):=\operatorname{Coh}^{\omega,B}(X)$ as follows:

[TABLE]

Then we can define the notion of $\nu_{\alpha,\beta}$-stability (or tilt-stability) as similar to the $\mu_{\omega,B}$-stability on $\operatorname{Coh}(X)$. Using the tilt-stability, the torsion pair $(\mathcal{T}^{{}^{\prime}}_{\alpha,\beta},\mathcal{F}^{{}^{\prime}}_{\alpha,\beta})$ on $\operatorname{Coh}^{\beta}(X)$ is also defined similarly to $(\mathcal{T}_{\omega,B},\mathcal{F}_{\omega,B})$ on $\operatorname{Coh}(X)$. Bayer, Macrì and Toda conjectured the following BG type inequality for $\nu_{\alpha,\beta}$-semistable objects:

Conjecture 2.4** (([8])).**

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a $\nu_{\alpha,\beta}$-semistable object with $\nu_{\alpha,\beta}(E)=0$. Then we have

[TABLE]

Moreover, they showed that the above inequality implies the existence of a stability condition with the central charge $Z_{\alpha,\beta}:=Z_{\omega,B}$. Let $\mathcal{A}_{\alpha,\beta}$ be a tilting heart of $\operatorname{Coh}^{\beta}(X)$ with respect to $\nu_{\alpha,\beta}$-stability, i,e,

[TABLE]

Theorem 2.5** (([8])).**

Assume that Conjecture 2.4 holds. Then the pair $(Z_{\alpha,\beta},\mathcal{A}_{\alpha,\beta})$ is a stability condition on $X$.

Hence the construction problem of stability conditions on $X$ is reduced to Conjecture 2.4. The main theorem of this paper is the following.

Theorem 2.6**.**

Let $X$ be $\mathbb{P}^{1}\times S,\mathbb{P}^{2}\times C$, or $\mathbb{P}^{1}\times\mathbb{P}^{1}\times C$, where $S$ is an Abelian surface and $C$ is an elliptic curve. Then for every ample divisor $H$ on $X$, $\alpha>0$, and $\beta\in\mathbb{R}$, Conjecture 2.4 holds.

Remark 2.7*.*

In [21] and [23], counter-examples for Conjecture 2.4 were obtained when $X$ is the blow-up of a smooth projective threefold at a point. Furthermore, there exists a counter-example even when $X$ is a Calabi-Yau threefold containing a plane. For the latter, see the appendix of this paper.

2.3. Reduction Theorem.

In this subsection, we recall the further reduction of Conjecture 2.4 due to [7]. First we recall the notion of $\bar{\beta}$-stability.

Definition 2.8**.**

Let $E\in\operatorname{Coh}^{\beta}(X)$ be a $\nu_{\alpha,\beta}$-semistable object.

1. (1)

We define

[TABLE]

where

[TABLE] 2. (2)

$E$ is $\bar{\beta}$-semistable (resp. stable) if there exists an open neighborhood $V$ of $(0,\bar{\beta}(E))$ in $(\alpha,\beta)$-plane such that for every $(\alpha,\beta)\in V$ with $\alpha>0$, $E$ is $\nu_{\alpha,\beta}$-semistable (resp. stable).

Remark 2.9*.*

In [8], it was shown that $\overline{\Delta}_{H}(E)$ is non-negative for every $\nu_{\alpha,\beta}$-semistable object $E$.

Then Conjecture 2.4 is reduced as follows:

Theorem 2.10** (([7], Theorem 5.4)).**

Assume that for every $\bar{\beta}$-stable object $E$ with $\operatorname{ch}_{0}(E)\geq 0$ and $\bar{\beta}(E)\in[0,1)$, we have

[TABLE]

Then Conjecture 2.4 holds for every $\alpha,\beta$.

3. Preparation for the Main Theorem

In this section, we collect key results which we will use in the proof of our main theorem. The first one is about the toric Frobenius push forward of line bundles:

Theorem 3.1** (([24])).**

Let $Y$ be a smooth projective toric variety, let $\{D_{\rho}\}_{\rho}$ be the torus invariant divisors. For $m\in\mathbb{Z}_{>0}$, denote the toric Frobenius morphism by $\underline{m}\colon Y\to Y$. Then for every divisor $D$ on $Y$, we have

[TABLE]

where

[TABLE]

Here, integers $0\leq a_{\rho}\leq m-1$ move so that $L_{j}$ becomes an integral divisor and $\eta_{j}$ counts the multiplicity of $\{a_{\rho}\}$ which defines the same $L_{j}$.

