Math 220 C - Lecture 18

- - --

June 5 ,2020

Last time-

• I is flabby if t VE u ,Icu) → Flu) surjective

a

pFemina

.

o - I → G → Je → o exact,I flabby

⇒ o - F Cx) - S (x) → Tecx) → o exact

theorem ( sheaf cohomology) .

F contravariant functors

1-IP : Sheaves↳- Ab. Groups for p zo

'Il Ho (x,F) = F- Cx)

.

I flabby ⇒ ttpcx,F) =o + pzl .

o - I - G → fl → o there are morphisms

Sp : HP (x , Te) - HP "'

(x,

F) functional in exact sequences

suds that

↳ HP Cx,F ) → HPCX

,S) → HPCX

,Tl)

-s Het ' ( x

,F) →

. ..

( these requirements determine the functors uniquely !

Question How do we define cohomology ?- -

Fundamental Example ( Last time)- -- . .

Short exact sequence of complexes

• → Ai - B.

-s C'

→ o

induces

→Ttkca' ) → titers. ) → tired ) ,-

& Hk" CA.

) → . . .

Ideas 3 steps :

Taf Replace F by a complex of sheaves I ?

I•

Canonical flabby resolution--- -

o - F - F° - F'→

. . .

172 Global sections : F°f×) → F' ( x) → . - -

I Define

HP ( x,F) = cohomology of F (x) .

=

Ker t"(x) → FP"f×)-

Im Ft-'

(x) - IPCx) .

PIpar ford proofed - sh-eafo-fdiscontrnuousseafror.se

For F → X construct a sheaf of I→ X

of F ( u) = { Cfa) * ⇐ u , fa E IE }.

II o → I → lot : s c- Fcu) - ( Sa) C- &F ( u).

A C-U

Ti, & F flabby : extend Flu) → fo Fcv) by zero .

my o → I → b → fl → o exact

I

o → of F → of b → & Tl → o exact

( since exactness holds on stalks),

tf o o o

i. l l

.

• - F - S - ye → •/ exact

t t t° - GI - 4S → of Td→ of

ex .at by

I.

I. I I

v

o- lofts → 491g → of The- o exact

t t tO

° 0

First two rows are exact & columns exact ⇒ third row is exact.

Tvt Canonical flabby resolution-

- -- - - -

←flab'sf flabby . . .

( *) o - F - I-

- F'→ . . .

Iterate- -

:

( o) o - F → off → doffs = I'

- o

ca) o → I ' → GE' - 4 EYE ' =I-→ o

- - -

( p) .o → It

→ IP → IT "'→ o

Inductively to = of F ⇒ I' =4¥ ,

I -= I

IP= ¢ It ⇒ FT' '

= of I%Ep

' How does it interact with exact sequences ?

o → I → b - Je → o

Inductively for all p we have diagram (by ⇒) .

o - IP→ g- P → Jets → o

l. t t

o → IP → GP → Jlp → o

t t t• → IPT '

→ TP + '

→ GIFT'→o

thus o - I → G - Te - o exact

t t t ±

(Y ) o →F-

- G.

- Seo - oexact

Proof of theorem For F → x construct- - - -

-

Tad o - F - F° → I? - . . .canonical resolution

Tbd II) → F' (x) -1 . . - global sections

II Define

HP (x.F) =Ker FP (x) → JP-14×3- - - -

Im F" (x) - F'Tx)

why does it work- - - -

?

PIPIL 'I H°(x. IS = Ker F9⇒ F' Cx)

( o) .

t

p - i

- o - F ( x) - F°( x) → F Cx)

z .

(1) .

.

o - F- ( x) - F'Cx) - E-

Cx).

Ken Cyp) = Ker13

since 2 is injective by c ,)

I F Cx) by ( o).

Property I flabby ⇒- - -

• IP = of IP flabby tpzo

• IP flabby t p Zo Wds? Inducton

p . using

f flabby 1 flabby⇒

f flabby( p) o → TIP

→ FP → IT PT'- •

Furthermore, by Lemma last time

• → IP (×) - F'Tx) → IT×) → oexaot

IF. ( x) - F' Cx) → F-Ex) → . . . exact

.

