Torus fibrations, gerbes, and duality. With an appendix by Dmitry Arinkin.

*(English)*Zbl 1140.14001
Mem. Am. Math. Soc. 901, 90 p. (2008).

If \(X\to B\) is an elliptic fibration with integral fibres and a section, one can identify \(X\) with its compactified relative Jacobian. Using this identification, one has a relative Poincaré sheaf and the induced relative integral functor is an auto-equivalence of the bounded derived category \(D^b(X)\) of coherent sheaves. This result was proven by T. Bridgeland and A. Maciocia [J. Algebr. Geom. 11, No. 4, 629–657 (2002; Zbl 1066.14047)] when \(X\) is smooth and by I. Burban and B. Kreussler [J. Reine Angew. Math. 584, 45–82 (2005; Zbl 1085.14018)] and A. C. López Martín, F. S. de Salas and the reviewer [Adv. Math. 211, 594–620 (2007; Zbl 1118.14022)] in the singular case.

One important consequence is the spectral construction (originally due to Friedman-Morgan-Witten) which gives and equivalence between vector bundles on \(X\) that are semistable of degree zero on the generic fibres and “spectral data”, that is, line bundles on a spectral divisor \(C\) in \(X\), finite over the base.

In many applications, both to physics and mathematics, one has to consider elliptic fibrations \(Y\to B\) without a section (also known as genus one fibrations). In this case the relative compactified Jacobian \(X\to B\) is no longer isomorphic to \(Y\), and it is not a fine moduli space of sheaves on \(X\), so that one does not have a relative Poincaré sheaf. There is also an asymmetry due to the fact that different genus one fibrations may have isomorphic relative compactified Jacobians.

The authors state that string theory gives a solution to this problem, namely that one needs a extra data which corresponds to what is called the B-field in physics. Mathematically, the holomorphic version of this data is encoded in a \(\mathcal O^\times\)-gerbe on \(X\), and a gerby spectral construction can be given as a special case of a gerby Fourier-Mukai transform. Related constructions are due to Căldăraru in the context of the derived categories of twisted sheaves, which can be understood as particular instances of coherent sheaves on gerbes.

For any elliptic fibration \(X\to B\) with a section, the twisted versions of \(X\) are parameterized by the analytic Tate-Shafarevich group \(\text{III}_{an}(X )\), so that any element \(\beta\in \text{III}_{an}(X )\) gives rise to a genus one fibration \(X_\beta \to B\). Moreover, there is a pairing on \(\text{III}_{an}(X )\) valued in \(H^3_{an}(\mathcal O_B^\times)\) such that, if tow elements \((\alpha,\beta)\) of the Tate-Shafarevich group are orthogonal with respect to this pairing (complementary in the terminology used in the paper), then it is possible to construct a gerbe \({}_\alpha X_\beta\) over \(X_\beta\). The derived category \(D^b({}_\alpha X_\beta)\) decomposes as the orthogonal sum of “pure weights” subcategories \(D^b_k({}_\alpha X_\beta)\) labeled by the integer numbers.

In the paper the following conjecture is stated: For any complementary pair as above, there exists an equivalence \(D^b_1({}_\alpha X_\beta)\simeq D^b_{-1}({}_\beta X_\alpha)\). The conjecture is proven in the paper in the following cases:

1) When \(X\) is a smooth surface and the fibres of \(X\to B\) are integral. In this case any pair \((\alpha,\beta)\) of elements of the Tate-Shafarevich is complementary.

2) For higher dimensional \(X\), when \(B\) is smooth with trivial Brauer group and \(X\to B\) is a smooth morphism. In this case it is also required that \(\alpha\) is \(m\)-divisible and \(\beta\) is \(m\)-torsion for some integer \(m\) (so that they are complementary). The second result is easier than the first and it proved using particular presentations (the lifting presentation and the extension presentation) of the gerbes and then inducing an integral functor between the derived categories of the presentations which is an equivalence and is compatible with the relations so that descends to an equivalence between the derived categories of the gerbes. The fact that the integral functor between presentations is an equivalence is proved using the Bondal-Orlov-Bridgeland criteria for smooth schemes. The second result is considered by Arinkin in the appendix with different techniques in the framework of commutative group stacks. The conjecture has been proved recently by O. Ben-Bassat [Twisting Derived Equivalences, arXiv:math/0606631] for higher dimensional \(X\) when the fibres of \(X\to B\) are integral.

