Vector bundles on curves

Lectures by Gerd Faltings held in Bonn

Notes by Michael Stoll

Comments remarks questions etc inserted by me are set in slanted

Contributions by other p eople are set in sans serif and marked by their

initials

ck Christian Kaiser

Since these notes are quite far from p erfection I very much appreci

ate any comments suggestions corrections or whatever Send me email to

michaelrheiniamunibonnde or put your message in my mailb ox in

the staircase of Beringstr

Note: The information above on email or mailbox is obsolete.

My current address is: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany [email protected] Thanks to Ulrich Görtz for providing me with a copy of the original postscript file.

Michael Stoll, September 12, 2008

May

Intro duction

Let C b e a smo oth pro jective connected algebraic curve over some eld

k A vector bund le always on C if not otherwise sp ecied is a lo cally free

sheaf E

Goal Construct a mo duli space of vector bundles over C

There are two invariants asso ciated to E

The rank rk E dim E k x for any x C

rkE

We have the determinant bund le of E det E E see b elow for the

construction it is a line bundle The degree of E is deg E deg det E

this is the degree of a divisor corresp onding to det E eg the divisor

of a global section if it exists

Construction of det E

Let r rk E and consider more generally a representation GLr

GLr for det E this is det GLr GL G Cho ose an op en

m

S

r

cover C O U of C such that there are isomorphisms E j

i i U

iI

i

U

i

On U U we have

i j

GLr U U O

ij i i j

j

and on U U U we have Then GLr U U O

i j k ij j k ik ij i j

denes a vector bundle E As another example we have the dual bund le

t t

E given by the contragredient representation

r

There exist vector bundles of any degree and rank O L with a line

C

bundle L has rank r and degree deg L

Theorem RiemannRoch

C E dim H C E dim H C E g rk E deg E

where g is the genus of C and C E is the Euler characteristic of E

Example Let k b e algebraically closed and lo ok at the case g ie

C P There is an exact sequence

O O O

O is the universal line and O is the universal quotient O has rank

and degree we have a global section coming from the global section

of O which has a unique and simple zero at the line k

n

We let O n O for n Z

Theorem On P every vector bund le E is isomorphic to a direct sum of

O ns

O m O m O m E

r

And this is essential ly unique

Proof Coherent sheaves on P essentially corresp ond to nitely generated

L

M graded mo dules over R k t t To a graded Rmo dule M

n

nZ

we asso ciate the coherent sheaf F that is dened by

D t F M D t F M

t t

where D t is the op en subvariety of P dened by t and M is

j j t

j

the homogeneous localization ie the elements of degree zero in the usual

We get a trivial F i M for n suciently big since lo calization M

n t

j

h

mt

m j

then for h and M M have the same image i there is

dh d

t

t

j

j

a common submo dule M of M and M coinciding with b oth in suciently

high degrees

Given a coherent sheaf F a suitable M is

M M

P F n M M

n

nn nn

where F n F O n with some arbitrary n Z M is nitely

generated by Serres Theorem H I I I For F O m we may take

M Rm dened by M R this is a free Rmo dule generated by

n mn

M

m

Now let E b e a vector bundle We claim that there are m m Z

r

such that

M

Rm Rm M P E n

r

nZ

Im particular P E n for n We rst show this By Serre

t t N

E m E m is globally generated for m hence we have a surjection O

P

N

Then dually there is an injection E m O which by tensoring with

P

N

O gives E m O and the latter has no nontrivial

P

global sections

Next we show that M is a pro jective Rmo dule R is regular of dimension

T

thus pro jective is equivalent to reexive ie M M where p runs

p

through the prime ideals of height one in R see b elow ck Consider

A A n fg U P

t t t t

L

O m Then O

U

mZ

t t

Sp ec k t t Sp ec k Sp ec k t t D t t Sp ec k

t t

t t

Sp ec k t t t t Sp ec k

t t

t t

Sp ec k t t Sp ec k Sp ec k t t t Sp ec k D t

t t

Now

M

m

t t

k t k t t k

