Lecture 6 Analysis on compact Riemann surfaces: meromorphic - - PowerPoint PPT Presentation

Lecture 6 Analysis on compact Riemann surfaces: meromorphic functions, differential forms and the Riemann–Roch Theorem

April 21, 2020

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Meromorphic functions

X Riemann surface A meromorphic function is a homomorphic map f : X Ñ P1pCq. Equivalent description: pUi, ziqiPI atlas of coordinate charts X “ Ť

iPI Ui,

zi : Ui – zipUiq Ă C wij “ zi ˝ z´1

j

: zjpUi X Ujq Ñ zjpUi X Ujq transition maps f ØtfipziquiPI fipziq :“ f ˝ z´1

i

: zipUiq Ñ P1pCq fipwijpzjqq “ fjpzjq MpXq “ tf : X Ñ P1pCqu field of meromorphic functions

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Meromorphic functions & compactness

X is compact ñ a non-constant f : X Ñ P1pCq has a well-defined degree d: f assumes every value exactly d times counting multiplicities d “ ř

PPf ´1pcq νPpf ´ cq,

@c P C “ ř

PPf ´1p8q νP

` 1

f

˘ DivpXq “ tř

j njPju free abelian group

degpř

j njPjq “ ř j nj,

DivpXq Ñ Z divpf q :“ ř

PPX νPpf qP P DivpXq

divisor of f divpf ¨ gq “ divpf q ` divpgq div : MpXqˆ Ñ DivpXq homomorphism (we put divpconstq “ 0) Observe: principal divisors have degree 0. degpdivpf qq “ # of zeroes ´# of poles (counting multiplicities!) “ d ´ d “ 0

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Differential forms

pUi, ziqiPI atlas of coordinate charts on X X “ Ť

iPI Ui,

zi : Ui – zipUiq Ă C wij “ zi ˝ z´1

j

: zjpUi X Ujq Ñ zjpUi X Ujq transition maps A differential form ω on X is a collection of meromorphic functions ω “ tgipziquiPI satisfying gjpzjq “ gipwijpzjqqw1

ijpzjq,

@i, j. This transition or glueing rule comes from differential calculus: gipziqdzi ˇ ˇ ˇ

zi“wijpzjq “ gipwijpzjqqw1 ijpzjqdzj “ gjpzjqdzj.

ΩpXq “ tdiff .formsu is an MpXq-vector space of dimension 1.

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Divisors of differential forms

ω “ tgipziquiPI gjpzjq “ gipwijpzjqqw1

ijpzjq

@P P X νPpωq :“ νzipPqpgiq @i : P P Ui w1

ijpzjq ‰ 0 ñ νPpωq is well-defined

divpωq :“ ř

PPX νPpωqP P DivpXq

ΩpXq “ tdiff .formsu is an MpXq-vector space of dimension 1 ñ divisors of differential forms belong to one linear equivalence class DivpXq{divpMpXqˆq, called the canonical class. When X is compact:

§ degpdivpωqq “ 2g ´ 2 § dimCtω : divpωq ě 0u “ g

(holomorphic diff. forms) These facts are consequences of the Riemann–Roch Theorem.

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Riemann–Roch Theorem

D “ ÿ niPi P DivpXq LpDq “ tf P MpXqˆ : divpf q ` D ě 0u Y t0u νPipf q ě ´ni @i, νPpf q ě 0, P ‰ Pi Exercises:

§ LpDq – LpD ` divpgqq for any g P MpXqˆ § X compact ñ LpDq “ t0u when degpDq ă 0 § X compact ñ ℓpDq :“ dimC LpDq ă 8

• Theorem. Assume that X is compact. Let K :“ divpωq for some

0 ‰ ω P ΩpXq. Then for all D P DivpXq ℓpDq “ degpDq ` 1 ´ g ` ℓpK ´ Dq.

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Residues of differential forms & compactness

ω “ tgipziquiPI gjpzjqdzj “ gipwijpzjqqw1

ijpzjqdzj “ gipziqdzi

ñ residues are well-defined @P P X ResPpωq :“

1 2πi

ű

P gjpzjqdzj @j : P P Uj (here i “ ?´1)

• Lemma. If X is compact, then for any ω P ΩpXq one has

ř

P ResPpωq “ 0.

As an application, one can get another proof of existence of degree (valence) for meromorphic functions. It is sufficient to show that for a non-constant f P MpXq one has degpdivpf qq “ 0. Note that ResP ` df

f

˘ “ νppf q, thus degpdivpf qq “ ÿ

P

νPpf q “ ÿ

P

ResP ˆdf f ˙ “ 0.

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Example from modular forms

Γ Ă SL2pZq of finite index A modular (resp. cusp) form of weight k for Γ is a function f : H Ñ C satisfying: (i) f is holomorphic (ii) pf |khqpzq “ f pzq for every h P Γ (iii) f is bounded (resp. vanishing) at cusps, that is for every g P SL2pZq |pf |kgqpzq| “ Op1q (resp. op1q) as Impzq Ñ `8 Recall: right action of g “ ˆa b c d ˙ in weight k pf |kgqpzq “ 1 pcz ` dqk f ˆaz ` b cz ` d ˙

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Example from modular forms

Let f pzq be a modular form of weight 2 for Γ. Then f pzqdz is Γ-invariant: f pzqdz ˇ ˇ ˇ

z“ aw`b

cw`d

“ f ˆaw ` b cw ` d ˙ ˆaw ` b cw ` d ˙1 dw “ f ˆaw ` b cw ` d ˙ dw pcw ` dq2 “ f pwqdw ñ f pzqdz descends to a holomorphic differential form ωf P ΩpY q

• n Y pΓq “ ΓzH.

Near cusps: q “ expp2πiz

h q is a local coordinate near r8s P XpΓq

dq “ 2πi h expp2πiz h qdz ñ dz “ h 2πi dq q ωf “ h 2πi f pqqdq q P ΩpXpΓqq!

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