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First Cousin gluing on the pseudoconvex domain $C$

Statement

Assume the Axiom of Choice (AC). On $C$ the sets $U_{1} : = {z : ∣ z ∣ < 2}, U_{2} : = {z : ∣ z ∣ > 1}$ and the functions $m_{1} : = 1 / z$ on $U_{1}$ and $m_{2} : = 0$ on $U_{2}$ form compatible first-Cousin data on the Hartogs pseudoconvex domain $C$ in the sense of First Cousin problem on a pseudoconvex domain: the cover is finite, each $m_{i}$ is meromorphic on $U_{i}$ , and $m_{1} - m_{2} = 1 / z$ is holomorphic on $U_{1} \cap U_{2} = {1 < ∣ z ∣ < 2}$ . The global meromorphic function $G : = 1 / z$ satisfies $G - m_{1} = 0 \in O (U_{1})$ and $G - m_{2} = 1 / z \in O (U_{2})$ , so it realizes the prescribed simple pole at $0$ .

Facts & Assumptions

Given: The Axiom of Choice; the plane $C$ ; the sets $U_{1}$ , $U_{2}$ ; the functions $m_{1} = 1 / z$ on $U_{1}$ and $m_{2} = 0$ on $U_{2}$ ; the candidate $G = 1 / z$ .

[F1]

(First Cousin problem on a pseudoconvex domain.) If $Ω$ is a Hartogs pseudoconvex domain, $(U_{i})_{i \in I}$ a locally finite open cover of $Ω$ and $m_{i}$ meromorphic on $U_{i}$ with $m_{i} - m_{j}$ holomorphic on $U_{i} \cap U_{j}$ for all $i, j$ , then there is a meromorphic $G$ on $Ω$ with $G - m_{i}$ holomorphic on $U_{i}$ for every $i$ .

[F2]

A meromorphic function on an open $U$ is a function on an open dense $D \subseteq U$ , holomorphic there, which near each point of $U$ equals a quotient $f / g$ of holomorphic functions with $g$ not identically zero on any component; every holomorphic function is meromorphic, and " $F - G$ is holomorphic on $V$ " means that the difference admits a holomorphic extension to $V$ (Meromorphic functions on an open set in complex Euclidean space).

[F3]

When $Ω = C^{m}$ one has $δ_{Ω} \equiv + \infty$ , the boundary function is by convention the constant function $0$ , and the whole space is Hartogs pseudoconvex (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[F4]

If $g : U \to C$ is complex differentiable at $a$ with $g (a) \neq 0$ , then $1 / g$ is complex differentiable at $a$ with $(1 / g)^{'} (a) = - g^{'} (a) / g (a)^{2}$ , and the identity function has derivative $1$ (Linearity, product, reciprocal, and quotient rules for complex derivatives); a function is holomorphic on $U$ when it is complex differentiable at every point of $U$ (Holomorphic functions on an open subset of $C^{m}$ ).

[F5]

The Euclidean ball $B (0, 2) \subseteq C$ is convex and hence path-connected, and path-connected sets are connected (Every convex subset of $R^{n}$ , in particular every ball and $R^{n}$ itself, is path-connected and hence connected, Every path-connected space is connected, and every path component lies inside a component), while the exterior ${z : ∣ z ∣ > 1}$ is open and path-connected (The exterior of a closed disc in the plane is path-connected); balls are open and closed balls are closed in a metric space (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed).

[F6]

AC is the assertion that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC is the ambient hypothesis recorded in the Statement and cited as [F6]; it is consumed only inside the supplier theorem [F1], whose proof carries its own choice hypotheses. The example exhibits $G$ by an explicit formula and selects nothing.

Proof

technique · direct

1.1F5givenalgebra

(The cover.) By [F5] the set $U_{1} = B (0, 2)$ is open, convex, path-connected and connected, and $U_{2} = {z : ∣ z ∣ > 1} = C ∖ \overline{B} (0, 1)$ is open and path-connected, hence connected; moreover $U_{1} \cup U_{2} = C$ and $U_{1} \cap U_{2} = {z : 1 < ∣ z ∣ < 2}$ , so $(U_{1}, U_{2})$ is a finite, hence locally finite, open cover of $C$ by domains.

1.2F2F4givenalgebra

(The meromorphic data.) By [F4] the identity $z \mapsto z$ is holomorphic on $C$ and $z \mapsto 1 / z$ is holomorphic on $C ∖ {0}$ ; hence $m_{1} = 1 / z$ is meromorphic on $U_{1}$ in the sense of [F2]: its domain $U_{1} ∖ {0}$ is open and dense in $U_{1}$ , $m_{1}$ is holomorphic there, and at every point of $U_{1}$ it equals $f / g$ with $f : = 1$ and $g : = z$ , the denominator $g$ not vanishing identically on any component of $U_{1}$ . Likewise $m_{2} = 0$ is holomorphic, hence meromorphic, on $U_{2}$ .

2.1F1F2F3F4step 1.2algebra

(Compatibility.) On the overlap $U_{1} \cap U_{2} = {1 < ∣ z ∣ < 2}$ , which does not contain $0$ , the difference $m_{1} - m_{2} = 1 / z$ is holomorphic by [F4]; this is the compatibility clause of [F1], and $C$ is Hartogs pseudoconvex by [F3], so the data satisfy the hypotheses of [F1].

3.1F1step 2.1

(Existence by the Cousin theorem.) By [F1] there is a meromorphic function $G$ on $C$ with $G - m_{i}$ holomorphic on $U_{i}$ for $i = 1, 2$ .

4.1F1F2F4step 1.2step 3.1algebra

(The explicit solution.) The function $G = 1 / z$ is meromorphic on $C$ with domain $C ∖ {0}$ by [F2] and [F4]; moreover $G - m_{1} = 0$ is holomorphic on $U_{1}$ , and $G - m_{2} = 1 / z$ is holomorphic on $U_{2}$ because $0 \notin U_{2}$ and $z \mapsto 1 / z$ is holomorphic on $C ∖ {0}$ by [F4]. So $G = 1 / z$ is a solution in the sense of [F1]: it differs from $m_{i}$ by a holomorphic function on each $U_{i}$ , and its only pole is the simple pole at $0$ with principal part $1 / z$ , which is exactly the pole prescribed by the data.

5.1F6step 4.1∎

(Conclusion.) The sets $U_{1}, U_{2}$ and the functions $m_{1}, m_{2}$ are compatible first-Cousin data on the Hartogs pseudoconvex domain $C$ , and the global meromorphic function $G = 1 / z$ realizes the prescribed simple pole at the origin, as claimed under the ambient Axiom of Choice [F6].

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Sources

Harald P. Boas, Lecture Notes on Several Complex Variables (standard reference, not scraped)

First Cousin gluing on the pseudoconvex domain CC