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**On the strange duality conjecture for abelian surfaces.**
*(English)*
Zbl 1322.14063

The authors prove two of the three versions of Le Potier’s strange duality conjectures in the case of abelian surfaces \(X=F\times B\) that decompose as (unpolarised) products of elliptic curves. These conjectures assert that (perhaps under weak conditions such as rank and degree being coprime) there is a jumping locus over the product \({\mathfrak M}^+_v\times{\mathfrak M}^+_w\) of moduli spaces of semi-stable sheaves with orthogonal Mukai vectors \(v\) and \(w\), and that this induces an isomorphism between the spaces sections of the determinant line bundles over the two moduli spaces. Furthermore, a similar dual statement holds on the moduli spaces of sheaves with fixed determinant of their Fourier-Mukai transforms.

Moduli spaces of sheaves on simply-connected elliptic surfaces are essentially Hilbert schemes, as was shown by T. Bridgeland [J. Reine Angew. Math. 498, 115–133 (1998; Zbl 0905.14020)], but the situation is slightly more complicated in this non-simply connected case. The first result of this paper describes the moduli spaces birationally: for example, \({\mathfrak M}_v\simeq \{(Z,b)\in X^{[d_v]}\times B\mid a_b(Z)=rb\}\), where \(a_b: X^{[d_v]}\to B\) is addition followed by projection on \(B\) and \(d_v=(v,v)/2\) is half the dimension of the moduli space. Then it is shown that the expected isomorphism between spaces of sections is obtained, in the case of fibre degree \(1\) and rank at least \(3\), and similarly for the Fourier-Mukai transformed version.

In fact this result still holds, with minor changes and extra conditions, even if \(B\) is replaced by a curve of higher genus.

The main technical difficulties are in establishing the first result, describing the moduli spaces. For fibre degree \(1\) there are two methods available, one based on K. G. O’Grady’s [J. Algebr. Geom. 6, No. 4, 599–644 (1997; Zbl 0916.14018)] description of the moduli space and one using Fourier-Mukai methods. The Fourier-Mukai methods, together with the results of Bridgeland, allow one to establish the result for arbitrary fibre degree coprime to the rank. The duality isomorphisms are established by giving explicit descriptions of the determinant bundles.

Moduli spaces of sheaves on simply-connected elliptic surfaces are essentially Hilbert schemes, as was shown by T. Bridgeland [J. Reine Angew. Math. 498, 115–133 (1998; Zbl 0905.14020)], but the situation is slightly more complicated in this non-simply connected case. The first result of this paper describes the moduli spaces birationally: for example, \({\mathfrak M}_v\simeq \{(Z,b)\in X^{[d_v]}\times B\mid a_b(Z)=rb\}\), where \(a_b: X^{[d_v]}\to B\) is addition followed by projection on \(B\) and \(d_v=(v,v)/2\) is half the dimension of the moduli space. Then it is shown that the expected isomorphism between spaces of sections is obtained, in the case of fibre degree \(1\) and rank at least \(3\), and similarly for the Fourier-Mukai transformed version.

In fact this result still holds, with minor changes and extra conditions, even if \(B\) is replaced by a curve of higher genus.

The main technical difficulties are in establishing the first result, describing the moduli spaces. For fibre degree \(1\) there are two methods available, one based on K. G. O’Grady’s [J. Algebr. Geom. 6, No. 4, 599–644 (1997; Zbl 0916.14018)] description of the moduli space and one using Fourier-Mukai methods. The Fourier-Mukai methods, together with the results of Bridgeland, allow one to establish the result for arbitrary fibre degree coprime to the rank. The duality isomorphisms are established by giving explicit descriptions of the determinant bundles.

Reviewer: G. K. Sankaran (Bath)

### MSC:

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14K99 | Abelian varieties and schemes |

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\textit{A. Marian} and \textit{D. Oprea}, J. Eur. Math. Soc. (JEMS) 16, No. 6, 1221--1252 (2014; Zbl 1322.14063)

### References:

[1] | Atiyah, M. F.: Vector bundles over an elliptic curve. Proc. London Math. Soc. 7, 414-452 (1957) · Zbl 0084.17305 · doi:10.1112/plms/s3-7.1.414 |

[2] | Bridgeland, T: Fourier-Mukai transforms for elliptic surfaces. J. Reine Angew. Math. 498, 115-133 (1998) · Zbl 0905.14020 · doi:10.1515/crll.1998.046 |

[3] | Bernardara, M., Hein, G: The Euclid-Fourier-Mukai algorithm for elliptic surfaces. · Zbl 1311.14044 |

[4] | Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles. Springer, New York (1998) · Zbl 0902.14029 |

[5] | Le Potier, J.: Dualité étrange sur le plan projectif. Lectures at Luminy (1996) |

[6] | O’Grady, K.: The weight-two Hodge structure of moduli spaces of sheaves on a K3 sur- face. J. Algebraic Geom. 6, 599-644 (1997) · Zbl 0916.14018 |

[7] | Marian, A., Oprea, D.: A tour of theta dualities on moduli spaces of sheaves. In: Curves and Abelian Varieties, Contemp. Math. 465, Amer. Math. Soc., Providence, RI, 175-202 (2008) · Zbl 1149.14301 |

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