t t

mZ

L

this corresp onds to O m it is glued over D t D t with the

mZ

corresp onding expression obtained by exchanging t and t We then have

M

E m E O E

U O

P

mZ

Taking global sections P we get

M

P E m M U E P E

mZ

Here E is a vector bundle on U A Sp ec R Let p R b e a prime

ideal of height it corresp onds to a curve in A A y M Quot R is

in M i the corresp onding section is regular on an op en subset of ck this

p

T

M the singular set can contain only discrete p oints but curve If y

p

p

then the section must b e regular hence y M

L

Rm M Now since M is nitely generated there is a surjection

Because M is pro jective it is a direct summand with the graded structures

hence M Rm Rm with suitable m To get the graded splitting

r i

L

Rm as Rmo dules not necessarily resp ecting take a splitting s M

the grading Let pr M M b e the projection of M onto its degreem

m m

part and dene

M X

s M spr a pr Rm by sa

m m

mZ

Then s is a homomorphism of graded Rmo dules identifying M with a direct

L

Rm ck summand of

Since M t t M k m k m where k m denotes a one

r

dimensional graded k vector space in degree m this representation of M is

unique

Going back to sheaves nishes the pro of 2

On the equivalence of projective and reexive used in the pro of ab ove

Let A b e a regular noetherian ring and let M b e a nitely generated

Amo dule Let K Quot A b e the quotient eld of A

Claim The following two prop erties of M are equivalent

i M M where M Hom M A

A

T

M M K where the intersection is over all prime ideals p of ii M

p A

p

height in A

If M has these prop erties M is called reexive

Proof i and ii b oth imply that M is torsion free ie M M K

A

Since M K M K canonically we have inclusions

A A

M M M K and M M M K

p A A

p

T

hence it is sucient to show M M

p

p

Let p b e a prime ideal of height then A is a discrete valuation

p

T

ring Since M is torsion free we have M M whence M M

p p

p

p p

T

M

p

p

T

Let x M Consider l M ie l M A l extends to

p

p

T T

l M K K Since x M l x A A whence x M

K A p K p

p p

2

Serre If A is a regular noetherian ring of dimension and M is a

nitely generated reexive Amo dule then M is projective

Proof Without loss of generality we may assume A to b e lo cal Then we have

to show that M is a free Amo dule which is equivalent to M b eing at over A

A

which in turn is equivalent to Tor M k for all j where k Am is

A

j

the residue eld of A

Let K Quot A Let n dim M K then since M is torsion free

K A

n n

there is an injection i M A Let Y A M and consider the long exact

Torsequence for

n

M A Y

A A

Y k for j A k for j it suces to show Tor Since Tor

j j

Since M is reexive all zero divisors of Y b elong to some prime ideal p of

A of height therefore the set of asso ciated prime ideals of Y ie maximal

elements in the set of annihilators of elements of Y consists of nitely many

prime ideals of height In particular m do es not b elong to this set Hence

A

there is some x m such that we have a short exact sequence

A

x

Y Y Y xY

Lo oking again at the long exact Torsequence we see that we only need to

A

show Tor Y xY k for j Note that multiplication by x is zero on

j

A

Tor Y k since it is zero on k But this follows from the fact that A is regular

j

of dimension 2

ck

General Observations

Supp ose a mo duli space M exists Let S b e some parameter space and

consider vector bundles on C S The isomorphism classes of these ob jects

should b e given by Map S M

M is smo oth

Lift over innitesimals Let R b e a ring I R an ideal with I and let

S Sp ec R S Sp ec RI We have to show that any morphism S M

extends to S This means that any vector bundle E on C S comes from

a vector bundle E on C S It will b e imp ortant that C has dimension

This criterion lo oks a bit like H I I I iii

S

Cho ose an op en ane cover C S U such that we have isomor

i

i

phisms

r

O E j

i U

U i

i

S

U Since S S as spaces the U also give an op en ane cover C S

i i

i

As usual we then have U U GLr with

ij i j

ij j k ik

It is always p ossible to lift to U U GLr but we have to