Remark 3.2*.*

Let $Y=\mathbb{P}^{n}$ be a projective space. Let $a_{\rho}$ be as in Theorem 3.1. Then we have

[TABLE]

and hence $-K_{Y}-\frac{\sum_{\rho}a_{\rho}D_{\rho}}{m}$ is ample on $Y$. This fact will be used in Section 4.

The next one is about the preservation of tilt-stability under the pull back by finite étale morphisms:

Proposition 3.3** (([7], Proposition 6.1)).**

Let $f\colon Y\to X$ be a finite étale surjective morphism between smooth projective threefolds. Let $\omega$ be an ample $\mathbb{R}$-divisor on $X$, $B$ an $\mathbb{R}$-divisor on $X$. Let $E\in D^{b}(X)$. Then

1. (1)

$\nu_{f^{*}\omega,f^{*}B}(f^{*}E)=\nu_{\omega,B}(E)$. 2. (2)

$f^{*}E\in\operatorname{Coh}^{f^{*}\omega,f^{*}B}(Y)$* if and only if $E\in\operatorname{Coh}^{\omega,B}(X)$.* 3. (3)

$f^{*}E$* is $\nu_{f^{*}\omega,f^{*}B}$-semistable (resp. stable) if and only if $E$ is $\nu_{\omega,B}$-semistable (resp. stable).*

Example 3.4*.*

Let $A$ be an Aberian variety of $\dim\leq 2$, let $X=Y\times A$ be a product threefold. Let $\underline{m}\colon A\to A$ be a multiplication map $(m\in\mathbb{Z}_{>0})$. Then $\operatorname{id}_{Y}\times\underline{m}\colon X\to X$ is a finite étale surjective morphism. Hence we can apply the above proposition to $\operatorname{id}_{Y}\times\underline{m}$.

The third one is about the tilt-stability of line bundles:

Lemma 3.5** (([7], Corollary 3.11)).**

Let $X$ be a smooth projective threefold, $H$ an ample divisor on $X$. Assume that for every effective divisor $D$ on $X$, we have $H.D^{2}\geq 0$. Then for every line bundle $L$ on $X$, $\alpha>0$, and $\beta\in\mathbb{R}$, $L$ or $L[1]$ is $\nu_{\alpha,\beta}$-stable.

Example 3.6*.*

1. (1)

Let $C$ be an elliptic curve, $S$ an Abelian surface. Let $X$ be $\mathbb{P}^{1}\times S,\mathbb{P}^{2}\times C$, or $\left(\mathbb{P}^{1}\times\mathbb{P}^{1}\right)\times C$. Then the assumption of the above lemma holds for every ample divisor on $X$, since there are no negative divisors on projective spaces or Abelian varieties. 2. (2)

Let $Y$ be any smooth projective toric surface other than $\mathbb{P}^{2},\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let $X=Y\times C$. Since there exists a negative curve on $Y$, the assumption in the above lemma does not hold for any ample divisor on $X$.

Remark 3.7*.*

The tilt-stability of line bundles is crucial in our proof of the main theorem. Hence our approach can not apply to threefolds in (ii) of Example 3.6.

The last one is about the approximation of dimensions of certain $\operatorname{Ext}$-groups due to [7].

Proposition 3.8** (([7])).**

Let $C$ be an elliptic curve, $S$ an Abelian surface. Let $X$ be $\mathbb{P}^{1}\times S,\mathbb{P}^{2}\times C$, or $\mathbb{P}^{1}\times\mathbb{P}^{1}\times C$. Let $f^{(m^{2},m)}:=\underline{m^{2}}\times\underline{m}\colon X\to X$ be the product of the toric Frobenius morphism and the multiplication map. Let $E\in D^{b}(X)$ be a two term complex concentrated in degree $-1$ and [math].

1. (1)

If there exists an ample divisor $H^{\prime}$ on $X$ such that

[TABLE]

then

[TABLE] 2. (2)

If there exists an ample divisor $H^{\prime}$ on $X$ such that

[TABLE]

then

[TABLE]

Proof.

