¹

²

1 Introduction↩︎

Let \(X\) be an abelian surface over \({\mathbb{C}}\) and let \((H^{2*}(X,{\mathbb{Z}}),\langle\;\;,\;\; \rangle)\) be the Mukai lattice of \(X\): Thus \(H^{2*}(X,{\mathbb{Z}}):={\mathbb{Z}} \oplus H^2(X,{\mathbb{Z}}) \oplus {\mathbb{Z}}\) and \[\langle x,y \rangle:=(x_1 \cdot y_1)-x_0 y_2-y_0 x_2 \in {\mathbb{Z}}\] for \(x=(x_0,x_1,x_2), y=(y_0,y_1,y_2) \in {\mathbb{Z}} \oplus H^2(X,{\mathbb{Z}}) \oplus {\mathbb{Z}}\). Mukai lattice has a natural Hodge structure and \(H^{2*}(X,{\mathbb{Z}})_{\operatorname{alg}}:={\mathbb{Z}} \oplus \operatorname{NS}(X) \oplus {\mathbb{Z}}\) is the algebraic part of \(H^{2*}(X,{\mathbb{Z}})\). For an object \(E\) of the bounded derived category \({\boldsymbol{D}}(X)\) of coherent sheaves, \(v(E):=\operatorname{ch}(E) \in H^{2*}(X,{\mathbb{Z}})\) is the Mukai vector of \(E\). We also say that \(v \in H^{2*}(X,{\mathbb{Z}})_{\operatorname{alg}}\) is a Mukai vector. For an ample divisor \(H\) and a Mukai vector \(v\), \({\mathcal{M}}_H(v)\) denotes the moduli stack of semi-stable sheaves \(E\) with \(v(E)=v\). There are many results on semi-stable sheaves and also their moduli stacks \({\mathcal{M}}_H(v)\). For examples, \({\mathcal{M}}_H(v) \ne \emptyset\) if and only if \(\langle v^2 \rangle \geq 0\) and \({\mathcal{M}}_H(v)\) is an irreducible normal stack of dimension \(\langle v^2 \rangle+1\), where \(H\) is general with respect to \(v\). For the proof of these results, we used the symmetry of the derived category of coherent sheaves, that is, the Fourier-Mukai transforms. Since they are the transforms of the derived categories of coherent sheaves, Gieseker stability is not preserved in general. So we need to find some conditions for the preservation of stability, or to replace the kernel of the Fourier-Mukai transform for the preservation of stability.

For the Fourier-Mukai transform \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}:{\boldsymbol{D}}(X) \to {\boldsymbol{D}}(\widehat{X})\), we proved in [1] that the stability is preserved if the Picard number is 1, where \(\widehat{X}\) is the dual abelian surface and \({\mathcal{P}}\) is the Poincaré line bundle on \(X \times \widehat{X}\). In this paper, we shall treat the case where the Picard number is not 1. The following generalization of [1] is our main result. .

In [2], we constructed a birational map of moduli space by a modification of the kernel. For the modification, we used a characterization of the Albanese map in [3] to study the Fourier-Mukai transform of a general member \(E\) of \({\mathcal{M}}_H(v)\). In particular we constructed a birational map of moduli stacks. In [4], [5], [6], we explained the birationality of moduli spaces in terms of wall crossing of Bridgeland stability conditions. In particular if the Picard number is 1, then the birational map in [2] is the same as the birational map induced by totally semistable walls.

For the preservation of Gieseker stability, we studied extensively if the Picard number is 1. We proved that the stability is preserved for a general member of moduli spaces. On the other hand, if the Picard number is not 1, not much is known.

On the other hand if \({\mathcal{P}}\) is a family of line bundles and the Picard number \(\rho(X)=1\), then the Fourier-Mukai transform preserves the stability for a general member of the moduli stack.

We would like to remark that the preservation of stability is related to the weak Brill-Noether property. Indeed if \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}(E)^{\vee} \in \operatorname{Coh}(\widehat{X})\) (resp. \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}(E)[1] \in \operatorname{Coh}(\widehat{X})\)), then there is a point \(\hat{x} \in \widehat{X}\) such that \(H^i(X,E \otimes {\mathcal{P}}_{|X \times \{\hat{x} \}}^{\vee})=0\) for \(i \ne 0\) (resp. \(H^i(X,E \otimes {\mathcal{P}}_{|X \times \{\hat{x} \}}^{\vee})=0\) for \(i \ne 1\)). In [7], Coskun, Nuer and the author proved that the weak Brill-Noether property does not hold in general. In particular there is a Mukai vector \(v\) and an ample divisor \(L \in \operatorname{NS}(X)\) such that the cohomology sheaves \(H^0(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}(E))\) and \(H^1(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}(E))\) are of positive rank for all \(E \in {\mathcal{M}}_L(v)\). On the other hand, our main result almost says that if we choose the polarization suitably, we have the weak Brill-Noether property. Indeed in the course of the proof of Theorem 1, we get an affirmative result on the weak Brill-Noether property.

