Twisted Fourier-Mukai number of a \(K3\) surface. (English) Zbl 1184.14068
The author gives a counting formula for the twisted Fourier-Mukai partners of a projective \(K3\) surface. If \(S\) is a \(K3\) surface and \(\alpha\) a class in the Brauer group of \(S\), one can consider the abelian category \(Coh(S,\alpha)\) of \(\alpha\)-twisted coherent sheaves on \(S\) and its derived category \(D(S,\alpha)\). Given a projective \(K3\) surface \(S\), a (twisted) Fourier-Mukai partner of \(S\) is a projective \(K3\) surface \(S'\) together with an equivalence between \(D(S)\) and \(D(S')\) (resp. \(D(S',\alpha)\) for some \(\alpha\) in \(Br(S')\)).
The number of Fourier-Mukai partners (up to isomorphism) of a \(K3\) surface \(S\) is finite and has been calculated by [S. Hosono, B. H. Lian, K. Oguiso and S.-T. Yau, Contemp. Math. 322, 43–55 (2003; Zbl 1058.14056)]. In the present paper, the author gives a formula for the number of twisted Fourier-Mukai partners of \(S\). This number depends in particular on the order \(d\) of the element \(\alpha\) and coincides with the known result for \(d=1\). As an application, the author considers a \(K3\) surface \(S\) whose Picard group is generated by an element \(H\) of self-intersection \(2n\) and relates the twisted Fourier-Mukai partners of \(S\) to coarse moduli spaces of semistable vector bundles on \(S\). This extends known results in the untwisted case [S. Hosono, B. H. Lian, K. Oguiso and S.-T. Yau, loc. cit. and P. Stellari, Geom. Dedicata 108, 1–14 (2004; Zbl 1072.14043)].
The number of Fourier-Mukai partners (up to isomorphism) of a \(K3\) surface \(S\) is finite and has been calculated by [S. Hosono, B. H. Lian, K. Oguiso and S.-T. Yau, Contemp. Math. 322, 43–55 (2003; Zbl 1058.14056)]. In the present paper, the author gives a formula for the number of twisted Fourier-Mukai partners of \(S\). This number depends in particular on the order \(d\) of the element \(\alpha\) and coincides with the known result for \(d=1\). As an application, the author considers a \(K3\) surface \(S\) whose Picard group is generated by an element \(H\) of self-intersection \(2n\) and relates the twisted Fourier-Mukai partners of \(S\) to coarse moduli spaces of semistable vector bundles on \(S\). This extends known results in the untwisted case [S. Hosono, B. H. Lian, K. Oguiso and S.-T. Yau, loc. cit. and P. Stellari, Geom. Dedicata 108, 1–14 (2004; Zbl 1072.14043)].
Reviewer: Marcello Bernardara (Essen)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
References:
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