Riemann Surfaces, Branched Maps, and Differentials: Examples and Counterexamples

(Plane domains give Riemann surfaces through the identity chart.) Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be nonempty, connected and open; as a subset of ${\mathbb{R}}^2$ it is Hausdorff and second countable by [F5] and [F6], and the one-chart atlas $\{\mathrm{i}\mathrm{d}:\mathrm{\Omega }\to \mathbb{C}\}$ has the holomorphic transition $\mathrm{i}\mathrm{d}\circ {\mathrm{i}\mathrm{d}}^{-1}=\mathrm{i}\mathrm{d}$ on $\mathrm{\Omega }$; hence $\mathrm{\Omega }$ is a Riemann surface by [F1].

(The sphere's two charts are a holomorphic atlas.) The domains $U_0,U_{\mathrm{\infty }}$ cover $\widehat{\mathbb{C}}$ and each chart is a homeomorphism onto $\mathbb{C}$; on the overlap the two transitions are $w\mapsto 1\mathrm{/}w$, holomorphic on ${\mathbb{C}}^{\times }$ [F2]; $\widehat{\mathbb{C}}$ is nonempty and, by [F3], compact Hausdorff, and it is connected because a separation of $\widehat{\mathbb{C}}$ would put the connected dense subspace $\mathbb{C}$ on one side while the other side, being open and nonempty and containing a point of the closure of $\mathbb{C}$, meets $\mathbb{C}$; it is second countable because $S^2\subseteq {\mathbb{R}}^3$ is second countable by [F5] and [F6] and a homeomorphism transports a countable base, so [F4] transfers this to $\widehat{\mathbb{C}}$; hence $\widehat{\mathbb{C}}$ is a Riemann surface by [F1].

(The listed plane domains satisfy the hypotheses of step 1.1.) The plane $\mathbb{C}$ and the unit disc $\mathbb{D}$ are convex, hence path-connected and therefore connected by [F7]; the annulus $A(0;r,R)$ with $0<r<R$ is nonempty: fix a finite radius $\rho \in (r,R)$, taking $\rho =(r+R)\mathrm{/}2$ if $R<\mathrm{\infty }$ and $\rho =r+1$ if $R=\mathrm{\infty }$. It is path-connected because for $u\in A(0;r,R)$ the radial segment from $u$ to $\frac{\rho }{\mathrm{\mid }u\mathrm{\mid }}u$ stays in the annulus (its modulus runs between $\mathrm{\mid }u\mathrm{\mid }$ and $\rho$, both in $(r,R)$) and any two points can be joined through the circle $\mathrm{\mid }z\mathrm{\mid }=\rho$; therefore step 1.1 applies to $\mathbb{C}$, $\mathbb{D}$ and $A(0;r,R)$.

(Conclusion.) Steps 1.1 and 2.1 exhibit the identity atlas on $\mathbb{C}$, on the unit disc and on every round annulus $A(0;r,R)$ with $0<r<R$, and step 1.2 exhibits the two-chart atlas on the sphere with transition $w\mapsto 1\mathrm{/}w$ on ${\mathbb{C}}^{\times }$; in each case the transition maps are holomorphic and the Riemann-surface axioms hold, as claimed.

(A gap constant exists.) The map $T(s,t)=s{\omega }_1+t{\omega }_2$ is an $\mathbb{R}$-linear isomorphism ${\mathbb{R}}^2\to \mathbb{C}$, because ${\omega }_1,{\omega }_2$ are $\mathbb{R}$-linearly independent, and $\mathrm{\Lambda }=T({\mathbb{Z}}^2)$; applying [F3] to the inverse of $T$ gives $c>0$ with $\mathrm{\mid }T(s,t)\mathrm{\mid }\ge c\max (\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }t\mathrm{\mid })$, hence $\mathrm{\mid }\lambda \mathrm{\mid }\ge c$ for every nonzero $\lambda \in \mathrm{\Lambda }$.