In order to explain our counter example for the weak Brill-Noether property in [7], let \({\mathcal{M}}(v)\) be the moduli stack of coherent sheaves \(E\) such that \(v(E)=v\) and \(E\) is semistable with respect to a general ample divisor. Thus \({\mathcal{M}}(v)=\cup_L {\mathcal{M}}_L(v)\), where \(L\) runs the set of general polarizations with respect to \(v\) (see Definition 9). Then the counter examples exists if \({\mathcal{M}}(v)\) is not irreducible. In this case, if the boundary \(\partial \! \operatorname{Amp}(X)\) of the ample cone contains an irrational ray, we have infinitely many irreducible components \({\mathcal{M}}_L(v)\) and the weak Brill-Noether property does not hold if \(L\) is close to \(\partial \! \operatorname{Amp}(X)\). In this paper, we shall give positive results on the preservation of stability and the weak Brill-Noether property for a fixed ample divisor.

(I) Under the irreducibility of moduli stacks, we do not need to care about polarizations. So we get following result from Theorem 1.

(II) It is natural to study the weak Brill-Noether property when the ample divisor is not close to \(\partial \! \operatorname{Amp}(X)\). For a Mukai vector \(v=(r,\xi,a)\) with \(\xi \in \operatorname{Amp}(X)\), a natural choice of the ample divisor is \(\xi\). In this case, we get the following nice result.

For the proof of these results, we use Bridgeland stability conditions as in [7]. Thus we study the wall crossing behavior for totally semistable walls in the space \(\operatorname{Stab}(X)\) of Bridgeland stability conditions. Under the assumption of Theorem 1, we shall find a stability condition such that a general stable object \(E\) is torsion free and its Fourier-Mukai transform is also torsion free. If a general stable object \(E\) is torsion free, then we can find an ample divisor such that \(E\) is Gieseker stable, and hence our main result follows.

Let us explain the organization of this paper. In section 2, we explain several notation and basic results. In subsection 2.2, we introduce totally semi-stable walls and study irreducible components parameterizing stable sheaves. We also recall a result on the irreducible components in [7]. In section 3, we recall Bridgeland stability conditions on an abelian surface. In particular, we explain totally semi-stable walls and the relation of adjacent chambers. In section 4, we prove our result. In subsection 4.1, we study wall crossing behavior along a line connecting two chambers, one is the chamber corresponding to the Gieseker semi-stability, and the other is the chamber whose transform by \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}\) is related to Gieseker semi-stability. In subsection 4.2, we prove Theorem 1 and Proposition 2. In subsection 4.3, we treat the case where \(v=(r,dH,a)\). In particular, we prove Theorem 4.

.

In Proposition 26, we divide the line into at most three parts by the property of semi-stable objects:

1. a general semi-stable object is a torsion free coherent sheaf,

2. a general semi-stable object is a coherent sheaf with a torsion subsheaf, and

3. a general semi-stable object is a two term complex of coherent sheaves.

2 Preliminaries.↩︎

In this paper, \(H\) denotes an ample divisor on an abelian surface \(X\). Let \(\operatorname{Amp}(X)\) be the ample cone of \(X\). Then \[\operatorname{Amp}(X)=\{L \in \operatorname{NS}(X) \mid (L^2)>0, (L \cdot H)>0 \}.\] We note that \(D \in \operatorname{NS}(X)\) is effective, that is, \(D\) is represented by an effective divisor iff

1. \((D^2)>0\) and \((D \cdot H)>0\) or

2. \((D^2)=0\) and \((D \cdot H)>0\).

2.1 Fourier-Mukai transforms↩︎

Let \(\widehat{X}\) be the dual abelian surface of \(X\) and \({\mathcal{P}}\) the Poincaré line bundle on \(X \times \widehat{X}\). Let \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}:{\boldsymbol{D}}(X) \to {\boldsymbol{D}}(\widehat{X})\) be the Fourier-Mukai transform whose kernel is \({\mathcal{P}}^{\vee}\) [8]: \[\begin{matrix} \Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}: & {\boldsymbol{D}}(X) & \to & {\boldsymbol{D}}(\widehat{X})\\ & E & \mapsto & {\boldsymbol{R}}p_{\widehat{X}*}({\mathcal{P}}^{\vee} \otimes p_X^*(E)), \end{matrix}\] where \(p_X:X \times \widehat{X} \to X\) and \(p_{\widehat{X}}:X \times \widehat{X} \to \widehat{X}\) are projections. We set \(\Phi:=\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}\) and \(\widehat{\Phi}:=\Phi_{\widehat{X} \to X}^{{\mathcal{P}}}\). We also set \[\begin{align} \Phi^i(E):=& H^i(\Phi(E)) \in \operatorname{Coh}(\widehat{X}),\; E \in \operatorname{Coh}(X)\\ \widehat{\Phi}^i(F):=& H^i(\widehat{\Phi}(F)) \in \operatorname{Coh}(X),\;F \in \operatorname{Coh}(\widehat{X}). \end{align}\]

