##
**The space of stability conditions on the local projective plane.**
*(English)*
Zbl 1238.14014

It is a major problem to study the space of Bridgeland stability conditions on the derived category \(D^{b}(Y)\) of a projective Calabi-Yau threefold \(Y\) containing a projective plane. In the paper under review, the authors study the space of stability conditions on the full subcategory \(D^{b}_{\mathbb{P}^{2}}(Y)\) of complexes concentrated on \(\mathbb{P}^{2}\). Several examples of stability conditions on local Calabi-Yau manifolds have been studied so far, like R. Thomas [Commun. Anal. Geom. 14, No. 1, 135–161 (2006; Zbl 1179.53084)], T. Bridgeland [Int. Math. Res. Not. 2009, No. 21, 4142–4157 (2009; Zbl 1228.14012)], A. Ishii and H. Uehara [J. Differ. Geom. 71, No. 3, 385–435 (2005; Zbl 1097.14013)] or E. Macri, S. Mehrotra and P. Stellari [J. Algebr. Geom. 18, No. 4, 605–649 (2009; Zbl 1175.14010)].

The local model of this situation is given by the total space \(X\) of the canonical bundle of \(\mathbb{P}^{2}\), and \(D^{b}_{\mathbb{P}^{2}}(Y)\) corresponds to the full subcategory \(\mathcal{D}_{0}\) of \(D^{b}(X)\) of complexes supported on the zero section. The space \(Stab(\mathcal{D}_{0})\) of Bridgeland stability conditions on \(\mathcal{D}_{0}\), which is a \(3-\)dimensional manifold with a local homeomorphism \(\mathcal{Z}:Stab(\mathcal{D}_{0})\longrightarrow\mathbb{C}^{3}\), was first studied by T. Bridgeland in [Commun. Math. Phys. 266, No. 3, 715–733 (2006; Zbl 1118.14045)]. The authors’ aim is to study a connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\), and to deduce from this some consequences about autoequivalences of \(\mathcal{D}_{0}\) and mirror symmetry.

In section 2 the authors introduce the notion of geometric stability condition, which is a stability condition \(\sigma\in\mathcal{D}_{0}\) for which all the skyscraper sheaves \(k(x)\) of closed points \(x\in\mathbb{P}^{2}\) are \(\sigma-\)stable and of the same slope, and such that the connected component of \(Stab(\mathcal{D}_{0})\) containing \(\sigma\) ha maximal dimension 3. The set \(U\) of all geometric stability conditions in \(Stab(\mathcal{D}_{0})\) is called geometric chamber.

The first goal is to produce a geometric stability condition: this is done by following the approach of T. Bridgeland [Duke Math. J. 141, No. 2, 241–291 (2008; Zbl 1138.14022)], and by using a reformulation of a result of existence of slope-stable sheaves on \(\mathbb{P}^{2}\) which is due to J.-M. Drezet and J. Le Potier [Ann. Sci. Ec. Norm. Super. (4) 18, 193–243 (1985; Zbl 0586.14007)]. This last result is explained in Appendix A.

First, the authors show that any geometric stability condition on \(\mathcal{D}_{0}\) is (up to the action of a unique element in \(\mathbb{C}\)) necessarily of the form \(\sigma_{a,b}=(Z_{a,b},\mathcal{P}_{a,b})\), where \(Z_{a,b}:K(\mathcal{D}_{0})\longrightarrow\mathbb{C}\) is the central charge defined on the Grothendieck group of \(\mathcal{D}_{0}\) by letting \(Z_{a,b}(E):=-c(E)+ad(E)+br(E)\). Here \(r(E)\), \(d(E)\) and \(c(E)\) are the rank and the Chern classes of \(E\), and \((a,b)\in\mathbb{C}^{2}\) verify some inequalities (see Definition 2.4) which are motivated by the result of Drezet and Le Potier. The set of pairs \((a,b)\in\mathbb{C}^{2}\) satisfying these inequalities is denoted \(G\).