Summarizing the arguments of Section 7 in [7], we get the result. ∎

4. Proof of the Main Theorem

In this section, we prove our main theorem, Theorem 2.6. Let $C$ be an elliptic curve and $S$ an Abelian surface. Let $X=Y\times Z$, where $(Y,Z)=(\mathbb{P}^{1},S),(\mathbb{P}^{2},C),(\mathbb{P}^{1}\times\mathbb{P}^{1},C)$. Let $H$ be an ample divisor on $X$. Then $H$ can be written as $H=h+f$, where $h,f$ are the pull back of some ample divisors on $Y,Z$, respectively. For integers $a,b\in\mathbb{Z}_{\geq 0}$, let $f^{(a,b)}:=\underline{a}\times\underline{b}$ be the product of the toric Frobenius morphism $\underline{a}$ on $Y$ and the multiplication map $\underline{b}$ on $Z$. Furthermore, let us denote by $D_{\rho}\in\operatorname{NS}(X)$ the pull backs of the torus invariant divisors on Y.

Remark 4.1*.*

Let $m\in\mathbb{Z}_{>0}$. Note that $f^{(m^{2},m)*}$ acts on the even cohomology as follows:

[TABLE]

We will use this property in the followings.

Let $E$ be a $\bar{\beta}$-stable object with $\operatorname{ch}_{0}(E)\geq 0$ and $\bar{\beta}:=\bar{\beta}(E)\in[0,1)$. To prove Theorem 2.6, it is enough to show that $\operatorname{ch}_{3}^{\bar{\beta}}(E)\leq 0$ by Theorem 2.10. We prove it in the following three subsections. We start with two easy lemmas which we will frequently use in the followings.

Lemma 4.2**.**

For every $E,F\in D^{b}(X)$, we have

[TABLE]

Proof.

Use Serre duality and the adjointness between $f^{(a,b)*}$ and $f^{(a,b)}_{*}$. Note that we do not need to take derived functors since $f^{(a,b)}$ is finite and flat. ∎

Lemma 4.3**.**

Let $E\in\operatorname{Coh}^{\bar{\beta}}(X)$ be a $\bar{\beta}$-stable object with $\operatorname{ch}_{0}(E)\geq 0$ and $\bar{\beta}=\bar{\beta}(E)\in[0,1)$, $L$ a line bundle on $X$.

1. (1)

If $\operatorname{ch}_{1}^{\bar{\beta}}(L)$ is ample, then

[TABLE] 2. (2)

If $\operatorname{ch}_{1}^{\bar{\beta}}(L)$ is anti-ample, then

[TABLE]

Proof.

We only prove the first statement. The second one also follows from the similar computation. Since $\operatorname{ch}_{1}^{\bar{\beta}}(L)$ is ample, we have

[TABLE]

By the first inequality, we have $L\in\operatorname{Coh}^{\bar{\beta}}(X)$. Moreover, by Proposition 3.5, $L$ is tilt-stable near $(0,\bar{\beta})$. On the other hand, the second inequality implies

[TABLE]

Hence by the tilt-stability of $L$ and $E$, we have

[TABLE]

∎

4.1. Integral case

Assume that $\bar{\beta}=0$. Let us consider $\chi(\mathcal{O},f^{(m^{2},m)*}E)$. By the Riemann-Roch Theorem and Remark 4.1, we have

[TABLE]

On the other hand,

[TABLE]

since $E$ is a two term complex concentrated in degree $-1$ and [math].

By Proposition 3.8, the following two lemmas show that the RHS of the inequality (4.1) is of order $m^{4}$. Hence we must have $\operatorname{ch}_{3}(E)\leq 0$ as required.

Lemma 4.4**.**

We have

[TABLE]

Proof.

By Theorem 3.1 and Lemma 4.2, we have

[TABLE]

where

[TABLE]

As remarked in Remark 3.2, $\mathcal{O}(-K_{X})\otimes L_{j}^{*}$ is the pull back of an ample line bundle on $Y$. Hence $\mathcal{O}(-K_{X}+f)\otimes L_{j}^{*}$ is ample on $X$. Note that $f^{(1,m)*}E$ is $\bar{\beta}$-stable with $\bar{\beta}(f^{(1,m)*}E)=0$ (with respect to the polarization $f^{(1,m)*}H$) by Proposition 3.3. Hence Lemma 4.3 implies that

[TABLE]

Summing up, we conclude that

[TABLE]

∎

Lemma 4.5**.**

We have

[TABLE]

Proof.