\(\Phi\) induces an isometry of Hodge structures \[\begin{matrix} H^{2*}(X,{\mathbb{Z}}) & \to & H^{2*}(\widehat{X},{\mathbb{Z}})\\ (r,\xi,a) & \mapsto & (a,-\widehat{\xi},r), \end{matrix}\] where \(\widehat{\xi}\) is the Poincaré dual of \(\xi\). If \(\Phi^i(E)\) is a torsion sheaf, then \(h^i(E \otimes {\mathcal{P}}_{|X \times \{ y \}}^{\vee})=0\) for a general \(y \in \widehat{X}\). Hence the Fourier-Mukai transform can be used to study cohomology groups of a general stable sheaf.

The following follows from the structure of Fourier-Mukai transforms on an abelian surface [9], [10]. See also [2].

.

.

For a divisor \(\xi\) with \((\xi^2)=0\), \(\Phi({\mathcal{O}}_X(\xi)) \ne 0\) implies there is \(t \in \widehat{X}\) such that \(h^0({\mathcal{O}}_X(\xi) \otimes {\mathcal{P}}_t)>0\) or \(h^2({\mathcal{O}}_X(\xi) \otimes {\mathcal{P}}_t)>0\). Hence \(\xi\) is nef or \(-\xi\) is nef.

2.2 The dependence of Gieseker semistability on polarizations↩︎

For the \(\mu\)-semi-stability of coherent sheaves, we have the notion of wall and chamber in \(\operatorname{Amp}(X)_{\mathbb{R}}\). An ample divisor \(H \in \operatorname{Amp}(X)\) is general, if \(H\) is in a chamber, which is equivalent to the following definition.

\({\mathcal{M}}(v)\) is a normal stack of dimension \(\langle v^2 \rangle+1\). Moreover \({\mathcal{M}}(v)\) is irreducible if \(v\) is not primitive. Assume that \(v\) is primitive. Then \({\mathcal{M}}(v)\) is smooth, but it is not irreducible in general.

.

For a general ample divisor \(L\) with respect to \(v\), \({\mathcal{M}}_L(v)\) consists of stable sheaves. By the openness of stability, \({\mathcal{M}}_L(v)\) is an open substack of \({\mathcal{S}}(v)\). Hence the closure of \({\mathcal{M}}_L(v)\) in \({\mathcal{S}}(v)\) is an irreducible component.

The following proposition shows that there are infinitely many totally semi-stable walls in general.

3 Bridgeland stability.↩︎

3.1 Stability conditions on an abelian surface↩︎

Let us briefly recall some notation on stability conditions of Bridgeland for an abelian surface \(X\). For more details, see [12], [13] and [14]. A stability condition \(\sigma=(Z_\sigma,{\mathcal{P}}_\sigma)\) on \({\boldsymbol{D}}(X)\) consists of a group homomorphism \(Z_\sigma: {\boldsymbol{D}}(X) \to {\mathbb{C}}\) and a slicing \({\mathcal{P}}_\sigma\) of \({\boldsymbol{D}}(X)\) such that if \(0 \ne E \in {\mathcal{P}}_\sigma(\phi)\) then \(Z_\sigma(E) =m(E)\exp(\pi \sqrt{-1} \phi)\) for some \(m(E) \in {\mathbb{R}}_{>0}\). The set of stability conditions has a structure of complex manifold. We denote this space by \(\operatorname{Stab}(X)\).

By [12], giving a stability condition \(\sigma\) is the same as giving a bounded \(t\)-structure on \({\boldsymbol{D}}(X)\) and a stability function \(Z_\sigma\) on its heart \({\mathcal{A}}_\sigma\) with the Harder-Narasimhan property. For \(\sigma=(Z_\sigma,{\mathcal{P}}_\sigma)\), we have the relation \({\mathcal{A}}_\sigma ={\mathcal{P}}_\sigma((0,1])\), where \({\mathcal{P}}_\sigma((0,1])\) is the subcategory of \({\boldsymbol{D}}(X)\) generated by semi-stable objects \(E \in {\mathcal{P}}_\sigma(\phi)\) with \(\phi \in (0,1]\). Since the pair \((Z_\sigma,{\mathcal{A}}_\sigma)\) defines a stability condition, we also use the symbol \(\sigma=(Z_\sigma,{\mathcal{A}}_\sigma)\) to denote a stability condition. For a \(\sigma\)-semi-stable object \(E \in {\mathcal{P}}_\sigma(\phi)\), we set \(\phi_\sigma(E)=\phi\). For \(E \in {\mathcal{P}}_\sigma((0,1])\), we also set \(\phi_\sigma(E) \in (0,1]\) by \(Z_\sigma(E)=m(E)\exp(\pi \sqrt{-1} \phi_\sigma(E))\).

