Wall crossing for moduli of stable sheaves on an elliptic surface

Abstract

Bridgeland studied moduli of stable sheaves on elliptic surfaces by using Fourier-Mukai transforms. In this paper, we shall study the wall crossing behavior of the moduli of stable sheaves which is a refinement of Bridgeland’s results.

Appendix

1.1 A category related to $\lambda$-stability

We give a remark on the $\lambda$-stability. We set $\lambda :=\frac{d_1-(\alpha \cdot r_1 f)}{(r_1 f \cdot H)}$.

Definition 6.1

• (1) Let ${\mathcal S}_\lambda$ be the full subcategory of ${\text {Coh}}(X)$ generated by $\alpha$-twisted stable sheaves S on a fiber such that $\frac{\chi _\alpha (S)}{(c_1(S) \cdot H)} \le \lambda$.

• (2) Let ${\mathcal T}_\lambda$ be the full subcategory of ${\text {Coh}}(X)$ whose object E satisfies ${\text {Hom}}(E, S)=0$ for all $S \in {\mathcal S}_\lambda$.

• (3) Let ${\mathcal F}_\lambda$ be the full subcategory of ${\text {Coh}}(X)$ whose objects E fits in an exact sequence

  $$\begin{aligned} 0 \rightarrow S \rightarrow E \rightarrow E_2 \rightarrow 0 \end{aligned}$$

  where $S \in {\mathcal S}_\lambda$ and $E_2$ is a torsion free sheaf such that $(E_2)_{\eta }$ is generated by $\mu$-semi-stable vector bundles $F_\eta$ with $\deg F_\eta /{\text {rk}}F_\eta \le d_1/r_1$.

Let $Y:=M_H^\alpha (0,r_1 f,d_1)$ be a fine moduli space and ${\mathcal P}$ a universal family on $X \times Y$, where $(\alpha \cdot f)=0$ and $(H,\alpha )$ is general. Let $\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}:\textbf{D}(X) \rightarrow \textbf{D}(Y)$ be the Fourier-Mukai transform by ${\mathcal P}$.

Proposition 6.2

We set $\lambda :=\frac{\chi _\alpha ({\mathcal P}_{|X \times \{ y\}})}{(r_1 f \cdot H)}=\frac{d_1}{r_1(f \cdot H)}$. Then $({\mathcal T}_\lambda ,{\mathcal F}_\lambda )$ is a torsion pair. Let ${\mathcal A}_\lambda$ be the tilting. Then $\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}({\text {Coh}}(Y))={\mathcal A}_\lambda$.

Proof

For $F \in {\text {Coh}}(Y)$ and an $\alpha$-twisted stable sheaf A on a fiber such that $A \in {\mathcal S}_\lambda$, Lemma 3.5 implies that

$$\begin{aligned} {\text {Hom}}(\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}(F),A)={\text {Hom}}(F,\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}(A(K_X))[1])=0. \end{aligned}$$

For an $\alpha$-twisted stable sheaf A on a fiber such that $\frac{\chi _\alpha (A)}{(c_1(A) \cdot H)} > \frac{\chi _\alpha ({\mathcal P}_{|X \times \{ y\}})}{(r_1 f \cdot H)}$,

$$\begin{aligned} {\text{ Hom }}(A,\Phi _{Y \rightarrow X}^{{\mathcal P}}(F))={\text{ Hom }}(\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}(A)[1],F[-1])=0. \end{aligned}$$

We also note that $H^i(\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}(F))=0$ for $i \ne -1,0$. By using these facts and [4, sect. 6.2], we shall prove that $({\mathcal T},{\mathcal F})$ is a torsion pair and ${\mathcal A}=\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}({\text {Coh}}(Y))$.

