Contractions of Symplectic Varieties

1999/10/25 by Jan Wierzba, Wierzba, Jan
Mathematics · #14B07 #14E15 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14B07 #msc:14E15

paper · pdf · doi:10.48550/arxiv.math/9910130

LaTeX, 26 pages with 7 figures. Section 2 contains new results

arxiv created 2000/06/12 · arxiv updated 2009/11/30

Abstract

We consider birational projective contractions f:X -> Y from a smooth symplectic variety X over the complex numbers. We first show that exceptional rational curves on X deform in a family of dimension at least 2n-2. Then we show that these contractions are generically coisotropic, provided X is projective. Then we specialize to contractions with 1-dimensional exceptional fibres. We classify them in a natural way in terms of (Γ, G), where Γis a Dynkin diagram of type A_l, D_l or E_l and G is a permutation group of automorphisms of Γ. The 1-dimensional fibres do not degenerate, except if the contraction is of type (A_2l,S₂). In that case they do not degenerate in codimension 1. Furthermore we show that the normalization of any irreducible component of Sing(Y) is a symplectic variety. We also provide examples for contractions of any type (Γ, G).

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