Let \(\operatorname{Stab}(X)\) be the space of stability conditions. For an equivalence \(\Phi:{\boldsymbol{D}}(X) \to {\boldsymbol{D}}(X')\), we have an isomorphism \(\Phi:\operatorname{Stab}(X) \to \operatorname{Stab}(X')\) such that \(\Phi(\sigma)\) \((\sigma \in \operatorname{Stab}(X))\) is a stability condition given by \[\label{eq:FM-action} \begin{align} Z_{\Phi(\sigma)}=& Z_\sigma \circ \Phi^{-1}:{\boldsymbol{D}}(X') \to {\mathbb{C}},\\ {\mathcal{P}}_{\Phi(\sigma)}(\phi)=& \Phi({\mathcal{P}}_\sigma(\phi)). \end{align}\tag{1}\] We also have an action of the universal covering \(\widetilde{\operatorname{GL}}_2^+({\mathbb{R}})\) of \(\operatorname{GL}_2^+({\mathbb{R}})\) on \(\operatorname{Stab}(X)\). Since \({\mathbb{C}}^{\times} \subset \operatorname{GL}_2^+({\mathbb{R}})\), we have an injective homomorphism \({\mathbb{C}} \to \widetilde{\operatorname{GL}}_2^+({\mathbb{R}})\). Thus we have an action of \(\lambda \in {\mathbb{C}}\) on \(\operatorname{Stab}(X)\). For a stability condition \(\sigma \in \operatorname{Stab}(X)\), \(\lambda(\sigma)\) is given by \[\label{eq:C-action} \begin{align} Z_{\lambda(\sigma)}=& \exp(-\pi \sqrt{-1} \lambda)Z_\sigma\\ {\mathcal{P}}_{\lambda(\sigma)}(\phi)=& {\mathcal{P}}_\sigma(\phi+\mathrm{Re}\lambda). \end{align}\tag{2}\]

3.2 Geometric stability conditions↩︎

We set \[{\mathcal{H}}:=\operatorname{NS}(X)_{\mathbb{R}} \times \operatorname{Amp}(X)_{\mathbb{R}}.\] For \((\beta,\omega) \in {\mathcal{H}}\), Bridgeland [13] constructed a stability condition \(\sigma_{(\beta,\omega)}=(Z_{(\beta,\omega)},{\mathcal{A}}_{(\beta,\omega)})\) which is characterized by the stability of \(k_x\) (\(x \in X\)), where \(Z_{(\beta,\omega)}(\bullet)= \langle e^{\beta+\sqrt{-1}\omega},v(\bullet) \rangle: {\boldsymbol{D}}(X) \to {\mathbb{C}}\) is the stability function and \({\mathcal{A}}_{(\beta,\omega)}\) is an abelian category which is a tilting of \(\operatorname{Coh}(X)\) by a torsion pair \(({\mathcal{T}}_{(\beta,\omega)},{\mathcal{F}}_{(\beta,\omega)})\):

1. \({\mathcal{T}}_{(\beta,\omega)}\) is a full subcategory of \(\operatorname{Coh}(X)\) generated by torsion sheaves and \(\mu\)-stable sheaves \(E\) with \(((c_1(E)-\operatorname{rk}E \beta) \cdot \omega)>0\) and

2. \({\mathcal{F}}_{(\beta,\omega)}\) is a full subcategory of \(\operatorname{Coh}(X)\) generated by \(\mu\)-stable sheaves \(E\) with \(((c_1(E)-\operatorname{rk}E \beta) \cdot \omega) \leq 0\).

This kind of stability conditions are called geometric. To be more precise, for a geometric stability condition \(\sigma=(Z_\sigma,{\mathcal{A}}_\sigma)\), we require that \(\mathrm{Re}\mho, \mathrm{Im}\mho\) span a positive definite 2-plane of \(H^*(X,{\mathbb{R}})\), where \(Z_\sigma(\bullet)=\langle \mho,\bullet \rangle\) with \(\mho \in H^*(X,{\mathbb{Q}})_{\operatorname{alg}} \otimes {\mathbb{C}}\). Up to the action of \(\widetilde{\operatorname{GL}}_2^+({\mathbb{R}})\), there is \((\beta,\omega)\) such that \(\sigma=\sigma_{(\beta,\omega)}\). We set \({\mathcal{M}}_{(\beta,\omega)}(v):= {\mathcal{M}}_{\sigma_{(\beta,\omega)}}(v)\). .