For $E \in {\text {Coh}}(X)$, [4, sect. 6.2] implies

$$\begin{aligned} \begin{aligned}&H^i(\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(E(K_X))) =0,\; i \ne 0,1,\\&E_1:=\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}(H^0(\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(E(K_X)))) \in {\text {Coh}}(X),\\&E_2:=\Phi _{Y \rightarrow X}^{{\mathcal P}}(H^1(\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(E))) \in {\text {Coh}}(X). \end{aligned} \end{aligned}$$

(6.1)

We also have an exact sequence

$$\begin{aligned} 0 \rightarrow E_1 \rightarrow E \rightarrow E_2 \rightarrow 0. \end{aligned}$$

Then by using [4, Lem. 6.2], we see that

$$\begin{aligned} E_1 \in {\mathcal T}_\lambda ,\; E_2 \in {\mathcal F}_\lambda . \end{aligned}$$

Thus we have a desired decomposition of E. We also get $\Phi _{Y \rightarrow X}^{{\mathcal P}[1]}({\text {Coh}}(Y))={\mathcal A}_\lambda$. $\square$

For an object $E \in {\mathcal A}_\lambda \cap {\text {Coh}}(X)$, the condition Definition 3.1 (3) implies E is an torsion free object of ${\mathcal A}_\lambda$.

1.2 Fourier-Mukai transforms and the $\lambda$-stability

In this subsection, we shall refine Proposition 3.9. Assume that $(\alpha \cdot f)=0$. Let $\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}:\textbf{D}(X) \rightarrow \textbf{D}(Y)$ be the Fourier-Mukai transform in subsection 1.1. As we explained in subsection 1.1, we take a locally free sheaf G such that $\frac{c_1(G)}{{\text {rk}}G}=\alpha +\frac{d_1 H}{r_1 (f \cdot H)}$. Then $\chi (G,{\mathcal P}_{|X \times \{ y\}})=0$ ($y \in Y$). We set $\tau ({\mathcal P}^{\vee }_{|\{ x \} \times Y}[1])=(0,r_1' f, d_1')$. We also define a locally free sheaf $G':=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}({\mathcal O}_H)[1]$ and set $H':=c_1(\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }}(G)[2])$. We set $\alpha ':=\frac{c_1(G')}{{\text {rk}}G'}-\frac{d_1' H'}{r_1' (f \cdot H')}$. Since $\frac{(c_1(G') \cdot f)}{{\text {rk}}G'}=\frac{d_1'}{r_1'}$, $(\alpha ' \cdot f)=0$.

For $\lambda _0=\frac{d_0-(\alpha \cdot r_0 f)}{(r_0 f \cdot H)}=\frac{d_0}{r_0 (f \cdot H)}$ such that $r_0 \in {\mathbb Z}_{>0}$, $d_0 \in {\mathbb Z}$ and $\gcd (r_0,d_0)=1$, we take an $\alpha$-twisted stable sheaf P with $\tau (P)=(0,r_0 f,d_0)$. We set $\varphi (\lambda _0):=\frac{\chi _{\alpha '}(P')}{(c_1(P') \cdot H')}= \frac{\chi (G',P')}{{\text {rk}}G' (c_1(P') \cdot H')}-\frac{d_1'}{r_1'(f \cdot H')}$, where $P'=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(P)$ and $\tau (P')=(0,r'_0 f,d_0')$. We regard $\lambda \in (-\infty ,\infty )$ as an element of ${\mathbb P}_{\mathbb R}^1 \cong S^1$. Then $\varphi$ is extened to an isomorphism of ${\mathbb P}_{\mathbb R}^1$, where $\varphi (\infty )= \frac{\chi _{\alpha '}({\mathcal P}^{\vee }_{|\{x \} \times Y}[1])}{(c_1({\mathcal P}^{\vee }_{|\{x \} \times Y}[1])) \cdot H')}=\frac{d_1'}{r_1' (f \cdot H')}$.

Proposition 6.3

Assume that $\gcd ((\xi \cdot f),r)=1$ and $r_1(\xi \cdot f)-rd_1>0$. We assume that $\lambda _0=\frac{d_0-(\alpha \cdot r_0 f)}{(r_0 f \cdot H)} >\lambda _1=\frac{d_1-(\alpha \cdot r_1 f)}{(r_1 f \cdot H)}$. Then for a $\lambda _0$-stable sheaf E, $E':=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(E)$ is $\varphi (\lambda _0)$-stable. Thus we have an isomorphism

$$\begin{aligned} \Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}: {\mathcal M}^{\lambda _0}(\textbf{e})^s \rightarrow {\mathcal M}^{\varphi (\lambda _0)}(\textbf{e}')^s, \end{aligned}$$

where $\textbf{e}'=\tau (E')$.

Proof

Let $A'$ be an $\alpha '$-twisted stable sheaf on a fiber. We shall prove the following.

