20 Riemann - Roch Theorem

Statement of Riemann-Roch theorem The well-known Riemann - Roch theorem is about computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles on a compact Riemann surface. It relates the complex analysis with the surface’s purely topological genus g.

It was initially proved as Riemann’s inequality by Riemann (1857), the theorem reached its deﬁnitive form for Riemann surfaces after work of Riemann’s short-lived student Gustav Roch 42 in 1865. It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.

On a Riemann surface M, a (Weil) divisor D on M is a locally ﬁnite sum of the form: Z ′ p∈M n(p)p where n(p) ∈ and n(p) = 0 only for a discrete subset of p s. For a divisor D = n(p)p, we deﬁne its degree

deg(D) := n(p). Let f be a meromorphic function on M. Let {pj} and {qj} be its zero and pole set with multiplies nj and mj, respectively. Then f induces a divisor (f):

(f)= njpj − mjqj = vp(f)p p∈M where 0 if f(p) =0 , finite v (f)= ord f if pisazero p  p  −ordpf if f(p)= ∞. As before, we denote by Div(M) the group of all divisors on M.  For any D ∈ Dvi(M), we denote D ≥ 0 (i.e., D is eﬀective) for D = n(p)p with n(p) ≥ 0. We deﬁne ℓ(D)= {f ∈ M(M) | (f)+ D ≥ 0}.

Since vp(f + g) ≥ min{vp(f), vp(g)}, we see that ℓ(D) is a vector space over C. In fact, by the map ℓ(D) → Γ(M, O[D]), f → {fφα} where the divisor D is deﬁned by {φα ∈ O(Uα)}, we get an isomorphism: ℓ(D) ≃ H0(M, O[D]). (72)

421839-1866, a German mathematician.

121 We also deﬁne

i(D)= {ω : meromorphic 1 − form over M, (ω) − D ≥ 0} = {(ω) ≥ D}. i(D) is also a vector space over C.

0 1 i(D) ≃ H (M, ΩM − O[D]) = H (M, O[D]). (73)

Here the last equality we used Serre’s duality theorem.

The famous Riemann-Roch theorem is to connect the meromorphic objects with topology. Theorem 20.1 Let M be a compact Riemann Surface. Let D be a divisor on M. Then

dimC ℓ(D) − dimC i(D)= deg(D)+(1 − g)

or dimC ℓ(D) − dimC l(KM − D)= deg(D)+(1 − g).

In terms of the modern terminology (72) and (73), the above can be written as

h0(M, O[D]) − h1(M, O[D]) = deg(D)+(1 − g) where hj(M, L) := dim Hj(M, L).

For any sheaf F over M, Hq(M, F ) is of ﬁnite dimension for any q. We may recall that the Euler number of a sheaf F is deﬁned by

∞ χ(F )= (−1)j dim Hj(M, F ). j=0

Applications of Riemann-Roch theorem

Lemma 20.2 Let M be a compact Riemann surface. There is a point p ∈ M such that 2 1 dimC ℓ((p)) ≥ 2 ⇐⇒ M is biholomorphic to S = CP . Proof:

ℓ((p)) = {f ∈ M(M) | (f)+(p) ≥ 0} = {f ∈ M(M) | f only has at most a simple pole at p}.

Then dimC ℓ(p) ≥ 2 ⇐⇒ ∃f ∈ ℓ(p) with f ≡ constant.

122 Hence 1 fˆ : M −→ CP f(z), f(z) ∈ C, . z → f(z) := ∞, f(z)= ∞. Claim: fˆ is a biholomorphic map. fˆ is one-to-one and onto: Since df 0= = #(zeros of f) − #(poles of f)=#(zeros of f) − 1, f C the equation f(z) = 0 has exactly one point. Similarly consider f − a instead of f for any a ∈ C. We know that the equation f(z) − a = 0 has exactly one solution. This proves one-to-one. Also, it shows that fˆ is surjective.

0 Theorem 20.3 Let M be a compact Riemann surface. Then dim H (M, ΩM )= g.

Proof: Applying the Riemann-Roch theorem with D = 0, we have

dimC ℓ(D) − dimC i(D) = deg(D) +(1 − g) dimC{f ∈ M(M) | (f) ≥ 0} − dimC{ω | (ω) ≥ 0} 0 dimC Hol(M) 0 1 − dimC H (M, ΩM )

Theorem 20.4 Let M be a compact Riemann surface with g(M)=0. Then M is biholomorphic to S2 = CP1.

Proof: Fixing any p ∈ M, we apply the Riemann-Roch theorem with D =(p)

dimC ℓ(D) − dimC i(D) = deg(D) +(1 − g) − dimC{ω | (ω) ≥ (p)} 1 1 0

123 0 to get dimC ℓ((p)) = 2. Here we use the fact that {ω | (ω) ≥ (p)} ⊂ H (M, ΩM ) and 0 dimC H (M, ΩM ) = g = 0 from Theorem 20.3. Next we apply Theorem 20.2 to conclude that M is biholomorphic to CP1.

Corollary 20.5 (1) deg(KM ) = −χ(M)=2g − 2 where χ(M) is the Euler characteristic 43 of M.

(2) Let ω be a holomorphic 1-form on M. Then the zeros of ω, counting multiplicity, is equal to −χ(M)=2g − 2.

Proof: (1) Applying the Riemann-Roch theorem with D = 0:

dimC ℓ(D) − dimC ℓ(K − D) = deg(D) +(1 − g) 1 − dimC ℓ(K) 0

Hence dimC ℓ(K)= g. Also, we apply the Riemann-Roch theorem with D = K:

dimC ℓ(K) − dimC ℓ(K − K) = deg(K) +(1 − g) g − dimC ℓ(0) 1

Here we used dimC ℓ(K)= g. Then deg(K)=2g − 2.

(2) By K = [(ω)], deg(K)= deg(ω)=#(zeroes of ω) and (1) above.

43The Euler characteristic χ was classically deﬁned for polyhedra, according to the formula: χ = V − E + F where V, E, and F are respectively the numbers of vertices’s (corners), edges and faces in the given polyhedron. χ is a topological invariant.

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