Twisted Fourier–Mukai transforms for holomorphic symplectic four-folds

Abstract

We apply the methods of Cȃldȃraru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its 'mirror' partner. As corollaries, we show that the two spaces are connected by a one-parameter family of deformations through Lagrangian fibrations, and we extend the original Fourier-Mukai transform to degenerations of abelian surfaces. * 2000 Mathematics Subject Classification. 14J60; 14D06; 18E30; 53C26. and Thaddeus' examples via the construction of Donagi, Ein, and Lazarsfeld [15] (namely, the compactified Hitchin system is a degeneration of the Beauville-Mukai system).The paper is organized as follows. In Section 2 we review results of Mukai, Bridgeland, Maciocia, and Cȃldȃraru on Fourier-Mukai and twisted Fourier-Mukai transforms. In Section 3 we introduce a pair of holomorphic symplectic four-folds which are fibred by abelian varieties and collect together some results about them. In Section 4 we construct a twisted Fourier-Mukai transform relating the derived category and twisted derived category of the pair of four-folds from Section 3. This is followed by our applications. This paper was begun during a visit to the Institut des HautesÉtudes Scientifiques and finished at the University of Kyoto; the author is grateful for the hospitality he received at both those places. The author has benefited from conversations with many people on the topics presented here: he thanks them all, particularly

Twisting derived equivalences

“…The conjectures made in [19] form the main geometric motivation for this paper, although the methods of proof will be closer to those found in [14]. More recently, these kinds of results have been used and expanded upon by other authors; for example see [11], [13], and [31]. An alternative, and in many ways stronger, version of the main theorem can be found as Theorem 5.12.…”

Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

“…It follows from Arinkin's autoduality theorem [5,Theorem B] that the moduli space Pic 0 (Pic n /PV ) is isomorphic to Pic 0 = Pic 0 (C/PV ) for any n. Inspecting the formula for the (1 α −n 0 )-twisted sheafP n0 on Pic n × PV Pic 0 , we see that it has degree 0 on the fibers of the projection to Pic 0 , and indeed that it is universal for this moduli problem; thus the Brauer class for this moduli problem is α −n 0 . Since α 1−g 0 = 1 but Pic g−1 ∼ = Pic 0 , this disagrees with a proposition of Sawon [40,Proposition 9]. He considers abelian fibrations X → B satisfying certain hypotheses, and lets P = Pic 0 (X/B) and X 0 = Pic 0 (P/B).…”

On the discriminant locus of a Lagrangian fibration

“…The reason is that there are ways to deform a Lagrangian fibration until it admits a section (see [23] and [24]) without changing the local structure of the fibration, and in particular, without changing the discriminant locus ∆. Now a Lagrangian fibration is projective if and only if it admits a rational section or multi-section (Proposition 5.1 of Oguiso [20]).…”

“…This suggests that we should assume d 1 = 1. Indeed if d 1 > 1 then Y is not primitive when restricted to a fibre, and in some circumstances one can use the methods described in [23] and [24] to deform …”