Summary: 4-DIMENSIONAL SYMPLECTIC CONTRACTIONS
MARCO ANDREATTA AND JAROSLAW A. WI“SNIEWSKI
Abstract. Four dimensional symplectic resolutions are (relative) Mori Dream
Spaces. Any two such resolutions are connected by a sequence of Mukai flops.
We discuss cones of their movable divisors with faces determined by curves
whose loci are divisors, we call them essential curves. These cones are divided
into nef chambers related to different resolutions, the division is determined
by classes of flopping 1-cycles. We also study schemes parametrizing minimal
essential curves and show that they are resolutions, possibly non-minimal, of
surface Du Val singularities.
In the paper we consider local symplectic contractions of 4-folds. That is, we deal
with maps : X Y where
· X is a smooth complex 4-fold with a closed holomorphic 2-form, non-
degenerate at every point,
· Y is an affine (or Stein) normal variety,
· is a birational projective morphism.
In dimension 2 symplectic contractions are classical and they are minimal resolu-
tions of Du Val singularities. In fact, any symplectic contraction can be viewed as
a special symplectic resolution of a symplectic normal singularity.