($q$ is continuous and open.) The quotient topology makes $q$ continuous, and for open $W\subseteq \mathbb{C}$ the preimage $q^{-1}(q(W))={\bigcup }_{\lambda \in \mathrm{\Lambda }}(W+\lambda )$ is open as a union of translates of an open set, so $q(W)$ is open by the definition of the quotient topology.

(Lattice points in a disc are finite.) If $\lambda \mathrm{\ne }{\lambda }^{\mathrm{\prime }}$ are in $\mathrm{\Lambda }$ then $\mathrm{\mid }\lambda -{\lambda }^{\mathrm{\prime }}\mathrm{\mid }\ge c>0$ by step 1.1, so $\mathrm{\Lambda }$ is discrete: each $\{\lambda \}$ is open in $\mathrm{\Lambda }$; for every $R>0$ the set $\mathrm{\Lambda }\cap \overline{D(0,R)}$ is closed and bounded in $\mathbb{C}={\mathbb{R}}^2$, hence compact by [F5] and discrete as a subspace, hence finite by [F6].

(Connectedness.) The space $\mathbb{C}$ is convex, hence connected, and $q$ is continuous and surjective, so $\mathbb{C}\mathrm{/}\mathrm{\Lambda }=q(\mathbb{C})$ is connected by [F4].

(Compactness.) The map $F(s,t)=q(T(s,t))$ is continuous on the compact square $[0,1{]}^2$, so its image is compact by [F4] and [F5]; every $z\in \mathbb{C}$ equals $T(s,t)$ for some $(s,t)\in {\mathbb{R}}^2$, and subtracting the integer parts of $s$ and $t$ exhibits $z$ as an element of $T([0,1{]}^2)+\mathrm{\Lambda }$, so $F([0,1{]}^2)=\mathbb{C}\mathrm{/}\mathrm{\Lambda }$; hence $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is compact.

(Small discs give charts.) Fix any $z\in \mathbb{C}$ and put $W=D(z,c\mathrm{/}4)$; if $q(w)=q(w^{\mathrm{\prime }})$ for $w,w^{\mathrm{\prime }}\in W$ then $w-w^{\mathrm{\prime }}\in \mathrm{\Lambda }$ and $\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }<c\mathrm{/}2<c$, so $w=w^{\mathrm{\prime }}$ by step 1.1; thus $q$ restricted to $W$ is an injective continuous open map onto the open set $q(W)$ of $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$, and its inverse $\phi =q{\mathrm{\mid }}_W^{-1}$ is a homeomorphism $q(W)\to W\subseteq \mathbb{C}$, that is, a chart; the sets $q(D(z,c\mathrm{/}4))$ over $z\in \mathbb{C}$ cover $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$.

(Second countability.) Let $\mathcal{B}$ be a countable base of $\mathbb{C}$; the family $\{q(B):B\in \mathcal{B}\}$ is countable and consists of open sets by step 1.2, and it is a base of $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$: given $[z]\in V$ with $V$ open, the set $q^{-1}(V)$ is an open neighbourhood of $z$, so some $B\in \mathcal{B}$ satisfies $z\in B\subseteq q^{-1}(V)$, whence $[z]\in q(B)\subseteq V$.

(Hausdorffness.) Let $[z]\mathrm{\ne }[z^{\mathrm{\prime }}]$, so $z-z^{\mathrm{\prime }}\mathrm{\notin }\mathrm{\Lambda }$; the distance $d=\inf \{\mathrm{\mid }z-z^{\mathrm{\prime }}-\lambda \mathrm{\mid }:\lambda \in \mathrm{\Lambda }\}$ is positive, because otherwise points of $\mathrm{\Lambda }$ would accumulate at $z-z^{\mathrm{\prime }}$ inside some disc $\overline{D(0,R)}$ containing $z-z^{\mathrm{\prime }}$ and infinitely many distinct lattice points, contradicting the finiteness in step 2.1; with $r=d\mathrm{/}3$, the open sets $q(D(z,r))$ and $q(D(z^{\mathrm{\prime }},r))$ of step 1.2 are disjoint, since $u-v\in \mathrm{\Lambda }$ for $u\in D(z,r)$, $v\in D(z^{\mathrm{\prime }},r)$ would give $\mathrm{\mid }z-z^{\mathrm{\prime }}-(u-v)\mathrm{\mid }<2d\mathrm{/}3<d$.

