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The Poincaré Dual of a Geodesic Algebraic Curve in a Quotient of the 2-Ball

Published online by Cambridge University Press:  20 November 2018

Stephen S. Kudla  and
John J. Millson
Stephen S. Kudla
Affiliation:
University of Maryland, College Park, Maryland
John J. Millson
Affiliation:
University of Toronto, Toronto, Ontario
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We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 485 - 499
Copyright
Copyright © Canadian Mathematical Society 1981

References

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