Inventiones mathematicae

, Volume 198, Issue 3, pp 505–590 | Cite as

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

  • Arend BayerEmail author
  • Emanuele Macrì


We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space \(M\) of stable sheaves on a K3 surface \(X\): (a) We describe the nef cone, the movable cone, and the effective cone of \(M\) in terms of the Mukai lattice of \(X\). (b) We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on \(M\) whenever \(M\) admits an integral divisor class \(D\) of square zero (with respect to the Beauville–Bogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions \(\mathop {\mathrm {Stab}}\nolimits (X)\) to the cone \(\mathop {{\mathrm {Mov}}}\nolimits (X)\) of movable divisors on \(M\); this map relates wall-crossing in \(\mathop {\mathrm {Stab}}\nolimits (X)\) to birational transformations of \(M\). In particular, every minimal model of \(M\) appears as a moduli space of Bridgeland-stable objects on \(X\).

Mathematics Subject Classification (2010)

14D20 (Primary) 18E30 14J28 14E30 (Secondary ) 



Conversations with Ralf Schiffler dissuaded us from pursuing a failed approach to the birationality of wall-crossing, and we had extremely useful discussions with Daniel Huybrechts. Tom Bridgeland pointed us towards Corollary 1.3, and Dragos Oprea towards the results in Sect. 15. We also received helpful comments from Daniel Greb, Antony Maciocia, Alina Marian, Eyal Markman, Dimitri Markushevich, Daisuke Matsushita, Ciaran Meachan, Misha Verbitsky, Kōta Yoshioka, Ziyu Zhang, and we would like to thank all of them. We would also like to thank the referee very much for an extremely careful reading of the paper, and for many useful suggestions. The authors were visiting the Max-Planck-Institut Bonn, respectively the Hausdorff Center for Mathematics in Bonn, while working on this paper, and would like to thank both institutes for their hospitality and stimulating atmosphere. A. B.  is partially supported by NSF grant DMS-1101377. E. M.  is partially supported by NSF grant DMS-1001482/DMS-1160466, Hausdorff Center for Mathematics, Bonn, and by SFB/TR 45.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghScotland, UK
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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