ij ij i j

preserve So take any lifting and write

ij

ij j k ij k

ik

r r

U U U I E nd E this with U U U I O

i j k ij k i j k

C S

identication is given by the trivialization via Question E is not yet

i

constructed can we use E nd E and here Because of I we get

i

on U U U U the relation by calculating in two ways ck

i j k l ij j k k l

ij k ij l ik l j k l

We may change our lifts by some with U U I E nd E

ij ij ij i j

Then changes by We have

ij k ij ik j k

H C S I E nd E f j holdsgf g

ij k ij ik j k

H since C is a curveH I I I and H I I I rememb er S is ane

Kunnethformula hence we can achieve

ij k

We see that H means that M is smo oth H also has a meaning

its dimension is that of the tangent space of M

Let Sp ec k x M b e a rational p oint It corresp onds to a vector

bundle E on C fxg C Try to extend Sp ec k M to S M with

S Sp ec k The following ob jects corresp ond to each other

m

x

C B

C B

O k

Mx

C B

Hom m m k C B Sp ec S M

k x

x

C B

A

x

k

and the dimension of the Hom is that of the tangent space of M in x

k

By the discussion ab ove the p ossible lifts corresp ond to changes by s

ij

with dierent lifts are isomorphic with an isomorphism

ij ik j k

that is the identity mo d I i the s dier by Hence the

ij i j

dierent lifts are parametrized by

H C S I E nd E H C E nd E

and the dimension of the tangent space is given by dim H C E nd E

And what is H Well it can b e interpreted as

H C I E nd E kerAut E Aut E f j C I E nd E g

Shouldnt it b e H C S I E nd E They are the same since I kills

I E nd E so that I E nd E is an O O I O mo dule ck

C C S C S

By RiemannRoch use that E nd E is a vector bundle of rank r and

degree

dim H C E nd E dim H C E nd E r g

A mo duli space for all vector bundles do es not exist but there will b e one

for those E with H C E nd E k its dimension then is by our discussion

ab ove g r

May

Raw notes by Michel Fontaine

Why do mo duli spaces not exist

There are diculties with automorphisms

Let E b e a vector bundle on C S S let L b e a line bundle on S and

S

take E E L There is an op en cover S U such that Lj is trivial

i U

i

E So if M exists there are maps for all i Hence on C U we have E

i

S M corresp onding to E and E resp Since E j E j

C U C U

i i

we should have j j whence But it is p ossible that E E

U U

i i

and so and should b e distinct a contradiction Let E E b e a

homomorphism Then

C S Hom E E L

S Hom E E L

End E S L

C

z

S E ndE

If eg S L cannot b e an isomorphism

Problem G Aut E

m

Need to consider lo cal isomorphism classes but then the mo duli space

will not b e very nice

Solutions

a Consider M as a mo duli stack see later lectures

b Restrict the set of vector bundles that are considered eg take only E

such that End E k or semistable bund les

deg E

For a vector bundle E dene its slope to b e E

rkE

E is called semistable stable if for every F E with F E

F E F E

Remark Any F E is torsion free and hence a vector bundle but E F

may have torsion There exists some F F E such that F F has nite

supp ort and such that E F is torsion free We have

rk F rk F and deg F deg F length F F

Explanation For our purp oses a torsion sheaf is a nite sum of coher

ent skyscrap er sheaves ie sheaves of nite supp ort with nitedimensional

P

dim T x stalks To such a sheaf T corresp onds the p ositive divisor D

x

xC

If we want the degree to b e additive on exact sequences which it should b e

det F det F F F F det F

P

then we are forced to dene deg T deg D dim T length T since we

x

x

have the exact sequence

LD O T

C

where deg LD deg D and deg O

C

Over an op en ane subset U Sp ec A C a coherent sheaf corresp onds

to a nitely generated Amo dule M For suitable A eg principal ideal

domains M decomp oses as

M M M

tors

with M a nitely generated free Amo dule This corresp onds to

F F E F E F

in the Remark ab ove rk F means the generic rank of F

Remark We can also consider quotients E G The condition for semi

n o

stability then is G E

Proof For every exact sequence F E G we have

deg E deg F deg G and rk E rk F rk G

