Legendrian tori and the semi-classical limit

Abstract

Let X be C P n or a compact smooth quotient of the n-dimensional complex hyperbolic space, n > 1 . Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X = C P n then L is the hyperplane bundle, and in the second case L is chosen so that L ⊗ ( n + 1 ) = K X ⊗ E , where K X is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L ∗ is a contact manifold. Let k ′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k ′ ) and sequences { u k } , k = k ′ m , m = 1 , 2 , … , of holomorphic sections of L ⊗ k associated to these tori. We study asymptotics of the norms ‖ u k ‖ k as m → + ∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n > 1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n = 1 ).