By Theorem 3.1, Serre duality, and the usual adjoint, we have

[TABLE]

where

[TABLE]

For all $j$, $\operatorname{ch}_{1}(\mathcal{O}(-f)\otimes L_{j}^{*})$ is anti-ample. Hence by Lemma 4.3,

[TABLE]

∎

4.2. Rational case

In this subsection, we assume that $\bar{\beta}=\frac{p}{q}\in\mathbb{Q}$, $q>0$, $p$ and $q$ are coprime. We consider $\chi\left(\mathcal{O},f^{(m^{2},m)*}\left(f^{(q^{2},q)*}E\otimes\mathcal{O}\left(-pqH\right)\right)\right)$. By Remark 4.1, we have

[TABLE]

and hence

[TABLE]

As in the previous subsection, we will check the assumption in Proposition 3.8.

Lemma 4.6**.**

We have

[TABLE]

Proof.

Using Theorem 3.1 and Lemma 4.2, we have

[TABLE]

where

[TABLE]

By Lemma 4.3, it is enough to show that

[TABLE]

is ample. We can compute it as

[TABLE]

Since $-K_{X}=\sum_{\rho}D_{\rho}$, $0\leq a_{\rho}\leq m^{2}-1$, and $0\leq b_{\rho}\leq q^{2}-1$, we have

[TABLE]

and it is ample on $X$. We conclude that

[TABLE]

∎

Lemma 4.7**.**

We have

[TABLE]

Proof.

By Serre duality, adjunction, and Theorem 3.1, we have

[TABLE]

Here,

[TABLE]

As before, it is enough to show that

[TABLE]

is anti-ample. Straightforward computation yields that

[TABLE]

This is anti-ample on $X$. Hence we get the required result. ∎

4.3. Irrational case

Assume that $\bar{\beta}$ is irrational. Define

[TABLE]

Take $\epsilon>0$ small enough so that for every $(\alpha,\beta)\in V_{\epsilon}$, $E$ is $\nu_{\alpha,\beta}$-stable. By the Dirichlet approximation theorem, we can take a sequence $\left\{\beta_{n}=\frac{p_{n}}{q_{n}}\right\}$ of rational numbers such that

[TABLE]

and $q_{n}\to+\infty$ as $n\to+\infty$. We compute $\chi\left(\mathcal{O},f^{(q_{n}^{2},q_{n})*}E\otimes\mathcal{O}\left(-p_{n}q_{n}H\right)\right)$. As before,

[TABLE]

We will show that the last line of the above inequalities is of order $q^{4}_{n}$.

Lemma 4.8**.**

Let $u,v\in\mathbb{Z}_{>0}$ such that $(u-2)h+K_{X}$ is effective and $v>2$. Then

[TABLE]

Proof.

As in the rational case, we have

[TABLE]

where

[TABLE]

Let $M_{j}:=\mathcal{O}\left(\left(p_{n}q_{n}+v\right)f\right)\otimes\mathcal{O}\left(-K_{X}\right)\otimes L_{j}^{*}$, $H^{(n)}:=f^{(1,q_{n})*}H$. For a while, assume that $\operatorname{ch}^{\beta_{n}H^{(n)}}_{1}(M_{j})-2\frac{1}{q_{n}^{2}}H^{(n)}$ is ample. Then we can compute as

[TABLE]

and

[TABLE]

which imply

[TABLE]

by the proof of Lemma 4.3. Hence it is enough to show that $\operatorname{ch}^{\beta_{n}H^{(n)}}_{1}(M_{j})-2\frac{1}{q_{n}^{2}}H^{(n)}$ is ample. As in the rational case, we have

[TABLE]

and hence

[TABLE]

As observed in Remark 3.2,

[TABLE]

is the pull back of an ample divisor on $Y$. Hence if we take $u,v$ so that $v>2$ and $(u-2)h+K_{X}$ is effective, then

[TABLE]

is ample on $X$. Note that these conditions does not depend on $n$. ∎

Lemma 4.9**.**

Let $u,v\in\mathbb{Z}_{>0}$, $u,v>2$. Then

[TABLE]

Proof.

