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 !