Old: \(\sigma \in \operatorname{Stab}(X)\) satisfies (1) \(Z_\sigma(\bullet)=\langle \mho,\bullet \rangle\) with \(\langle \mathrm{Re}\mho^2 \rangle= \langle \mathrm{Im}\mho^2 \rangle\), \(\langle \mathrm{Re}\mho,\mathrm{Im}\mho \rangle=0\) and (2) \(k_x\) (\(x \in X\)) are \(\sigma\)-stable with the phase \(\phi_{\sigma}(k_x)=1\). Then \(\sigma=\sigma_{(\beta,\omega)}\) for some \((\beta,\omega)\).

We collect some results in [5] and [6].

For a fixed Mukai vector \(v \in H^{2*}(X,{\mathbb{Z}})\), there exists a locally finite set of walls (real codimension one submanifolds with boundary) in \({\mathcal{H}}\), depending only on \(v\), with the following properties (see [13]):

1. When \(\sigma\) varies in a chamber, that is, a connected component of the complement of the union of walls, the sets of \(\sigma\)-semistable and \(\sigma\)-stable objects of class \(v\) do not change. If \(v\) is primitive, then \(\sigma\)-stability coincides with \(\sigma\)-semistability for \(\sigma\) in a chamber for \(v\).

2. When \(\sigma\) lies on a wall \(W \subset {\mathcal{H}}\), there is a \(\sigma\)-semistable object of class \(v\) that is unstable in one of the adjacent chambers and semistable in the other adjacent chamber. If \(\sigma= (Z_\sigma, {\mathcal{A}}_\sigma)\) lies on a wall, there exists a \(\sigma\)-semistable object \(E\) of Mukai vector \(v\) and a subobject \(F \subset E\) in \({\mathcal{A}}_\sigma\) with the same \(\sigma\)-slope but \(v(F) \not \in {\mathbb{R}} v\).

3. Given a polarization \(H\in\operatorname{Amp}(X)\) and the Mukai vector \(v\) of an \(H\)-Gieseker semistable sheaf, there exists a chamber \({\mathcal{C}}\) for \(v\), the Gieseker chamber, where the set of \(\sigma\)-semistable objects of class \(v\) coincides with the set of \(H\)-Gieseker semistable sheaves [13].

For a wall \(W\) in \({\mathcal{H}}\), there is \(v_1\) such that \(W= W_{v_1}\). We note that a Mukai vector \(v_1\) defines a wall in the space of stability conditions if and only if \[\label{eq:def-wall} \langle v_1^2 \rangle \geq 0, \langle (v-v_1)^2 \rangle \geq 0, \langle v_1,v-v_1 \rangle>0, \langle v_1^2 \rangle \langle v^2 \rangle>\langle v_1,v \rangle^2\tag{3}\] (see [6]).

3.3 Totally semi-stable walls↩︎

We set \[{\mathcal{I}}_1:=\{ v_1 \mid \langle v_1^2 \rangle=0, \langle v,v_1 \rangle=1 \}.\]

.

A chamber \({\mathcal{C}}\) is a connected component of \[{\mathcal{H}} \setminus \bigcup_{v_1 \in {\mathcal{I}}_1} W_{v_1}\] where \(v_1\) defines a totally semi-stable wall. By [6], \(W_{v_1}\) does not intersect with other walls.

For a totally semi-stable wall \(W_{v_1}\) defined by \(v_1 \in {\mathcal{I}}_1\), we have a decomposition \[\label{eq:tss-eq} v=\ell v_1+v_2, \; \langle v_1^2 \rangle=\langle v_2^2 \rangle=0,\; \langle v_1,v_2 \rangle=1.\tag{4}\]

Then Proposition 19 and Proposition 22 imply that the following corollary holds.

By [5], we have the following result.

4 Preservation of stability.↩︎

4.1 Wall crossing along a line.↩︎

We shall study totally semi-stable walls along the line \[\label{eq:line} {\mathcal{L}}:=\{(0,tH) \mid t>0 \} \subset {\mathcal{H}}.\tag{5}\] We set \[v=(r,\xi,a),\; \xi=dH+D, D \in H^\perp.\] Then \[Z_{(0,tH)}(v)=(rt^2n-a)+2ndt \sqrt{-1},\; n=\frac{(H^2)}{2}.\] Hence \[\label{eq:mu} -\frac{\mathrm{Re}Z_{(0,tH)}(v)}{\mathrm{Im}Z_{(0,tH)}(v)}= \frac{a-rt^2 n}{2ndt}.\tag{6}\]