⇒ HP (x , I) = o .

it pz i. ,F flabby

Property o - I - G - Je → o- - -

exact

fly by (f) above

o → F• → 3.°

→ It.

→ o exact

µ F.

is flabby & Lemma

o → F. (x) → g. (x) → H.

(x) → o exact

µ "

fundamental example"

above

→ HP (x,F) → HP Cx

,S) → H'Tx

,Te)

-> Htt'( ×

,t) → . . -

CozHAI H" (x

,F) = o for pzi if I skyscraper sheaf .

Sheaves of Ox - modules I is ①× -

module if# # #

t u EX,Tcu) is a module over Gx (u)

.

and

it U Z V,

Ox Cu) x Fcu) - Fcu)

t a.

IG× Cv) x Fsu) - FG)

Remark I = G× CD) is a sheaf of Ox - modules Indeed if- -

n

g e O (u) , f- e Fcu) as gf C- Fcu)

⇐s div Cgf) + D) u Zo

⇐S drug + divf +Dfa Zo is correct.

-

-

Zo Zo since f e Fcu)

-

Renard,

I is Ox - module =3 F- (x) is a - rector space ⇒ Hocx,I)

is a - vector space . Also HP (x,F) is Q - vector space .

Then X compact Riemann surface , I coherent Ox - module ( e - g . QCD)).

=> dim HP Cx,

F) < is .& HP (x

,F) = - if p > I .

¢

Exempt g - olim H' ( x,Q)

.

Define X (x ,F) = E C -D' dam HPCX

,F) .

Eixample . I = skyscraper sheaf

Xcx,F) = I

• F = G. Xcx

,6) = n - demH) = , - g .

g

R o → Vo → V , → . . .→ Ve → 0 (vector spaces) .

=> E C - ish dem Vq = 0 .

"

G

' setting

o - I - G - Se → o

=3 o - Ho Cx,F) → HEX

,

b) → H'( x,Je)

-H'( ×

,I) - H

'(x

,3) → . . .

⇒ x ( x, b ) = X (x , F) 1- Xcx,Tl)

.

Riemann - Roch- - - -

X compact

X ( x, 0×677 = 1 - g t deg D;

Quick proof when D Zo D= Enp Ep] .

- - -

o - ①×→ Ox CD) - IT QpnP → o

.

=3 X ( x , Ox ( DD = Xcx , 0×7 t E np XCX , Qp) .

-

= e - g t Emp'

= i - g t degD'

General case-

(*) o - Ox CD) → Gx (Dtp)→ Gp → o

.

⇒ X ( Q, CDtpD = X ( Ox Cb)) -11 .

Define

f CD) = X ( Ox CD)) - Cr- g t deg D) .

I f co) = o .

El f CD) = f (Dtp) ⇒ f- Cst E) - fcp) t E Zo

d-ff constant . Indeed Dr Z Dz⇐ b

,- D2 20 .

Note f CD,) = f (D2) .

if D,z Dz

.

In general , any DisDa s fend

Dz Z D, & Dez Z D, ⇒ f- (Dz) = f- CD,) & f- ( Dz) = f- (Dz) =, f const .

Thus f constant - s f EO =3 QED .

hop .

←Proof of sequence (* ) .

If D= E + up .

we show- - - -

Po → G -CE

trip) - G (E + anti) p)→ Ap → °

For f with du f t E t Cn +Dp zo , pick local coordinates near p and

A-n -7

f Cz) = - t - - -Laurent

expansion .n-11

( 2 - p)

be fine 13 Cf ) = a - n - , . Exactness follows easily .

Corollary dem Ho Cx, Ox (DD Z

' -

g + deg D .

( Riemann inequality) .

Remade The difference isgiven by dim H

' (x , CDD.

When togo from here ?-- -

II sheaf cohomology in more detail

ITI GenusI

- g = dim H'(x ,G×)

- topological genus 2g = dim H'Cx

,z)

- via I - forms g = dim H-

( x,R'× ) .

I

II.Serre duality

HOS x,F ) Z H

'

(x,

F-

⑦ sz!),

F"

- Hom CF, Ox)

Gx

III Line bundles 2 → x d Jacobian

II ample , very ample , projective embeddings

X - e?

I the moduli space of cu-res Mg , Mj . . .

Many directions are possible !