The second step is to show that geometric stability conditions exist: the main problem is to show that the stability function \(Z_{a,b}\), where \((a,b)\in G\), takes values in the upper half complex plane \(\mathbb{H}\) and has a Harder-Narasimhan filtration. While the first of these two properties is easy to check, the Harder-Narasimhan filtration is first shown in the case where \(a\) and \(b\) are rational, then extended to every \((a,b)\in G\) using a deformation result due to Bridgeland.

Once the geometric chamber \(U\) is described, the authors study the boundary \(\partial U\) of \(U\). More precisely, they show that every exceptional vector bundle \(\mathcal{E}\) determines two codimension-one walls \(W^{+}_{\mathcal{E}}\) and \(W^{-}_{\mathcal{E}}\) in \(\partial U\), and that the family of all the possible \(W^{+}_{\mathcal{E}}\) and \(W^{-}_{\mathcal{E}}\) gives \(\partial U\). As a consequence of this, the connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\) containing \(U\) is covered by the translates of the closure \(\overline{U}\) of \(U\) under the group of autoequivalences of \(\mathcal{D}_{0}\) generated by the spherical twists.

A further step is to study the open subset \(Stab_{a}\) of \(Stab(\mathcal{D}_{0})\) of algebraic stability conditions, which is indeed an open subset of \(Stab^{+}(\mathcal{D}_{0})\): its complement is the union of all the \(\Phi(U)\), where \(\Phi\) is an autoequivalence of \(\mathcal{D}_{0}\) in the subgroup generated by the spherical twists associated to exceptional vector bundles on \(\mathbb{P}^{2}\). The properties of \(Stab_{a}\) are used to show that \(Stab^{+}(\mathcal{D}_{0})\) is simply connected: this is reduced to show that \(Stab_{a}\) is simply connected using a simple topological argument. Then the authors show that \(Stab_{a}\) is simply connected by associating to any loop in \(Stab_{a}\) a word in the generators of the affine braid group \(B_{3}\).

As a consequence, the authors calculate the group \(Aut^{+}(\mathcal{D}_{0})\) of autoequivalences of \(\mathcal{D}_{0}\) preserving the connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\). More precisely, they show that \(Aut^{+}(\mathcal{D}_{0})\) is isomorphic to the product \(\mathbb{Z}\times\Gamma_{1}(3)\times Aut(\widehat{X})\): here \(\mathbb{Z}\) is identified with the subgroup generated by the shift \([1]\), \(\Gamma_{1}(3)\) is a congruence subgroup of \(SL(2,\mathbb{Z})\) identified with the subgroup generated by the tensor product with \(\mathcal{O}_{X}(1)\) and by the spherical twist at the structure sheaf \(\mathcal{O}_{\mathbb{P}^{2}}\) of the zero section of \(X\), and \(Aut(\widehat{X})\) is the group of automorphisms of the formal completion \(\widehat{X}\) of \(X\) along \(\mathbb{P}^{2}\).

As conclusion of the paper, the authors outline the relation with mirror symmetry for the local \(\mathbb{P}^{2}\): the mirror partner for the local \(\mathbb{P}^{2}\) is the universal family over the moduli space \(\mathcal{M}_{\Gamma_{1}(3)}\) of elliptic curves with \(\Gamma_{1}(3)-\)level structure, and \(\Gamma_{1}(3)\) is its fundamental group. The authors show that the universal cover \(\widetilde{\mathcal{M}}_{\Gamma_{1}(3)}\) of \(\mathcal{M}_{\Gamma_{1}(3)}\) with \(\Gamma_{1}(3)\) acting as the group of deck transformations has an embedding in \(Stab^{+}(\mathcal{D}_{0})\) which is \(\Gamma_{1}(3)-\)equivariant.