• (i) If $\frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')}>\varphi (\lambda _0)$, then ${\text {Hom}}(A',E')=0$.

• (ii) If $\frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')} \le \varphi (\lambda _0)$, then ${\text {Hom}}(E',A')=0$.

(i) If $\frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')}>\varphi (\infty )=\frac{d_1'}{r_1' (f \cdot H')}$, then there is an $\alpha$-twisted stable 1-dimensional sheaf A such that $A'=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(A[1])$ and $\frac{\chi _{\alpha }(A)}{(c_1(A) \cdot H)}<\lambda _1$ (see Lemma 1.5 and Remark 1.6). Hence ${\text {Hom}}(A',E')={\text {Hom}}(A[1],E)=0$. If $\varphi (\infty )> \frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')}>\varphi (\lambda _0)$, then there is an $\alpha$-twisted stable 1-dimensional sheaf A such that $A'=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(A)$ and $\frac{\chi _{\alpha }(A)}{(c_1(A) \cdot H)}>\lambda _0$. Hence ${\text {Hom}}(A',E')={\text {Hom}}(A,E)=0$. If $\varphi (\infty )=\frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')}$, then ${\text {Hom}}(A',E')={\text {Hom}}({\mathbb C}_x,E)=0$.

(ii) If $\frac{\chi _{\alpha '}(A')}{(c_1(A') \cdot H')} \le \varphi (\lambda _0)$, then there is an $\alpha$-twisted stable 1-dimensional sheaf A such that $A'=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(A)$ and $\frac{\chi _{\alpha }(A)}{(c_1(A) \cdot H)} \le \lambda _0$. Hence ${\text {Hom}}(E',A')={\text {Hom}}(E,A)=0$. $\square$

We also have the following result.

Proposition 6.4

Assume that $r_1(\xi \cdot f)-rd_1>0$. We set $\psi (\lambda _0):=\frac{\chi _{-\alpha '}({P'}^{\vee }[1])}{(c_1({P'}^{\vee }[1]) \cdot H')} =-\frac{\chi _{\alpha '}(P')}{(c_1(P') \cdot H')}$. Assume that $\lambda _0<\lambda _1$. Then for a $\lambda _0$-stable sheaf E, $E'':=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(E)^{\vee }$ satisfies the following.

• (i) For a $(-\alpha ')$-stable fiber sheaf $A''$ with $\psi (\lambda _0) \le \frac{\chi _{-\alpha '}(A'')}{(c_1(A'') \cdot H)}$, ${\text {Hom}}(A'',E'')=0$.

• (ii) For a $(-\alpha ')$-stable fiber sheaf $A''$ with $\psi (\lambda _0) > \frac{\chi _{-\alpha }(A'')}{(c_1(A'') \cdot H)}$, ${\text {Hom}}(E'',A'')=0$.

In particular if $\lambda _0$ is general, then $E''$ is $\psi (\lambda _0)$-stable.

Proof

(i) Assume that $\psi (\lambda _0) \le \frac{\chi _{-\alpha '}(A'')}{(c_1(A'') \cdot H)}< \psi (\infty )=\frac{-d_1'+(\alpha ' \cdot r_1' f)}{r_1' (f \cdot H)}$. Then $A''=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[2]}(A)^{\vee }[1]$ and $\frac{\chi _\alpha (A)}{(c_1(A) \cdot H)} \le \lambda _0$. Hence ${\text {Hom}}(A'',E'')={\text {Hom}}(E,A)=0$. Assume that $\psi (\infty ) \le \frac{\chi _{-\alpha '}(A'')}{(c_1(A'') \cdot H)}$. Then $A''=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[1]}(A)^{\vee }[1]$. Hence ${\text {Hom}}(A'',E'')={\text {Hom}}(E[1],A)=0$.

(ii) Assume that $\psi (\lambda _0) > \frac{\chi _{-\alpha }(A'')}{(c_1(A'') \cdot H)}$. Then $A''=\Phi _{X \rightarrow Y}^{{\mathcal P}^{\vee }[2]}(A)^{\vee }[1]$ and $\frac{\chi _\alpha (A)}{(c_1(A) \cdot H)} > \lambda _0$. Hence ${\text {Hom}}(E'',A'')={\text {Hom}}(A,E)=0$. $\square$