Adapted complex structures and the geodesic flow

Brian Hall and William Kirwin

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Submission date: 24. Nov. 2008
Pages: 15
Bibtex
MSC-Numbers: 53D25, 32D15, 32Q15, 53D50, 81S10
Keywords and phrases: adapted complex structures, Grauert tube, geodesic flow, geometric quantization, Kähler structure, polarization
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Abstract:
In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the "imaginary-time geodesic flow" to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by "composition with the imaginary-time geodesic flow." We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.