Assume that \(r>0\) and \((\xi \cdot H)>0\). Let \(W_u\) be a totally semi-stable wall defined by \(u \in {\mathcal{I}}_1\). Thus \[\label{eq:u} v=\ell_1 v_1+\ell_2 v_2, \; \langle v_1^2 \rangle=\langle v_2^2 \rangle=0,\; \langle v_1,v_2 \rangle=1,\; \{ \ell_1,\ell_2 \}=\{ \ell,1 \},\; u \in \{v_1,v_2 \}.\tag{7}\] Assume that \((0,t_0 H) \in W_u\). We take \((0,t_\pm H)\) from adjacent chambers. Thus \[t_0-\epsilon<t_-<t_0 <t_+ <t_0+\epsilon, \;\;(0<\epsilon \ll 1).\] Let us study \({\mathcal{M}}_{(0,t_\pm H)}(v)\). For \(E \in {\mathcal{M}}_{(0,t_+ H)}(v)\), we have an exact triangle \[\label{eq:HNF-} E_1 \to E \to E_2 \to E_1[1]\tag{8}\] which is the Harder-Narasimhan filtration with respect to \(\sigma_{(0,t_- H)}\)-semistability, where \(E_1 \in {\mathcal{M}}_{(0,t_0 H)}(\ell_1 v_1)\) and \(E_2 \in {\mathcal{M}}_{(0,t_0 H)}(\ell_2 v_2)\). We set \[v_i=(r_i,\xi_i,a_i),\;\xi_i=d_i H+D_i,\; D_i \in H^\perp.\] Then \(d_1,d_2>0\) and \[\frac{a_1 d-ad_1}{r_1 d-rd_1}=t^2 \frac{(H^2)}{2}.\]

.

\[\frac{r_1 d-rd_1}{a_1 d-ad_1}=(nt)^{-2} \frac{(\widehat{H}^2)}{2}.\]

We next study the opposite chamber.

By Remark 20, we have a contravariant equivalence \(\Lambda: {\boldsymbol{D}}(X) \to {\boldsymbol{D}}(X)\) such that \(\Lambda(E_1)=E_1\), \(\Lambda(E_2)=E_2\) and \(\Lambda\) induces an isomorphism \[{\mathcal{M}}_{(0,t_+ H)}(v){ \cong \mathcal{M}}_{(0,t_- H)}(v).\] Proposition 24 shows that the birational maps in [2] are the birational maps constructed by wall-crossing along the line \({\mathcal{L}}\) in 5 .

The following proposition will play an important role in the proof of our main result (Theorem 34).

.

\[0 \to E_1 \to E \to E_2 \to 0\] Assume that \(\ell_1=\ell\). Let \(\Phi_{X \to X'}^{{\boldsymbol{E}}^{\vee}}:{\boldsymbol{D}}(X) \to {\boldsymbol{D}}(X')\) be a Fourier-Mukai transform such that \(\Phi_{X \to X'}^{{\boldsymbol{E}}^{\vee}}(E)[1]\) is a torsion free sheaf of rank 1 and \[0 \to \Phi_{X \to X'}^{{\boldsymbol{E}}^{\vee}}(E)[1] \to \Phi_{X \to X'}^{{\boldsymbol{E}}^{\vee}}(E_2)[1] \to \Phi_{X \to X'}^{{\boldsymbol{E}}^{\vee}}(E_1)[2] \to 0.\] Let \(F\) be a subsheaf of \(E\) with \(\frac{(c_1(F) \cdot (H+\eta))}{\operatorname{rk}F}> \frac{(c_1(E) \cdot (H+\eta))}{\operatorname{rk}E}\). We may assume that \(\frac{(c_1(F) \cdot H)}{\operatorname{rk}F}= \frac{(c_1(E) \cdot H)}{\operatorname{rk}E}\) We note that \(F \cap E_1\) and \(E_1/F \cap E_1\) are \(\mu\)-semi-stable sheaves. Hence \(v(E_1 \cap F)=kv_1\). If \(0<k \leq \ell\), then we see that \(\frac{(c_1(F) \cdot (H+\eta))}{\operatorname{rk}F}< \frac{(c_1(E) \cdot (H+\eta))}{\operatorname{rk}E}\). Hence \(F \subset E_1=0\). Then \(F \to E_2\) is isomorphic, which is a contradiction.

For the wall crossing along the line \[{\mathcal{L}}':=\{(0,t' \widehat{H}) \mid t'>0 \},\] Proposition 26 is restated as follows.

4.2 Proof of Theorem 1.↩︎

Let \(E_1\) and \(E_2\) be semi-stable objects in 8 with Mukai vectors \[v(E_i)=v_i=(r_i,\xi_i,a_i),\;(i=1,2).\] We first study the Fourier-Mukai transforms \(\Phi(E_1),\Phi(E_2)\) of \(E_1,E_2\) by using Lemma 6.

.

If \(X\) does not contain an elliptic curve, then \(a_2 \ne 0\).

