Deformations of singular symplectic varieties and termination of the log minimal model program

Christian Lehn (1) , Gianluca Pacienza (2)

1 Chemnitz University of Technology / Technische Universität Chemnitz
2 IECL - Institut Élie Cartan de Lorraine

We generalize Huybrechts' theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of one another. As an application we show the termination of any log-minimal model program for a pair (X, ∆) of a projective irreducible symplectic manifold X and an effective R-divisor ∆. To prove this result we follow Shokurov's strategy and show that LSC and ACC for mlds hold for all the models appearing along any log-MMP of the initial pair.

Mots clés: minimal model program; deformation; local triviality; minimal log discrepancy; termination of flips; ACC and LSC conjectures; irreducible symplectic manifolds; symplectic variety

Domaines: Géométrie algébrique [math.AG]

Christian Lehn, Gianluca Pacienza. Deformations of singular symplectic varieties and termination of the log minimal model program. Algebraic Geometry, 2016, 3 (4), pp.392-406. ⟨10.14231/AG-2016-018⟩. ⟨hal-03256222⟩