Legendrian tori and the semi-classical limit |
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Authors: | Tatyana Foth |
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Affiliation: | Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada |
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Abstract: | Let X be CPn 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=CPn then L is the hyperplane bundle, and in the second case L is chosen so that L⊗(n+1)=KX⊗E, where KX 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 {uk}, k=k′m, , of holomorphic sections of L⊗k associated to these tori. We study asymptotics of the norms ‖ukk‖ 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). |
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Keywords: | 32N05 32N15 53D05 53D10 |
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