The local model of this situation is given by the total space \(X\) of the canonical bundle of \(\mathbb{P}^{2}\), and \(D^{b}_{\mathbb{P}^{2}}(Y)\) corresponds to the full subcategory \(\mathcal{D}_{0}\) of \(D^{b}(X)\) of complexes supported on the zero section. The space \(Stab(\mathcal{D}_{0})\) of Bridgeland stability conditions on \(\mathcal{D}_{0}\), which is a \(3-\)dimensional manifold with a local homeomorphism \(\mathcal{Z}:Stab(\mathcal{D}_{0})\longrightarrow\mathbb{C}^{3}\), was first studied by T. Bridgeland in [Commun. Math. Phys. 266, No. 3, 715–733 (2006; Zbl 1118.14045)]. The authors’ aim is to study a connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\), and to deduce from this some consequences about autoequivalences of \(\mathcal{D}_{0}\) and mirror symmetry.

In section 2 the authors introduce the notion of geometric stability condition, which is a stability condition \(\sigma\in\mathcal{D}_{0}\) for which all the skyscraper sheaves \(k(x)\) of closed points \(x\in\mathbb{P}^{2}\) are \(\sigma-\)stable and of the same slope, and such that the connected component of \(Stab(\mathcal{D}_{0})\) containing \(\sigma\) ha maximal dimension 3. The set \(U\) of all geometric stability conditions in \(Stab(\mathcal{D}_{0})\) is called geometric chamber.

The first goal is to produce a geometric stability condition: this is done by following the approach of T. Bridgeland [Duke Math. J. 141, No. 2, 241–291 (2008; Zbl 1138.14022)], and by using a reformulation of a result of existence of slope-stable sheaves on \(\mathbb{P}^{2}\) which is due to J.-M. Drezet and J. Le Potier [Ann. Sci. Ec. Norm. Super. (4) 18, 193–243 (1985; Zbl 0586.14007)]. This last result is explained in Appendix A.

First, the authors show that any geometric stability condition on \(\mathcal{D}_{0}\) is (up to the action of a unique element in \(\mathbb{C}\)) necessarily of the form \(\sigma_{a,b}=(Z_{a,b},\mathcal{P}_{a,b})\), where \(Z_{a,b}:K(\mathcal{D}_{0})\longrightarrow\mathbb{C}\) is the central charge defined on the Grothendieck group of \(\mathcal{D}_{0}\) by letting \(Z_{a,b}(E):=-c(E)+ad(E)+br(E)\). Here \(r(E)\), \(d(E)\) and \(c(E)\) are the rank and the Chern classes of \(E\), and \((a,b)\in\mathbb{C}^{2}\) verify some inequalities (see Definition 2.4) which are motivated by the result of Drezet and Le Potier. The set of pairs \((a,b)\in\mathbb{C}^{2}\) satisfying these inequalities is denoted \(G\).

The second step is to show that geometric stability conditions exist: the main problem is to show that the stability function \(Z_{a,b}\), where \((a,b)\in G\), takes values in the upper half complex plane \(\mathbb{H}\) and has a Harder-Narasimhan filtration. While the first of these two properties is easy to check, the Harder-Narasimhan filtration is first shown in the case where \(a\) and \(b\) are rational, then extended to every \((a,b)\in G\) using a deformation result due to Bridgeland.

Once the geometric chamber \(U\) is described, the authors study the boundary \(\partial U\) of \(U\). More precisely, they show that every exceptional vector bundle \(\mathcal{E}\) determines two codimension-one walls \(W^{+}_{\mathcal{E}}\) and \(W^{-}_{\mathcal{E}}\) in \(\partial U\), and that the family of all the possible \(W^{+}_{\mathcal{E}}\) and \(W^{-}_{\mathcal{E}}\) gives \(\partial U\). As a consequence of this, the connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\) containing \(U\) is covered by the translates of the closure \(\overline{U}\) of \(U\) under the group of autoequivalences of \(\mathcal{D}_{0}\) generated by the spherical twists.

