meromorphic function on projective space must be rational
To define a rational function on complex projective space , we just take two homogeneous polynomials of the same degree and on and we note that induces a meromorphic function on In fact, every meromorphic function on is rational.
Let be a meromorphic function on . Then is rational.
Note that the zero set of and the pole set are analytic subvarieties of and hence algebraic by Chow’s theorem. induces a meromorphic function on . Let and be two homogeneous polynomials such that are the poles and are the zeros of We can assume we can take and such that if we multiply by we have a holomorphic function outside the origin. Hence extends through the origin by Hartogs’ theorem. Further since was constant on complex lines through the origin, it is not hard to see that is homogeneous and hence a homogeneous polynomial, by the same argument as in the proof of Chow’s theorem. Since it is not zero outside the origin, it can’t be zero at the origin, and hence must be a constant, and the proof is finished. ∎
|Title||meromorphic function on projective space must be rational|
|Date of creation||2013-03-22 17:52:13|
|Last modified on||2013-03-22 17:52:13|
|Last modified by||jirka (4157)|