(Transition functions are translations.) Let $\phi =q{\mathrm{\mid }}_W^{-1}$ and ${\phi }^{\mathrm{\prime }}=q{\mathrm{\mid }}_{W^{\mathrm{\prime }}}^{-1}$ be two charts as in step 2.4; on the overlap of their images, ${\phi }^{\mathrm{\prime }}({\phi }^{-1}(w))=w+\lambda (w)$ with $\lambda (w)={\phi }^{\mathrm{\prime }}(q(w))-w\in \mathrm{\Lambda }$, and $w\mapsto \lambda (w)$ is continuous because both ${\phi }^{-1}$ and $q$ are; since distinct lattice points are at distance at least $c$ by step 1.1, $\lambda$ is locally constant, so near each point the transition is $w\mapsto w+\lambda$ for a fixed $\lambda \in \mathrm{\Lambda }$, which is holomorphic.

(Conclusion.) The quotient $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is nonempty, connected by step 2.2, compact by step 2.3, Hausdorff by step 3.1 and second countable by step 2.5, and the charts of step 2.4 have the holomorphic transition functions $w\mapsto w+\lambda$ of step 3.2, so they form a holomorphic atlas; by [F1], $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is a compact Riemann surface.

(The conic is nonsingular.) The curve is cut out by the single polynomial $f(x,y)=x^2+y^2-1$ with gradient $\mathrm{\nabla }f=(2x,2y)$; if $\mathrm{\nabla }f$ vanished at a point of $C_0$ then $x=y=0$ and $x^2+y^2=0\mathrm{\ne }1$, so $\mathrm{\nabla }f$ is nowhere zero on $C_0$ and the Jacobian-rank condition of [F1] holds at every point.

(The two factors are reciprocal.) For $(x,y)\in C_0$ one has $(x+iy)(x-iy)=x^2+y^2=1$, so $u=x+iy$ satisfies $u\mathrm{\ne }0$ and $x-iy=u^{-1}$; adding and subtracting the two equations gives $x=(u+u^{-1})\mathrm{/}2$ and $y=(u-u^{-1})\mathrm{/}(2i)$.

($u$ is a bijection onto the punctured plane.) If $u(x,y)=u(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$ then $x+iy=x^{\mathrm{\prime }}+i{y}^{\mathrm{\prime }}$ and, by step 1.2, also $x-iy=u^{-1}=x^{\mathrm{\prime }}-i{y}^{\mathrm{\prime }}$, so $(x,y)=(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$ and $u$ is injective; conversely, for $u\in {\mathbb{C}}^{\times }$ the point $v(u)=((u+u^{-1})\mathrm{/}2,(u-u^{-1})\mathrm{/}(2i))$ satisfies $v(u)\in C_0$ because its coordinates give $x+iy=u$ and $x-iy=u^{-1}$, whose product is $1$, and $u(v(u))=u$, so $u$ is surjective with the displayed inverse.

($u$ and its inverse are continuous.) The map $u$ is the restriction of the polynomial map $(x,y)\mapsto x+iy$, which is complex differentiable and hence continuous by [F6]; the inverse $v$ has components that are rational functions of $u$ and $u^{-1}$, complex differentiable on ${\mathbb{C}}^{\times }$ by [F6] and so continuous; hence $u:C_0\to {\mathbb{C}}^{\times }$ is a continuous bijection with continuous inverse, that is, a homeomorphism.

(Transport of the complex structure.) Give $C_0$ the one-chart atlas $\{u\}$, the chart $u$ being the homeomorphism of step 3.1 onto the plane domain ${\mathbb{C}}^{\times }$; its only transition with itself is the identity, which is holomorphic, and $C_0$ is nonempty, connected as a continuous image of the connected ${\mathbb{C}}^{\times }$ under $v$ by [F5], Hausdorff and second countable by [F4], so $C_0$ is a Riemann surface by [F3]; with these charts the chart expression of $u$ is the identity and the chart expression of its inverse $v$ is also the identity, so $u$ is a biholomorphism $C_0\to {\mathbb{C}}^{\times }$ by [F3].