A further step is to study the open subset \(Stab_{a}\) of \(Stab(\mathcal{D}_{0})\) of algebraic stability conditions, which is indeed an open subset of \(Stab^{+}(\mathcal{D}_{0})\): its complement is the union of all the \(\Phi(U)\), where \(\Phi\) is an autoequivalence of \(\mathcal{D}_{0}\) in the subgroup generated by the spherical twists associated to exceptional vector bundles on \(\mathbb{P}^{2}\). The properties of \(Stab_{a}\) are used to show that \(Stab^{+}(\mathcal{D}_{0})\) is simply connected: this is reduced to show that \(Stab_{a}\) is simply connected using a simple topological argument. Then the authors show that \(Stab_{a}\) is simply connected by associating to any loop in \(Stab_{a}\) a word in the generators of the affine braid group \(B_{3}\).

As a consequence, the authors calculate the group \(Aut^{+}(\mathcal{D}_{0})\) of autoequivalences of \(\mathcal{D}_{0}\) preserving the connected component \(Stab^{+}(\mathcal{D}_{0})\) of \(Stab(\mathcal{D}_{0})\). More precisely, they show that \(Aut^{+}(\mathcal{D}_{0})\) is isomorphic to the product \(\mathbb{Z}\times\Gamma_{1}(3)\times Aut(\widehat{X})\): here \(\mathbb{Z}\) is identified with the subgroup generated by the shift \([1]\), \(\Gamma_{1}(3)\) is a congruence subgroup of \(SL(2,\mathbb{Z})\) identified with the subgroup generated by the tensor product with \(\mathcal{O}_{X}(1)\) and by the spherical twist at the structure sheaf \(\mathcal{O}_{\mathbb{P}^{2}}\) of the zero section of \(X\), and \(Aut(\widehat{X})\) is the group of automorphisms of the formal completion \(\widehat{X}\) of \(X\) along \(\mathbb{P}^{2}\).

As conclusion of the paper, the authors outline the relation with mirror symmetry for the local \(\mathbb{P}^{2}\): the mirror partner for the local \(\mathbb{P}^{2}\) is the universal family over the moduli space \(\mathcal{M}_{\Gamma_{1}(3)}\) of elliptic curves with \(\Gamma_{1}(3)-\)level structure, and \(\Gamma_{1}(3)\) is its fundamental group. The authors show that the universal cover \(\widetilde{\mathcal{M}}_{\Gamma_{1}(3)}\) of \(\mathcal{M}_{\Gamma_{1}(3)}\) with \(\Gamma_{1}(3)\) acting as the group of deck transformations has an embedding in \(Stab^{+}(\mathcal{D}_{0})\) which is \(\Gamma_{1}(3)-\)equivariant.

Reviewer: Arvid Perego (Nancy)

### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

18E30 | Derived categories, triangulated categories (MSC2010) |

### Citations:

Zbl 1179.53084; Zbl 1228.14012; Zbl 1097.14013; Zbl 1175.14010; Zbl 1118.14045; Zbl 1138.14022; Zbl 0586.14007### Software:

SageMath
PDF
BibTeX
XML
Cite

\textit{A. Bayer} and \textit{E. Macrì}, Duke Math. J. 160, No. 2, 263--322 (2011; Zbl 1238.14014)

### References:

[1] | M. Aganagic, V. Bouchard, and A. Klemm, Topological strings and (almost) modular forms , Comm. Math. Phys. 277 (2008), 771-819. · Zbl 1165.81037 |

[2] | D. Arcara, A.Bertram, and M. Lieblich, Bridgeland-stable moduli spaces for K-trivial surfaces , preprint, [math.AG] · Zbl 1259.14014 |

[3] | P. S. Aspinwall, “D-branes on Calabi-Yau manifolds” in Progress in String Theory , World Sci., Hackensack, N.J., 2005, 1-152. · Zbl 1084.81058 |

[4] | P. S. Aspinwall and M. R. Douglas, D-brane stability and monodromy , J. High Energy Phys. 5 (2002), 35. |

[5] | P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Measuring small distances in N = 2 sigma models , Nuclear Phys. B 420 (1994), 184-242. · Zbl 0990.81689 |