One important consequence is the spectral construction (originally due to Friedman-Morgan-Witten) which gives and equivalence between vector bundles on \(X\) that are semistable of degree zero on the generic fibres and “spectral data”, that is, line bundles on a spectral divisor \(C\) in \(X\), finite over the base.

In many applications, both to physics and mathematics, one has to consider elliptic fibrations \(Y\to B\) without a section (also known as genus one fibrations). In this case the relative compactified Jacobian \(X\to B\) is no longer isomorphic to \(Y\), and it is not a fine moduli space of sheaves on \(X\), so that one does not have a relative Poincaré sheaf. There is also an asymmetry due to the fact that different genus one fibrations may have isomorphic relative compactified Jacobians.

The authors state that string theory gives a solution to this problem, namely that one needs a extra data which corresponds to what is called the B-field in physics. Mathematically, the holomorphic version of this data is encoded in a \(\mathcal O^\times\)-gerbe on \(X\), and a gerby spectral construction can be given as a special case of a gerby Fourier-Mukai transform. Related constructions are due to Căldăraru in the context of the derived categories of twisted sheaves, which can be understood as particular instances of coherent sheaves on gerbes.

For any elliptic fibration \(X\to B\) with a section, the twisted versions of \(X\) are parameterized by the analytic Tate-Shafarevich group \(\text{III}_{an}(X )\), so that any element \(\beta\in \text{III}_{an}(X )\) gives rise to a genus one fibration \(X_\beta \to B\). Moreover, there is a pairing on \(\text{III}_{an}(X )\) valued in \(H^3_{an}(\mathcal O_B^\times)\) such that, if tow elements \((\alpha,\beta)\) of the Tate-Shafarevich group are orthogonal with respect to this pairing (complementary in the terminology used in the paper), then it is possible to construct a gerbe \({}_\alpha X_\beta\) over \(X_\beta\). The derived category \(D^b({}_\alpha X_\beta)\) decomposes as the orthogonal sum of “pure weights” subcategories \(D^b_k({}_\alpha X_\beta)\) labeled by the integer numbers.

In the paper the following conjecture is stated: For any complementary pair as above, there exists an equivalence \(D^b_1({}_\alpha X_\beta)\simeq D^b_{-1}({}_\beta X_\alpha)\). The conjecture is proven in the paper in the following cases:

1) When \(X\) is a smooth surface and the fibres of \(X\to B\) are integral. In this case any pair \((\alpha,\beta)\) of elements of the Tate-Shafarevich is complementary.

2) For higher dimensional \(X\), when \(B\) is smooth with trivial Brauer group and \(X\to B\) is a smooth morphism. In this case it is also required that \(\alpha\) is \(m\)-divisible and \(\beta\) is \(m\)-torsion for some integer \(m\) (so that they are complementary). The second result is easier than the first and it proved using particular presentations (the lifting presentation and the extension presentation) of the gerbes and then inducing an integral functor between the derived categories of the presentations which is an equivalence and is compatible with the relations so that descends to an equivalence between the derived categories of the gerbes. The fact that the integral functor between presentations is an equivalence is proved using the Bondal-Orlov-Bridgeland criteria for smooth schemes. The second result is considered by Arinkin in the appendix with different techniques in the framework of commutative group stacks. The conjecture has been proved recently by O. Ben-Bassat [Twisting Derived Equivalences, arXiv:math/0606631] for higher dimensional \(X\) when the fibres of \(X\to B\) are integral.

Reviewer: Daniel Hernandez Ruiperez (Salamanca)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

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

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

14A20 | Generalizations (algebraic spaces, stacks) |

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

14J81 | Relationships between surfaces, higher-dimensional varieties, and physics |