A generalisation of the Morse inequalities

In this paper we construct a family of variational families for a Legendria n embedding, into the 1-jet bundle of a closed manifold, that can be obtain ed from the zero section through Legendrian embeddings, by discretising the action functional. We compute the second variation of a generating functio n obtained as above at a nondegenerate critical point and prove a formula r elating the signature of the second variation to the Maslov index as the me sh goes to zero. We use this to prove a generalisation of the Morse inequal ities thus refining a theorem of Chekanov.