(Conclusion.) The curve $x^2+y^2=1$ is nonsingular at every point by step 1.1, and step 4.1 exhibits it as a Riemann surface biholomorphic to ${\mathbb{C}}^{\times }$ through $u=x+iy$, with inverse $x=(u+u^{-1})\mathrm{/}2$, $y=(u-u^{-1})\mathrm{/}(2i)$, exactly as claimed.

(The affine curve is a nice topological space.) The set $C\subseteq {\mathbb{C}}^N$ is closed, so as a subspace of the Hausdorff second-countable space ${\mathbb{C}}^N$ it is itself Hausdorff and second countable by [F5] and [F6]; every connected component of $C$ is open by [F7], hence a nonempty connected Hausdorff second-countable space.

(The projective space is compact and second countable.) The quotient map $q:{\mathbb{C}}^{N+1}\setminus \{0\}\to {\mathbb{C}\mathbb{P}}^N$ has continuous restriction to $S^{2N+1}$ by [F3], it is surjective because every nonzero vector is a positive multiple of a unit vector, so ${\mathbb{C}\mathbb{P}}^N$ is compact by [F4]; its finitely many standard affine charts are homeomorphic to ${\mathbb{C}}^N$ by [F1], hence second countable by [F6], and a finite union of open second-countable subspaces is second countable, since the union of their countable bases is a countable base.

(The affine curve has a holomorphic atlas.) At each $p\in C$ the local parameter supplied by [F1] is a chart from a neighbourhood of $p$ onto a plane domain; over two such neighbourhoods the transition map between local parameters is holomorphic by [F1]; hence these charts form a holomorphic atlas on $C$, and restricting the charts to a connected component exhibits that component as a Riemann surface in the sense of [F2].

(The projective space is Hausdorff.) Let $[\zeta ]\mathrm{\ne }[\eta ]$ in ${\mathbb{C}\mathbb{P}}^N$. If some $U_i$ contains both, then the chart homeomorphism $U_i\to {\mathbb{C}}^N$ separates them, since ${\mathbb{C}}^N$ is Hausdorff and $U_i$ is open. Otherwise the supports of $\zeta$ and $\eta$ are disjoint. Put $S=\{i:{\zeta }_i\mathrm{\ne }0\}$ and define $g([\xi ])=({\sum }_{i\in S}\mathrm{\mid }{\xi }_i{\mathrm{\mid }}^2)\mathrm{/}({\sum }_{k=0}^N\mathrm{\mid }{\xi }_k{\mathrm{\mid }}^2)$. This is well defined under nonzero complex scaling; its composite with the quotient map is continuous, so it is continuous by [F3]. Since $g([\zeta ])=1$ and $g([\eta ])=0$, the open sets $\{g>2\mathrm{/}3\}$ and $\{g<1\mathrm{/}3\}$ are disjoint neighbourhoods of the two points.

(The conic is a nonsingular affine instance.) For $f=x^2+y^2-1$ one has $\mathrm{\nabla }f=(2x,2y)$, which vanishes only at the origin, and $(0,0)\mathrm{\notin }{C}_0$ because $-1\mathrm{\ne }0$; hence the Jacobian of the single equation has rank $1=N-1$ at every point of $C_0$ with $N=2$, so $C_0$ satisfies the hypothesis of steps 1.1 and 2.1 and is a complex curve whose components are Riemann surfaces.

(The projective curve has a holomorphic atlas.) Let $C\subseteq {\mathbb{C}\mathbb{P}}^N$ be the zero set of homogeneous polynomials $F_1,\dots ,F_r$ and suppose it is nonsingular in the Jacobian-rank sense in each standard affine chart, i.e. each piece $C\cap U_i$ is, under the chart identification, an affine curve satisfying the hypothesis of step 2.1; by step 2.1 each piece therefore carries the local-parameter atlas, and the charts coming from two different affine charts are compatible because the standard chart transitions are holomorphic and, by [F1], all local parameters of the curve transform holomorphically; hence these charts form one holomorphic atlas on $C$, and its connected components are Riemann surfaces.

