The derived category of coherent sheaves on

We consider an even-dimensional complex vector space V , dim V = 2 n , and a non-degenerate skew-symmetric form f on it, f ∈ (cid:27) 2 V ∗ . The set of isotropic subspaces of maximal (half) dimension of f forms a smooth projective manifold (the Lagrangian Grassmannian) LG C n . In this paper we study the derived category of coherent sheaves on LG C 3 , the six-dimensional Fano manifold of index 4. In it we construct a complete exceptional set (see [1] for the definition of a complete exceptional set in a triangulated category). from rk K LG S LG and the positive generator of the group Pic { LG C 3 O LG