.

Relation of parameter: For \(v'=(-a,\widehat{\xi},-r)\), we take \(t_1' \geq t_2' \geq 0\). Then \(\frac{1}{t_2}>\frac{1}{t_1}>t_1' \geq t_2' \geq 0\). In particular if there is no totally semistable wall in \(\operatorname{Amp}(X)_{\mathbb{R}}\), then \(t_1'=t_2'=0\).

.

.

We don’t need the following anymore since \(t<t_1\).

4.3 Wall crossing for \({\mathcal{M}}_H(r,dH,a)\).↩︎

.

The choice of polarization is important. In particular if the polarization is very close to the boundary of the ample cone, we need to change the polarization. We show that \(H=\xi\) is a good polarization if \(\xi\) is ample. In this subsection, we treat the case where \(\xi\) is ample and the polarization is \(\xi\). Thus we assume that \(v=(r,dH,a)\). Let us study wall crossing for \(v=(r,dH,a)\) along \({\mathcal{L}}\). We keep the notation in subsection 4.2. Thus \(E_1\) and \(E_2\) be semi-stable objects in 8 with Mukai vectors \[v(E_i)=v_i=(r_i,\xi_i,a_i),\;(i=1,2).\]

.

\[\label{eq:tss-cond} v=\ell_1 v_1+\ell_2 v_2,\;\langle v_1,v_2 \rangle=1,\; \langle v_1^2 \rangle=\langle v_2^2 \rangle=0,\; \{\ell_1,\ell_2 \}=\{ \ell,1 \}\tag{13}\]

We set \[v_1=(r_1,\xi_1,a_1),\;v_2=(r_2,\xi_2,a_2).\] Then we have \[\label{eq:dH1} r=\ell_1 r_1+\ell_2 r_2,\;dH=\ell_1 \xi_1+\ell_2 \xi_2,\; a=\ell_1 a_1+\ell_2 a_2\tag{14}\] and \[\label{eq:dH2} (\xi_1^2)=2r_1 a_1,\;(\xi_2^2)=2r_2 a_2,\; (\xi_1 \cdot \xi_2)=r_1 a_2+r_2 a_1+1.\tag{15}\]

By Lemma 23, \(r_1>0\) and \[\label{eq:d1} r_2 d(\xi_1 \cdot H)-r_1 d(\xi_2 \cdot H)<0,\; a_2 d(\xi_1 \cdot H)-a_1 d(\xi_2 \cdot H)<0.\tag{16}\]

.

Assume that \(v=(r,0,-1)e^{kH}\), where \(rk=d\). If \(k<0\), then \(\Phi_{X \to \widehat{X}}^{{\mathcal{P}}^{\vee}}(E)\) is not a sheaf see [7].

As a corollary, we get the following result which shows that the claim [17] holds for any polarized abelian surface.

.

\[\begin{align} v=&((\ell_1 k_1+\ell_2 k_2)p,\ell_1 \xi_1+\ell_2 \xi_2,(\ell_1 k_1+\ell_2 k_2)q)\\ =& \ell_1(k_1 p,\xi_1,k_1 q)+\ell_2 (k_2 p,\xi_2,k_2 q) \end{align}\] where

\[\gcd(p,q)=1 (p>0),\; (\xi_i^2)=2k_i^2 pq,\; (\xi_1 \cdot \xi_2)=2k_1 k_2 pq+1.\]

\(p,q\) and \(\ell_1 k_1+\ell_2 k_2\) are determined by \(r\) and \(a\).

.

If \(k_1 \ell_1=k_2 \ell_2\), then \(v_1\) shows that \(E\) is a properly semi-stable sheaf.

4.3.1 A relation to [18].↩︎

In [18], we studied the movable cone of a fiber of the albanese map \(M_H(r,dH,a) \to X \times \widehat{X}\) by studying Bridgeland walls, where \(d>0\) and \(a<0\). From the arguments, we can deduce a refinement of Lemma 48 under the assumption \(r \geq 2, d>0,a \geq 2\).

Let \({\mathcal{I}}\) be the set of isotropic Mukai vectors \(u\) such that \(\langle u,v \rangle=1,2\). .