(The projective curve is compact, Hausdorff and second countable.) The curve $C\subseteq {\mathbb{C}\mathbb{P}}^N$ is closed, because its preimage under the quotient map is the zero set of the continuous functions $F_i$ on ${\mathbb{C}}^{N+1}\setminus \{0\}$ by [F3]; being a closed subset of the compact space ${\mathbb{C}\mathbb{P}}^N$ of step 1.2 it is compact by [F5], and being a subspace of the Hausdorff second-countable space ${\mathbb{C}\mathbb{P}}^N$ of steps 1.2 and 2.2 it is Hausdorff and second countable by [F5]; hence, with the atlas of step 3.2, each component is a Riemann surface.

(Conclusion.) Steps 2.1 and 3.1 give the affine curves, in particular the nonsingular conic $x^2+y^2=1$, the local-parameter complex structure, and steps 3.2 and 4.1 do the same for a projective curve while proving that it is compact in its analytic topology; in each case the components inherit a holomorphic atlas, Hausdorffness and second countability, so they are Riemann surfaces.

(The finite-chart expression of $\omega$.) In the chart ${\varphi }_0$ the local expression of $\omega =\frac{dz}{z}$ is $h_{{\varphi }_0}(z)=1\mathrm{/}z$, holomorphic on ${\mathbb{C}}^{\times }$ with Laurent expansion $z^{-1}$ at $0$; hence ${\mathrm{ord}}_0(\omega )=-1$ and ${\mathrm{Res}}_0(\omega )=1$, and $0$ is the only pole of $\omega$ in the chart $U_0$.

(The infinity-chart expression of $\omega$.) Put $w={\varphi }_{\mathrm{\infty }}(z)=1\mathrm{/}z$, so that $z=1\mathrm{/}w$ and $z^{\mathrm{\prime }}(w)=-1\mathrm{/}{w}^2$ by the reciprocal rule; the transition law gives $h_{{\varphi }_{\mathrm{\infty }}}(w)=h_{{\varphi }_0}(z(w))z^{\mathrm{\prime }}(w)=w\cdot (-1\mathrm{/}{w}^2)=-1\mathrm{/}w$. Thus the chart expression at infinity has a simple pole at $w=0$, the point $z=\mathrm{\infty }$, so ${\mathrm{ord}}_{\mathrm{\infty }}(\omega )=-1$ and ${\mathrm{Res}}_{\mathrm{\infty }}(\omega )=-1$, and it is holomorphic on $U_{\mathrm{\infty }}\setminus \{\mathrm{\infty }\}$.

(The differential $\eta =dz$ in both charts.) In the chart ${\varphi }_0$ the local expression of $\eta$ is $h_{{\varphi }_0}\equiv 1$, holomorphic on all of $\mathbb{C}$, so $\eta$ has no pole in $U_0$ and ${\mathrm{ord}}_p(\eta )=0$ for $p\in \mathbb{C}$; in the chart ${\varphi }_{\mathrm{\infty }}$ the transition law with the same factor $z^{\mathrm{\prime }}(w)=-1\mathrm{/}{w}^2$ gives $h_{{\varphi }_{\mathrm{\infty }}}(w)=1\cdot (-1\mathrm{/}{w}^2)=-1\mathrm{/}{w}^2$, holomorphic on ${\mathbb{C}}^{\times }$; hence ${\mathrm{ord}}_{\mathrm{\infty }}(\eta )=-2$ and ${\mathrm{Res}}_{\mathrm{\infty }}(\eta )=0$, the Laurent expansion $-w^{-2}$ having no $w^{-1}$ term.