As in the rational case,

[TABLE]

where

[TABLE]

Let $M_{j}:=\mathcal{O}\left(p_{n}q_{n}f-vf\right)\otimes L_{j}^{*},$ $H^{(n)}:=f^{(1,q_{n})*}H$. Assume that

[TABLE]

is anti-ample. Then

[TABLE]

and

[TABLE]

Then the stability of $M_{j}$ and $f^{(1,q_{n})*}E$ shows that $\hom(f^{(1,q_{n}*)}E,M_{j}[1])=0$ as required. Hence it is enough to show that

[TABLE]

is anti-ample. We can compute it as

[TABLE]

For $u,v>2$, this is anti-ample. ∎

Appendix A Counter-example for Conjecture 2.4

In this appendix, we propose a counter-example for the original BG type inequality conjecture. More precisely, we show the following proposition using the argument of [21]:

Proposition A.1**.**

Let $X$ be a Calabi-Yau threefold containing a plane $\mathbb{P}^{2}\cong D\subset X$. Then there exists an ample divisor $H$ on $X$, $\alpha>0$, and $\beta\in\mathbb{R}$ such that $\mathcal{O}_{D}[1]\in\mathcal{A}_{\alpha,\beta}$ and

[TABLE]

This proves the pair $(Z_{\alpha,\beta},\mathcal{A}_{\alpha,\beta})$ is not a stability condition on $X$. In particular, Conjecture 2.4 does not hold by Theorem 2.5.

First we explain how to take the ample divisor $H$. Let $l\subset D$ be a line. Then

[TABLE]

Let $H^{\prime}$ be any ample divisor on $X$ and put $L:=3H^{\prime}+(H^{\prime}.l)D$. Then $L$ is nef and big. Hence by the Kawamata-Shokurov basepoint-free theorem, some multiple of $L$ defines a birational morphism $f:X\to Y$, which only contracts $D$. In particular, there exists an ample $\mathbb{Q}$-divisor $A$ on $Y$ such that $L=f^{*}A$. On the other hand, $-D$ is $f$-ample since $\mathcal{O}_{D}(-3)=\omega_{D}\cong\mathcal{O}_{D}(D)$. Hence $H:=mL-\frac{1}{2}D$ is ample on $X$ for all $m\gg 0$.

The Chern character of $\mathcal{O}_{D}$ is computed as :

[TABLE]

Similarly,

[TABLE]

Before beginning the proof of Proposition A.1, we recall the following aspect of the structure theorem of walls in tilt-stability:

Lemma A.2**.**

Let $v\in\Lambda:=\operatorname{Image}(H.\operatorname{ch}\colon K(X)\to\mathbb{Q}^{4})$. Let $E\in\operatorname{Coh}^{\beta_{0}}(X)$ be an object with $H.\operatorname{ch}(E)=v$. Let $0\to A\to E\to B\to 0$ be an exact sequence in $\operatorname{Coh}^{\beta_{0}}(X)$ with $H.\operatorname{ch}(E)=v$ which defines a wall $\mathcal{W}$ at $(\alpha_{0},\beta_{0})\in\mathcal{W}$. Then for every $(\alpha,\beta)\in\mathcal{W}$, we have $A,E,B\in\operatorname{Coh}^{\beta}(X)$.

Proof.

See for example, Lemma 6.3 of [2]. ∎

Now we can prove Proposition A.1:

of Proposition A.1.

The argument is exactly same as [21]. Since

[TABLE]

$Z_{\alpha,1}(\mathcal{O}_{D}[1])>0$ if and only if $\alpha<1$. On the other hand, since $\nu_{\alpha,1}(\mathcal{O}_{D})=0$, $\mathcal{O}_{D}[1]\in\mathcal{A}_{\alpha,1}$ if $\mathcal{O}_{D}$ is $\nu_{\alpha,1}$-semistable. Hence it is enough to show that there exists $\alpha\in(0,1)$ such that $\mathcal{O}_{D}$ is $\nu_{\alpha,1}$-stable.

Let us consider the wall $\mathcal{W}$ which is defined by a short exact sequence

[TABLE]

Let us denote $(r(A),c(A),d(A),e(A)):=H.\operatorname{ch}(A)$, etc. The center of this semicircular wall $\mathcal{W}$ is

[TABLE]

Let $R$ be the radius of the wall $\mathcal{W}$. We will bound $R$ from above. Since $\mathcal{H}^{-1}(B)\in\mathcal{F}_{\beta}$ and $A\in\mathcal{T}_{\beta}$ for all $(\alpha,\beta)\in\mathcal{W}$, we have

[TABLE]

By using the exact sequence

[TABLE]

we get

[TABLE]

Using these inequalities, we have

[TABLE]

for $m>1$. Since $\mathcal{O}_{D}$ is Gieseker stable, it is $\nu_{\alpha,\beta}$-stable for every $\alpha\gg 0,\beta\in\mathbb{R}$. By the bound of the radius of semicircular walls, we conclude that $\mathcal{O}_{D}$ is $\nu_{\alpha,1}$-stable for

[TABLE]

∎