1 – Riemann Surfaces
279
Next consider the open subset G obtained by removing the discs D k (r)
from X. Its border is the union of the “ circles ” limiting the “ discs ” D k (r).
In the neighbourhood of a border point of D k (r) ⊂ D k (r
), the situation is
similar to that of two concentric discs in C. It is, therefore, clear that, on the
one hand, the border of G is a (real) one-dimensional submanifold of of the
union S of circles |ϕ k (x)| = r, and that, on the other, in the neighbourhood
of a border point of G, the open subset G is located on only one side of its
boundary.
ω is holomorphic and a fortiori C
∞ on the manifold X
= X−{a 1 , . . . , a n }, .
G is open in X
, with compact closure in X
since the latter being the complement in X of the open discs |ϕ k (x)| < r, is closed in the compact set X
and does not contain any a k . Stokes’ theorem can, therefore, be applied:
∂G
ω =
G
dω .
dω = 0 since ω is holomorphic, so that the sum of the integrals of ω along
the “ circles ” limiting D k (r) is zero.
Using the chart (U k , ϕ k ) which transforms ω into a form ω k = h k (ζ)dζ
in the disc |ζ| < r
, it becomes clear that (U k , ϕ k ) transforms the orientation
of X, hence of X
, into the standard orientation of C ; it also transforms
the boundary of D k (r), oriented in conformity with Stokes’ formula, into the
circle |ζ| = r oriented counterclockwise. So the diffeomorphism ϕ k transforms
the extended integral ω over the boundary of D k (r) into the extended integral
of ω k over the circumference |ζ| = r, equal to 2πi Res 0 (h k ). But the residue
of h k at 0 = ϕ k (a k ) is, by definition, the residue of ω at a k . The theorem
follows.
Statement (b) has important corollaries. First, if f is a meromorphic
function on X, for any c ∈ C, the functions f and f − c clearly have the same
poles with same multiplicities. In conclusion, as
v a (f ) is the difference
between the number of zeros and the number of poles of f (counted with their
multiplicities), for a compact surface, the number of solutions of f (x) = c is
independent of c and equal to to the number of poles of f .
On the other hand, if ω is a meromorphic form on X and if (U, ϕ) is a
holomorphic local chart at a, with ϕ(a) = 0, the image ω ϕ = h ϕ (ζ)dζ of ω in
ϕ(U ) is meromorphic on ϕ(U ). The order v a (ω) at a is then defined by the
relation
v a (ω) = v 0 (h ϕ ) .
Since the coefficient ρ
(ζ) in (3) is holomorphic and = 0 at a, the definition
does not depend on the choice of the chart(U, ϕ). The a ∈ X, where v a (ω) = 0
form a discrete and hence a finite set if X is compact. Obviously, v a (hω) =
v a (h) + v a (ω) for every meromorphic function h on S. Hence setting
v(ω) =
v a (ω)
279
Next consider the open subset G obtained by removing the discs D k (r)
from X. Its border is the union of the “ circles ” limiting the “ discs ” D k (r).
In the neighbourhood of a border point of D k (r) ⊂ D k (r
), the situation is
similar to that of two concentric discs in C. It is, therefore, clear that, on the
one hand, the border of G is a (real) one-dimensional submanifold of of the
union S of circles |ϕ k (x)| = r, and that, on the other, in the neighbourhood
of a border point of G, the open subset G is located on only one side of its
boundary.
ω is holomorphic and a fortiori C
∞ on the manifold X
= X−{a 1 , . . . , a n }, .
G is open in X
, with compact closure in X
since the latter being the complement in X of the open discs |ϕ k (x)| < r, is closed in the compact set X
and does not contain any a k . Stokes’ theorem can, therefore, be applied:
∂G
ω =
G
dω .
dω = 0 since ω is holomorphic, so that the sum of the integrals of ω along
the “ circles ” limiting D k (r) is zero.
Using the chart (U k , ϕ k ) which transforms ω into a form ω k = h k (ζ)dζ
in the disc |ζ| < r
, it becomes clear that (U k , ϕ k ) transforms the orientation
of X, hence of X
, into the standard orientation of C ; it also transforms
the boundary of D k (r), oriented in conformity with Stokes’ formula, into the
circle |ζ| = r oriented counterclockwise. So the diffeomorphism ϕ k transforms
the extended integral ω over the boundary of D k (r) into the extended integral
of ω k over the circumference |ζ| = r, equal to 2πi Res 0 (h k ). But the residue
of h k at 0 = ϕ k (a k ) is, by definition, the residue of ω at a k . The theorem
follows.
Statement (b) has important corollaries. First, if f is a meromorphic
function on X, for any c ∈ C, the functions f and f − c clearly have the same
poles with same multiplicities. In conclusion, as
v a (f ) is the difference
between the number of zeros and the number of poles of f (counted with their
multiplicities), for a compact surface, the number of solutions of f (x) = c is
independent of c and equal to to the number of poles of f .
On the other hand, if ω is a meromorphic form on X and if (U, ϕ) is a
holomorphic local chart at a, with ϕ(a) = 0, the image ω ϕ = h ϕ (ζ)dζ of ω in
ϕ(U ) is meromorphic on ϕ(U ). The order v a (ω) at a is then defined by the
relation
v a (ω) = v 0 (h ϕ ) .
Since the coefficient ρ
(ζ) in (3) is holomorphic and = 0 at a, the definition
does not depend on the choice of the chart(U, ϕ). The a ∈ X, where v a (ω) = 0
form a discrete and hence a finite set if X is compact. Obviously, v a (hω) =
v a (h) + v a (ω) for every meromorphic function h on S. Hence setting
v(ω) =
v a (ω)