[6] | A. Bayer, Polynomial Bridgeland stability conditions and the large volume limit , Geom. Topol. 13 (2009), 2389-2425. · Zbl 1171.14011 |

[7] | A. I. Bondal, Representations of associative algebras and coherent sheaves , Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 25-44. · Zbl 0692.18002 |

[8] | C. Brav and H. Thomas, Braid groups and Kleinian singularities , preprint, [math.AG] · Zbl 1264.14026 |

[9] | T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms , Bull. London Math. Soc. 31 (1999), 25-34. · Zbl 0937.18012 |

[10] | T. Bridgeland, t-structures on some local Calabi-Yau varieties , J. Algebra 289 (2005), 453-483. · Zbl 1069.14044 |

[11] | T. Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold , Comm. Math. Phys. 266 (2006), 715-733. · Zbl 1118.14045 |

[12] | T. Bridgeland, Stability conditions on triangulated categories , Ann. of Math. (2) 166 (2007), 317-345. · Zbl 1137.18008 |

[13] | T. Bridgeland, Stability conditions on K 3 surfaces , Duke Math. J. 141 (2008), 241-291. · Zbl 1138.14022 |

[14] | T. Bridgeland, “Spaces of stability conditions” in Algebraic Geometry-Seattle 2005 , part 1, Proc. Sympos. Pure Math. 80 , Amer. Math. Soc., Providence, 2009, 1-21. · Zbl 1169.14303 |

[15] | T. Bridgeland, Stability conditions and Kleinian singularities , Int. Math. Res. Not. 2009 , no. 21, 4142-4157. · Zbl 1228.14012 |

[16] | T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories , J. Amer. Math. Soc. 14 (2001), 535-554. · Zbl 0966.14028 |

[17] | T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for K 3 and elliptic fibrations , J. Algebraic Geom. 11 (2002), 629-657. · Zbl 1066.14047 |

[18] | A. Canonaco and P. Stellari, Fourier-Mukai functors in the supported case , preprint, [math.AG] · Zbl 1326.14039 |

[19] | T. Coates, On the crepant resolution conjecture in the local case , Comm. Math. Phys. 287 (2009), 1071-1108. · Zbl 1200.53081 |

[20] | T. Coates, A. Corti, H. Iritani, and H.-H. Tseng, Computing genus-zero twisted Gromov-Witten invariants , Duke Math. J. 147 (2009), 377-438. · Zbl 1176.14009 |

[21] | A. Craw and A. Ishii, Flops of G -Hilb and equivalences of derived categories by variation of GIT quotient , Duke Math. J. 124 (2004), 259-307. · Zbl 1082.14009 |

[22] | D.-E. Diaconescu and J. Gomis, Fractional branes and boundary states in orbifold theories , J. High Energy Phys. 2000 , no. 10, paper 1. · Zbl 0965.81055 |

[23] | M. R. Douglas, “Dirichlet branes, homological mirror symmetry, and stability” in Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) , Beijing, Higher Ed. Press, 2002, 395-408. · Zbl 1008.81074 |

[24] | J.-M. Drezet and J. Le Potier, Fibrés stables et fibrés exceptionnels sur P 2 , Ann. Sci. École Norm. Sup. (4) 18 (1985), 193-243. · Zbl 0586.14007 |

[25] | A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces , Duke Math. J. 54 (1987), 115-130. · Zbl 0646.14014 |

[26] | D. Happel, I. Reiten, and O. SmaløSverre, Tilting in abelian categories and quasitilted algebras , Mem. Amer. Math. Soc. 120 (1996), no. 575. · Zbl 0849.16011 |

[27] | R. P. Horja, Derived category automorphisms from mirror symmetry , Duke Math. J. 127 (2005), 1-34. · Zbl 1075.18006 |

[28] | D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry , Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, Oxford, 2006. · Zbl 1095.14002 |

[29] | D. Huybrechts, E. Macrî, and P. Stellari, Stability conditions for generic K 3 categories , Compos. Math. 144 (2008), 134-162. · Zbl 1152.14037 |