n o n o

If G E then deg G rk E deg E rk G hence

n o n o

deg E rk F deg F rk E ie E F contradiction

Let F as ab ove Then the same argument in reverse direction shows

o n

E Since F F the claim follows 2 F

Remark If E and E are semistable vector bundles with E E

then Hom E E

E F E E F E contradiction

If E E we have F F and the map is strict ie it has constant

rank at each p oint

If E and E are even stable and E E then every nontrivial map

E E is an isomorphism In particular End E is a division algebra for

stable E and for k algebraically closed we have End E k

The HarderNarasimhan Filtration

For a semistable bundle E this is given by

E

If E is not semistable take F E such that F has maximal slop e

F and among the subvector bundles with maximal slop e has maxi

mal rank rk F This exists since the slop e is b ounded dim H C E

dim H C F deg F g rk F by RiemannRoch hence F

g dim H C E We now have

a F is semistable by construction

b Any G F E F has G F F Otherwise G F and

rk G rk F

c F F otherwise F F

We take E F as the rst step in the HarderNarasimhan ltration

E E E E

r

of E The construction is continued with E F replacing E We get a ltration

such that E E is semistable with slop e where

i i i r

We reindex the ltration as follows For R let

HN E E

maxfiji or g

i

Then HN HN if

Remark If f E F is a homomorphism then f HN E f HN F

for all In particular HN do es not dep end on the choices made in the

denition

Proof Let E HN E and cho ose maximal ie j minimal such that

i

f E HN F F Then we have maps

i j

f

F F F E E

j j j i i

and the comp osite f is not zero since j was minimal Supp ose that

HN F HN F then Now F F is semistable with slop e

j j

and for i E E is semistable with slop e hence f

induces the trivial map on E E for i Hence f on

E a contradiction 2

i

Consider a semistable vector bundle E of slop e Then

X M

F F with I nite

i

F E

iI

F stable

F

P

For F stable with slop e Hom F F tells us how many copies of F

o ccur in the sum Continue with this in the quotient We see that we get

the semistable bundles from the stable ones by direct sums and extensions

If E and E are semistable with the same slop e and we have an extension

E E E

then E is also semistable lo ok at the HN ltration

Let t b e a section of O on P which has its only and simple zero in

Consider the following bundles on C P

E pr E pr E pr O pr E

where pr E by pr t

E pr E pr O

E pr E

We get an extension of bundles on C P

E E E

For x P x we have E j E from the inclusion of E as the

C fxg

E E Lo oking at the map to mo duli space rst summand but E j

C fg

corresp onding to E we get

P mo duli space of semistable bundles

j

A E

E E

so E has to b e the same class as E E Hence we can only exp ect a

corresp ondence

Semistable bundles which are

Points in mo duli space

direct sums of stable bundles

Result

Consider E on C S for each s S lo ok at the HN ltration of E in C fsg

This gives a collection of slop es and ranks

Claim This is a constructible function That is S S with con

i

structible S constructible nite union of intersections of an op en and a

i

closed set such that the function is constant on each S

i

Proof Inductive criterion We have to show that if T S is irreducible

the function is constant on a nonempty op en subset U T Hence we

may assume S T and if necessary replace S by an op en subset such that

S Sp ec R with an integral domain R Let S b e the generic p oint Over

k E has a HN ltration which is dened over a nite extension K C

of k in fact even over k Replace R k by its normalization R

in K then S Sp ec R S Assume the assertion holds over S then the

with some f R and hence it is also HN ltration is constant on R

f

constant on R with f Normf Therefore without loss of generality we

f

S

may assume that the HN ltration is dened over k R

f

f

We can assume that the E HN E extend to subbundles E of E on

i i

C S Then it is enough to show that E E is semistable of constant slop e

i i

on each b er over an nonempty op en subset of S Hence we have reduced

to showing

