  In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E,
a "Koszul duality" equivalence between derived categories of G_m-equivariant
coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and
the corresponding derived intersection of the orthogonals of F_1 and F_2 inside
the dual vector bundle E^*. We also propose applications to Hecke algebras.
