Franz Pedit UMass Amherst Riemann Surfaces Problem Sheet 2 September 4, 2020

Problem 1. Show in great detail that the unit sphere S2 ⊂ R3 is a 2-dimensional manifold using the two charts given by stereographic projections from the north and south poles. Identify the topology induced by this atlas and verify that it is the subspace topology from R3. Thus S2 is compact , 2nd countable, and Hausdorff. Finally show that there is a smooth diffeomorphism between CP1 and S2. Additional question: if you stare at the transition map between the stereo- projection charts, can you see a way to modify the charts so that this map becomes holomorphic (when identifying R2 = C)? Then show that in this case S2 will become a Riemann surface and the previous diffeomorphism with CP1 will be holomorphic. Problem 2. Show that the Riemann sphere C ∪ {∞}, the complex projective line CP1, and thus (by the previous example) also S2, are equivalent (i.e. holo- morphically diffeomorphic) Riemann surfaces. Problem 3 (The torus). Show that the abelian coset group C/Γ has the structure of a Riemann surface (whose topology is the quotient topology). Here Γ = Zv1 ⊕ Zv2 ⊂ C is a rank 2 lattice, i.e, v1, v2 are R-linearly independent. Then show that C/Γ is smoothly diffeomorphic to the real 2-dimensional manifold S1 × S1 (which, as a product of two 1-dimensional manifolds, is a 2-dimensional manifold—generally the products of manifolds are manifolds w.r.t. the obvious product charts and the dimensions just add). Problem 4. Give a detailed verification that CP2 with the affine charts given in class is a 2-dimensional, compact, complex manifold (in particular show that the topology induced from the atlas is 2nd countable and Hausdorff). Suggestion: for the topological part show that CP2 can be viewed as C3 \{0}/C× and the latter is homeomorphic to S5/S1. Problem 5. Show that the chart maps of a complex (or real) manifold are holomorphic (or smooth) diffeomorphisms.

Recall that a subset M ⊂ C2 given as the zero set of a holomorphic function P : C2 → C is a regular (or non-singular) complex curve, if for each p ∈ M we ∂P ∂P have ∂z (p) 6= 0 or ∂w (p) 6= 0. If P ∈ C[Z,W ] is a polynomial, then we call M 2 a regular (complex) algebraic curve. We have shown in class that under those assumptions one can apply the implicit function theorem to construct holomor- phic charts and make M into a Riemann surface (whose topology is the subspace topology M ⊂ C2). Problem 6. Let M ⊂ C2 be a regular curve. Show that M can never be compact. Suggestion: you may first want to prove a version of the maximum principle, namely let D be a disk and f : D → C be holomorphic such that |f| has an interior maximum. Then f is constant. Problem 7. Go through the “Renaissance idea” of projective geometry one more time: the “affine view point” starts with geometry in Cn and then “projectivizes” n −1 n n n C by viewing it as ϕ0 (C ) = U0 ⊂ CP . Thus, points v ∈ C become lines −1 n+1 n ϕ0 (v) = C(e0 + v) in C , which are (projective) points in CP . The “points at infinity” in Cn, i.e. v 7→ ∞, then become the lines in Cn+1 contained in n n n−1 C ⊂ Ce0 ⊕ C , a projective space CP of one dimension less, usually called the “hyperplane” H∞ at infinity. Do the following example: consider the (affine) quadric (“conic section” as the Greeks would have said) 2 2 2 2 M = {(z, w) ∈ C ; z + w = 1} ⊂ C Calculate the projectivization M¯ ⊂ CP2 by constructing a homogeneous poly- ¯ nomial P (z0, z1, z2) whose zero set M “coincides” with M on the “finite” part ¯ 2 ϕ0(M ∩ U0) = M of CP . ¯ ¯ ¯ 2 Calculate all the “points at infinity” M ∩ H∞ of M and show that M ⊂ CP is ¯ 2 regular (by which we mean that ϕ(M ∩ Uk) ⊂ C is regular for all three affine charts) and thus a Riemann surface. In fact, show that M¯ is equivalent to the Riemann sphere. Draw a—necessarily—conceptual picture of how you view M. Problem 8 (Hyperelliptic Riemann surfaces). Consider F (z, w) := w2 − P (z) for an odd degree polynomial P (z). The zero set 2 2 M := {(z, w) ∈ C ; F (z, w) = 0} ⊂ C is called an affine hyperelliptic curve. (i) Show that if P (z) has simple roots then M is a non-singular curve (and thus a non-compact Riemann surface). We will now assume P has simple roots. (ii) Projectivize M to M¯ ⊂ CP2. What is the homogeneous polynomial ¯ ¯ ¯ F (z0, z1, z2) describing M? Why is M compact? (iii) How many “points at infinity” does M¯ have? Here we think of CP2 = 1 1 2 U0 ∪ CP with hyperplane H∞ = CP ⊂ CP at infinity (in this case −1 2 2 a projective line), U0 = ϕ0 (C ), and M ⊂ C . Would the number of points at infinity change, and how, if P (z, w) were a polynomial of even degree (with simple zeros)? 3

(iv) Show that M¯ is non-singular at infinity if and only if deg P ≤ 3. In this case M¯ is a compact Riemann surface. (v) Show that if deg P = 1 then M¯ is equivalent to the Riemann sphere. (If deg P = 3 then M¯ is called an elliptic curve; we will later show that in this case M¯ is a torus). (vi) Even though M¯ ⊂ CP2 will in general be singular, we can still “ab- stractly” compactify M to a Riemann surface Mˆ by adding the point(s) at infinity. How does one have to choose charts around those point(s)? Note: this may initially look contradictory, but the issue here is the following: compactifying by projectifying unfortunately does not give a non-singular projective curve (if deg P > 3), thus we cannot use the implicit function theorem to provide charts around the points at infinity. But the projectification tells us how many points at infinity we should add to get a compact object. Adding those points abstractly, we then define a compact Riemann surface by prescribing charts around these abstract points at infinity so that the transition functions with the charts (from the implicit function theorem) on the non-singular M are holomorphic. Problem 9. The following gives a criterion when a projective curve M ⊂ CP2 described as the zero set of a homogeneous polynomial P (z0, z1, z2) of degree d is d non-singular (homogeneous of degree d means P (λz0, λz1, λz2) = λ P (z0, z1, z2) for all λ ∈ C and all (z0, z1, z2) ∈ C). Assume that for all p ∈ M with p 6= 0 one has ( ∂P (p), ∂P (p), ∂P (p)) 6= 0. Show that under this assumption M is a ∂z0 ∂z1 ∂z2 2 non-singular projective curve (meaning that each affine part ϕi(M ∩ Ui) ⊂ C , i = 0, 1, 2, is a non-singular curve). Suggestion: Verify the following identity for a homogenous polynomial of degree d: ∂P ∂P ∂P z0 + z1 + z2 = d · P ∂z0 ∂z1 ∂z2 Obviously, this problem can be generalized verbatim to provide a criterion when a projective hypersurface in CPn is non-singular.