Contractions of Symplectic Varieties

Details Contractions of Symplectic Varieties Contractions of Symplectic Varieties

1999-10-25

arxiv.org/abs/math/9910130v3

Mathematics

Algebraic Geometry

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

Scientific paper

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 (\Gamma, G), where \Gamma is a Dynkin diagram of type A_l, D_l or E_l and G is a permutation group of automorphisms of \Gamma. The 1-dimensional fibres do not degenerate, except if the contraction is of type (A_{2l},S_2). 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 (\Gamma, G).

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