[30] | H. Iritani, “Ruan’s conjecture and integral structures in quantum cohomology” in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008) , Adv. Stud. Pure Math. 59 , Math. Soc. Japan, Tokyo, 2008, 111-166. · Zbl 1231.14046 |

[31] | A. Ishii, K.Ueda, and H. Uehara, Stability conditions on A n - singularities , J. Differential Geom. 84 (2010), 87-126. · Zbl 1198.14020 |

[32] | A. Ishii and H. Uehara, Autoequivalences of derived categories on the minimal resolutions of A n - singularities on surfaces , J. Differential Geom. 71 (2005), 385-435. · Zbl 1097.14013 |

[33] | A. D. King, Moduli of representations of finite-dimensional algebras , Quart. J. Math. Oxford Ser. 45 (1994), 515-530. · Zbl 0837.16005 |

[34] | M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations , preprint, [math.AG] · Zbl 1214.14014 |

[35] | J. Le Potier, Lectures on Vector Bundles , Cambridge Stud. Adv. Math. 54 , Cambridge Univ. Press, Cambridge, 1997. · Zbl 0872.14003 |

[36] | V. A. Lunts and D. O. Orlov, Uniqueness of enhancement for triangulated categories , J. Amer. Math. Soc. 23 (2010), 853-908. · Zbl 1197.14014 |

[37] | E. Macrî, Stability conditions on curves , Math. Res. Lett. 14 (2007), 657-672. · Zbl 1151.14015 |

[38] | E. Macrî, S. Mehrotra, and P. Stellari, Inducing stability conditions , J. Algebraic Geom. 18 (2009), 605-649. · Zbl 1175.14010 |

[39] | R. Ohkawa, Moduli of Bridgeland semistable objects on \Bbb P 2 , Kodai Math. J. 33 (2008), 329-366. · Zbl 1198.14040 |

[40] | S. Okada, On stability manifolds of Calabi-Yau surfaces , Int. Math. Res. Not. 2006 , no. 16, Art. ID 58743. · Zbl 1122.14015 |

[41] | R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category , Invent. Math. 178 (2009), 407-447. · Zbl 1204.14026 |

[42] | A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves , Mosc. Math. J. 7 (2007), 109-134. · Zbl 1126.14021 |

[43] | A. N. Rudakov, A description of Chern classes of semistable sheaves on a quadric surface , J. Reine Angew. Math. 453 (1994), 113-135. · Zbl 0805.14023 |

[44] | A. N. Rudakov, Versal families and the existence of stable sheaves on a del Pezzo surface , J. Math. Sci. Univ. Tokyo 3 (1996), 495-532. · Zbl 0883.14021 |

[45] | P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves , Duke Math. J. 108 (2001), 37-108. · Zbl 1092.14025 |

[46] | W. A. Stein, Sage Mathematics Software (Version 4.1) , the Sage Development Team, 2009, available at . |

[47] | R. P. Thomas, “Stability conditions and the braid group” in Superstring Theory , Adv. Lect. Math. 1 , Int. Press, Somerville, Mass., 2008, 209-233. · Zbl 1179.53084 |

[48] | Y. Toda, “Generating functions of stable pair invariants via wall-crossings in derived categories” in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008) , Adv. Stud. Pure Math. 59 , Math. Soc. Japan, Tokyo, 2008, 389-434. · Zbl 1216.14009 |

[49] | Y. Toda, Stability conditions and crepant small resolutions , Trans. Amer. Math. Soc. 360 (2008), 6149-6178. · Zbl 1225.14030 |

[50] | Y. Toda, Stability conditions and Calabi-Yau fibrations , J. Algebraic Geom. 18 (2009), 101-133. · Zbl 1157.14025 |

[51] | K. Yoshioka, Stability and the Fourier-Mukai transform, II , Compos. Math. 145 (2009), 112-142. · Zbl 1165.14033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.