E semistable for s op en subset semistable E j

fsg C

Tensor with a suitable line bundle L E E L such that E g

where g is the genus of C

Remark Any semistable F with F g has H C F and is

generated by global sections

is a line bundle of degree g hence semistable with g Proof

C

Let V b e a semistable vector bundle with V g Then Hom V

C

But this is dual to H C V hence H C V as well Take x C and

let F x F Lx Lx is a line bundle of degree consisting of

regular functions vanishing in x then F x F g hence

H C F x and therefore H C F H C F F x F ie

x

F is generated by global sections 2

Supp ose that E j is not semistable Then there is some F E j

C fsg C fsg

which is semistable of slop e E g F is generated by

W C fsg F V C E k s

We therefore have to test all subspaces W V if they generate a vector

bundle F with F E

Let G Grassmannians b e the pro jective variety parametrizing all

p ossible W s Over G we have a universal bundle W C E O

G

It generates a subsheaf F pr E on C G S Find a decomp osition

is at over T G S T into constructible subsets T such that F j

i i i C T

i

Then the rank and degree are lo cally constant on T hence by taking a

i

ner decomp osition we may assume them constant in particular we have

constant slop e on T

i i

Consider those T with E The by Chevalley constructible

i i

subset pr T of S do es not contain the generic p oint of S b ecause E is

S i

semistable hence lies in a prop er closed subset of S Now if we replace S

by the op en complement of these prop er closed subsets then these T must

i

disapp ear and the pro of is complete 2

Semicontinuity

Prop osition Take some vector bund le E and look at the pairs rk F deg F

for al l subbund les F E Then we get the fol lowing picture

deg F

6

E

E

A

A

A

A

al l other F in here

A

A

E

-

rk F

Under specialization the domain may at most become bigger

Proof The picture is clear To prove the statement ab out sp ecialization

Let V b e a discrete valuation ring and S Sp ec V Consider E on C S

Let S b e the generic p oint and E the generic HN ltration dened

i

over Extend E to a subsheaf E E over C S Reexive hence

i i

these are vector bundles and E coincides on b oth b ers Hence on

i i

the sp ecial b er we have subbundles E k s E k s mapping to the

i

same p oints as E Hence the region for E contains that for E

i s

If S is arbitrary with s t S such that s ftg there is a discrete

valuation ring V and a map Sp ec V S mapping to t and the sp ecial

p oint of Sp ec V to s Is this p ossible when dim ftg dim fsg Mayb e

one has to break this up into several steps Now use the ab ove argument

By sp ecializing step by step we may assume S to b e lo cal and of dimension

By pulling E back to the normalization S S we may assume S Sp ec V

with V a discrete valuation ring ck 2

May

Raw notes by Bernd Steinert

Geometric Invariant Theory

Why do we need this

Approach to mo duli spaces First construct mo duli space for bundles with

additional structure Then eliminate additional structure by taking invari

ants see b elow

Let E b e a semistable bundle with deg E d rk E r We may assume

dr g then C E generates E and H C E see last lecture

We have by RiemannRoch

dim C E d r g N

We want to parametrize E e e where e e is a basis of the

N N

global sections C E

Prop osition This gives a representable functor Grothendieck

Proof or rather Explanation E is a quotient of V O with a vector

space V of dimension dim V N There is an algebraic scheme H Hilbert

scheme and a universal quotient sheaf V O F which is at over

C H

O

H

One has

a

H H

dr

dr

where on H the degree d and the generic rank r are xed

dr

The hard part in the pro of is to show that H is of nite typ e

dr

Over a suitable op en subset H of H we have that

dr

a F is a vector bundle

b F is semistable on bres F is a vector bundle of rank N

C H H

c V O F is an isomorphism

Then H is representing the functor describ ed ab ove 2

Now try to get rid of the additional data

Two dierent bases of C E are related by GLN So GLN acts on