(Pole set of $\omega$.) The two chart domains cover $\widehat{\mathbb{C}}$, the only pole of $h_{{\varphi }_0}$ is $z=0$ and the only pole of $h_{{\varphi }_{\mathrm{\infty }}}$ is $w=0$, corresponding to $z=\mathrm{\infty }$; hence the pole set of $\omega$ is exactly $\{0,\mathrm{\infty }\}$, both poles simple, with residues $+1$ and $-1$.

(Pole set of $\eta$.) Since $h_{{\varphi }_0}$ is holomorphic on $U_0$ and the only pole of $h_{{\varphi }_{\mathrm{\infty }}}$ is at $w=0$, the pole set of $\eta$ is exactly $\{\mathrm{\infty }\}$, a single pole of order $2$ with residue $0$.

(Agreement with the residue theorem.) The sphere is a compact Riemann surface [F2], $\omega$ and $\eta$ are meromorphic differentials on it [F3], and each has a finite pole set, so [F6] applies to both; the totals are ${\mathrm{Res}}_0(\omega )+{\mathrm{Res}}_{\mathrm{\infty }}(\omega )=1+(-1)=0$ and ${\mathrm{Res}}_{\mathrm{\infty }}(\eta )=0$, and in each case only finitely many residues are nonzero, so both computations agree with the residue theorem.

($\exp$ is a holomorphic local biholomorphism onto ${\mathbb{C}}^{\times }$.) Being entire, $\exp$ is holomorphic as a map of Riemann surfaces in the identity atlases of [F1]; by [F2] its derivative is nowhere zero, so [F4] makes it a biholomorphism between a neighbourhood of each $z$ and a neighbourhood of $\exp z$, and by [F3] it maps $\mathbb{C}$ onto ${\mathbb{C}}^{\times }$.

({1} is compact.) The singleton $\{1\}\subseteq {\mathbb{C}}^{\times }$ is a one-point space, so by [F6] every open cover of it has a one-member subcover; hence $\{1\}$ is compact.

(The fibre of $1$ is infinite.) By [F5], ${\exp }^{-1}(\{1\})=\ker (\exp )=2\pi i\mathbb{Z}=\{2\pi ik:k\in \mathbb{Z}\}$; the assignment $k\mapsto 2\pi ik$ is injective on $\mathbb{Z}$ and $\mathbb{Z}$ is infinite, so the fibre is infinite.

(The fibre of $1$ is not compact.) Suppose $2\pi i\mathbb{Z}$ were compact. For each $k\in \mathbb{Z}$ let $U_k$ be the open disc of radius $2\pi$ about $2\pi ik$; the sets $U_k\cap 2\pi i\mathbb{Z}$ form an open cover of the subspace $2\pi i\mathbb{Z}$ in the sense of [F6]. By compactness finitely many of them cover $2\pi i\mathbb{Z}$, say for $k$ in a finite set $F$. But distinct points $2\pi im,2\pi ik$ of the fibre satisfy $\mathrm{\mid }2\pi im-2\pi ik\mathrm{\mid }=2\pi \mathrm{\mid }m-k\mathrm{\mid }\ge 2\pi$, so the disc $U_k$ contains exactly one point of the fibre, namely $2\pi ik$; the finitely many discs with $k\in F$ therefore cover at most the finitely many points $2\pi ik$, $k\in F$, contradicting the infinitude of the fibre from step 1.3. Hence ${\exp }^{-1}(\{1\})$ is not compact.

($\exp$ is not proper, and the degree theorem does not apply.) Since $\{1\}$ is compact by step 1.2 while ${\exp }^{-1}(\{1\})$ is not compact by step 2.1, the exponential is not proper; consequently the hypothesis of [F7] fails, no finite degree exists, and every fibre ${\exp }^{-1}(w)$, $w\in {\mathbb{C}}^{\times }$, is infinite: by [F3] write $w=\exp z_0$ and by [F5] the fibre is $z_0+2\pi i\mathbb{Z}$. Thus a holomorphic local biholomorphism of Riemann surfaces need not be proper and need not have finite fibres, so the properness hypothesis in Degree of a proper holomorphic map of Riemann surfaces cannot be dropped.