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Meromorphic differential

  1. Mar 22, 2008 #1
    This semester i'm taking "Introduction to Algebraic Curves" course. Up to now, the only problems i have with this course are the notion of meromorphic(and holomorphic) differentials, and coordinate charts. I'm good with the algebraic ideas. Here is one question from the book we are studying:

    Let w be a meromorphic differential on a Riemann surface C. Show that we can choose an appropriate coordinate chart so that w = (z^u)dz, u is an integer, in some neighborhood of a pole. Prove also that this integer u is independent of the coordinate chart selected.

    Actually i could prove the uniqueness of the number u for the function version of the statement(in a nbd. of a pole, a meromorphic function f on a Riemann surface can be expressed as f = (z^u')g(z), where u' is an integer, g(z) is a holomorphic function and z(q)=0 for this pole p). But i have trouble with proving the statement for the meromorphic differentials.

    I know that a meromorphic differential f dg is represented on the coordinate chart
    phi_i :U_i -> V_i by the meromorphic function f((phi_i)^(-1)) g((phi_i)^(-1)) but don't know how to use it for this question.

    Instead i tried to prove the result as follows:

    Let w be a meromorphic differential which have the local representation

    w = p(z)dz = ((a_n) z^n + (a_(n+1)) z^(n+1) + ...)dz, where a_n is non-zero and n is a negative integer. Then w has a pole of order n at the point q. Let z = f(w) be a mapping such that f(0)=0 and f'(0) is nonzero.Then we get gbar(w) = g(z)dz = g(f(w))(df/dw). So,

    lim (w->0) (w^(-n))*gbar(w) = lim (w/f(w))^(-n)*[f(w)^(-n) g(f(w))(df/dw)
    = f'(0)* lim (z->0) (z^(-n)) p(z).

    Here f'(0) is nonzero by construction, and thus the last expression is finite and non-zero when so is lim (z^(-n)) p(z) (this can be proved easily). So this number n, aka u in the original quesion is independent of the local chart.

    Does this prove the statement? Is there another good way of showing the desired result? Thanks!
    Last edited: Mar 22, 2008
  2. jcsd
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