In [18] we classified \(u \in {\mathcal{I}}\) separating \((0,rH,d(H^2))\) and \((-1,\tfrac{-a}{d(H^2)}H,0)\). Thus we assume that \[\label{eq:condition1} \begin{align} 0 > \langle u,(0,rH,d(H^2)) \rangle \langle u,(-1,\tfrac{-a}{d(H^2)}H,0) \rangle =(r(H,\eta)-pd(H^2))(\tfrac{-a}{d(H^2)}(H,\eta)+q). \end{align}\tag{18}\] It is the same as the classification of wall \(W_u\) intersecting \({\mathcal{L}}\). In particular there is no wall unless \(X\) is a product of two elliptic curve and \(d=1\). Under this condition we have .

\[-w_0^{\vee}=(0,rH,(H^2)d),\; -w_1^{\vee}=(-(H^2)d,-aH,0).\] Hence \(\langle w_i,u^{\vee} \rangle=-\langle -w\i^{\vee},u \rangle\). Let \(W\) be a wall and take \((0,t_\pm H)\) from adjacent chambers. If \[\dim ({\mathcal{M}}_{(0,t_\pm H)}(v) \setminus {\mathcal{M}}_{(0,t_\mp H)}(v)) \geq \dim {\mathcal{M}}_{(0,t_\pm H)}(v)-1,\] then there is \(u \in {\mathcal{I}}\) such that \(W=W_u\). It is easy to see that \((0,tH) \in W_u\) if and only if \[\frac{t^2}{2}(H^2) \langle u,(0,rH,d(H^2)) \rangle+\langle u,(-d(H^2),-aH,0) \rangle=0.\] By the proof of [18], \(W_u \cap {\mathcal{L}} =\emptyset\) for all \(u \in {\mathcal{I}}\).

5 Appendix.↩︎

5.1 Relation of Mukai vectors defining totally semi-stable walls.↩︎

Let \(W\) and \(W'\) be totally semi-stable walls with \((0,t_0 H) \in W\) and \((0,t_0' H) \in W'\). We assume that \(t_0<t_0'\) and there is no totally semi-stable wall containing \((0,tH)\) with \(t_0<t<t_0'\). Let \[v=\ell_1 v_1+\ell_2 v_2,\;\langle v_1,v_2 \rangle=1,\; \langle v_1^2 \rangle=\langle v_2^2 \rangle=0,\; \{\ell_1,\ell_2 \}=\{ \ell,1 \}\] and \[v=\ell_1 ' v_1'+\ell_2' v_2',\; \langle v_1',v_2' \rangle=1,\, \langle {v_1'}^2 \rangle=\langle {v_2'}^2 \rangle=0,\; \{\ell_1',\ell_2' \}=\{ \ell,1 \}\] be the corresponding decompositions of \(v\). For a general \(E\), we have an exact sequence \[0 \to E_1 \to E \to E_2 \to 0\] and \[0 \to E_2' \to E \to E_1' \to 0,\] where \(E_i \in {\mathcal{M}}_{(0,tH)}(\ell_i v_i)\) and \(E_i' \in {\mathcal{M}}_{(0,tH)}(\ell_i' v_i')\) are direct sum of stable objects. We set \[\begin{align} v_i=& (r_i,\xi_i,a_i),\; \xi_i=d_i H+D_i,\; D_i \in H^\perp, \; (i=1,2),\\ v_i'=& (r_i',\xi_i',a_i'),\; \xi_i'=d_i' H+D_i',\; D_i' \in H^\perp, \; (i=1,2). \end{align}\] By our assumption \(t_0<t_0'\), we have \[\frac{a_1' d_2'-a_2' d_1'}{r_1' d_2'-r_2' d_1'}>\frac{a_1 d_2-a_2 d_1}{r_1 d_2-r_2 d_1}.\] We shall prove that \[\begin{align} & r_1d_2-r_2 d_1> r_1' d_2'-r_2' d_1'>0,\\ & a_1' d_2'-a_2' d_1' > a_1 d_2-a_2 d_1>0. \end{align}\]

We note that \[r_1 d-r d_1>0,\, r_1' d-r d_1'>0,\; r_1>0,\;r_1'>0\] (Lemma 23).

.

\(r_2 d_1-r_1 d_2<0\) and \(r_2' d_1'-r_1' d_2'<0\).

Then \[\frac{d_2'}{r_2'}<\frac{d_2}{r_2},\;\frac{d_1}{r_1}<\frac{d_1'}{r_1'}.\] \(r_1 d_2-r_2 d_1>r_1' d_2'-r_2' d_1'>0\).

\(\operatorname{Hom}(E_2',E_2) \ne 0\). Hence \(\langle v_2',v_2 \rangle<0\). Since \(\operatorname{Hom}(E_2,E_1')=0\), \(\operatorname{Hom}(E_1,E_1') \ne 0\). Hence \(\langle v_1,v_1' \rangle<0\).

Assume that \(r_2<0\) and \(r_2'<0\). Then \[r_2' d_2-r_2 d_2'>0,\; r_1 d_1'-r_1' d_1>0.\]

In order to prove the other inequality, we prepare some lemmas.

.

References↩︎

---

1. The author is supported by the Grant-in-aid for Scientific Research (No. 21H04429, 23K03053, 26K06742), JSPS↩︎

2. 2010 Mathematics Subject Classification. Primary 14D20.↩︎