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



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 ) 


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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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