H Our mo duli space M should b e the quotient GLN nH The scalars act

trivially whence an action of PGLN this refers to the rst formulation

where bases mo d scalars were to b e parametrized

Remark H is pro jective hence H is quasipro jective

dr

Proof Sketch Use some Grassmannian

Take a line bundle L of degree deg L such that for any V O E

C

with deg E d rk E r with kernel K K L is generated by H C K L

and H C K L b ounded family

Note that

H C K L V H C L H C E L

z

has xed dimension

denes an emb edding of H into some Grassmannian G and the ample

rd

line bundle det H C E L on H corresp onds to the ample line bundle

dr

on G given by the dual of detsubspace ie detquotient space 2

How to construct quotients

In char k we use Invariant Theory

Let G b e a reductive group eg GL SL PGL

N N N

N

Problem X P quasipro jective G acts on X via G GL What

N

is XG

L L

G

Try R X O n R Is XG Pro j R

n

n n

G

Hilbert R is nitely generated over k

The key prop erty of G is that for any Gsurjection V W of vector

G G

spaces the induced map V W is again surjective This is equivalent

to H G V for every vector space V with Gaction It is equivalent

to H G V H G W b eing an injection for every Ginjection of vector

spaces V W Now emb ed V into an injective k Gmo dule W cf Brown

Cohomology of Groups Then H G W whence H G V

Since G is reductive G acts semisimply on inductive limits of algebraic repre

sentations This implies the lifting prop erty ab ove ck

In characteristic p the induced map is no longer surjective but at

least some pp ower can always b e lifted

L

G

R this is an ideal of R I R is a homogeneous ideal Let I

n

n

G G

hence I R f f with homogeneous f I Then the f generate

r i i

G G

I as an R mo dule

P

G

Take a homogeneous f I There are r R with f r f ie

i i i

r G

R Hence by the lifting prop erty ab ove we have a Gsurjection R I

G G G G r G

which means that the r can b e taken in R R I R I

i

G

This implies that the f generate R as a k algebra We now have an

i

emb edding

G M

Pro j R P

dened via all monomials in f of degree lcm fdeg f g

i i

There are maps

G

fhomog ideals of R g fhomog ideals of R g

I IR

G G

I I R p I

where id

Recall Pro j is given by the homogeneous prime ideals except I

L

R This means that on the Pro j level there is in general no map

n

n

corresp onding to ab ove

Remark and Denition One can dene a quotient map only on homoge

L

G

neous prime ideals which do not contain R The corresp onding p oints

n

n

x are called semistable they are given by the prop erty that there is some f

G

in some R n with f x Sometimes we will also call a p oint

n

N N

x A mapping to x X P semistable

ss G

Let X b e the complement in X of the zeroset of all R for n This

n

is an op en subset of X

Remark

Gx x semistable

for someany x x

G

Proof There is a f R with n and f f x Then

n

Gx but not in f f vanishes on

L

Let J R b e the ideal dening Gx then J I R and

n

n

hence J k RI where the Gaction on the right hand side is trivial

G G

Hence J k ie there is a f f f J with f But then

since f x some f x for some i 2

i

We get a map

ss G

X Pro j R

G

I I

What are the b ers of this map

G

for some x and y or x and y lie in the same b er i some f R

n

n vanishes at b oth of them this should read i for all n and

G

f vanishes on x i f vanishes on y We hop e that each b er all f R

n

ss

contains precisely one closed in X Gorbit This is indeed true

Proof

a Any b er contains a closed orbit take some x with dim Gx minimal

ss

b We have to show If Gx and Gy are closed in X and x and y have

G

the same image in Pro j R then Gx Gy

G

with d such that f x and f y Cho ose some f R

d

Let S R b e the lo calization and Y Sp ec S the op en subset dened by

f

N ss

f in X A Y pro jects to X Gx and Gy dene Ginvariant ideals

I J S resp Assume Gx Gy then Gx Gy and hence I J S

G

By the lifting prop erty I SJ

On the other hand let I I S b e the homogeneous ideals corresp ond

x y

G G G G G

ing to x and y resp Then I I I and I I I J I This

x y x y

implies I J I I S a contradiction

x

