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.


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)


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