Better Let I J I J b e the homogeneous ideals in R corresp onding to Gx

x y

L

Gy x and y resp ectively Since Gx Gy fg we have I J I R

n

n

Hence

G G G G G G

I I J I J I

x y x

which contradicts the fact that x is semistable The rst equality comes from

G G

the lifting prop erty the last one from the fact that I J by assumption

x y

ck 2

x is called properly stable if it is semistable and dim G for all y in

y

s

an op en neighb orho o d of x X stable p oints of X

L

N N

V op eration of G on P PV is Example Let V k V

m m

mZ

P

m

given by v x is v for v V Then x

m m

semistable m x and m x

m m

stable in addition x x

To test semistability in general it turns out to b e enough to do it for

all G in G

m

Theorem x is semistable for G x is semistable for al l G G

m

Proof Gx G x

m

Use valuative criterion If Gx then there is a discrete valua

tion ring V k t with quotient eld K such that there is a commutative

diagram

Sp ec K G g

y y y

N

Sp ec V A g x

and such that for g GK g x is integral and go es to as t For

G GL SL PGL

N N N

a

t

C B

C B

g g g

A

a

N

t

a

t

C B

C B

g x is integral Write g with g g GV hence

A

a

N

t

g higher order terms and replace x by g x We may assume that

g mo d t

X X

n n

n n n

t x g x x t x

N

n n

n

a n For n x x The integrality condition implies x

i

i

since g mo d t hence for x we have a This means that x

i

i

is not semistable for this parameter subgroup of G 2

Theorem x is stable for G x is stable for al l nontrivial G

m

G

Proof Clear

s s s

i G G G X X X is a prop er map even nite

m

s

Can cover X by invariant anes where x x with a b

a b

or a b Invert x G onane space one co ordinate is x x

a m b a

deg invert this

ss

ii Consider U X where the stabilizer has dimension U is constructible

and op en stbilizer will jump under sp ecialization Claim G U U U

prop er 2

May

Raw notes by Bernd Steinert

Recall

x semistable Gx

G

d f R f x

d

G G parameter subgroup

m

the weights of this G action o ccurring in x

m

are neither all nor all

N

x stable G A g g x is prop er

Gx is closed and G is nite

x

Remark

G

for some d with f x If G a Assume x semistable f R

y

d

the stabilizer of y is nite for all y with f y then all orbits

in there in where mayb e in X have dimension dim G and are

f

S

Gx n Gx orbits of smaller dimension closed

b Assume Gx closed G nite Then fy j dim G g is closed

x y

Gx Hence there is some h Ginvariant and do es not meet Gx

G

R n fg such that h vanishes on Y fy j dim G g but not at

y

x Let I and I b e the homogeneous ideals corresp onding to Y

Y Gx

and Gx resp Since Y Gx we have I I I therefore

Y Gx

G G G G G

I I I If we had I I we would get

Y Gx Y Gx

G G G

I I I contradicting semistability of x ck Since

Gx x

Y is G invariant we can take h to b e homogeneous Then change f

m

to f h Thus we have the situation of a This implies that the set of

stable p oints is op en ck

parameter subgroup criterion for semistability

For all G G nontrivial p ositive as well as negative weights o ccur

m

N

in x Ie take a basis b b of A diagonalizing the G action the

N m

w

weight of b is w with b b w o ccurs in x x x written

j j j N

wrt the basis b if there is a j such that x and b has weight w

j j j

Valuative criterion for what

V K discrete valuation ring with quotient eld g GK

B C

N

B C

g g g with g g GV g x A V g GV

A

z

really

Example V W vector spaces G SLV acting on Grass V W Here

d

Grass V W classies quotients V W E with dim E d

d

o n

semistable

i for all nontrivial subspaces V V and Criterion E is

stable

E imV W E we have

n o

dim E dim V

dim E dim V

Proof

Write V V V with dim V a dim V b and let

a b

t id G SLV t t id

V m V

d d

We obtain a pro jective emb edding of Grass V W from V W E

d

d

det E Elements of V W give global sections of det E which is an

ample line bundle on Grass V W unique up to p owers We will lo ok

d

at the weights of E under in this emb edding

We cho ose a basis of V W consisting of v w with v basis of V

basis of V and w basis of W Then

d

v w v w V W

d d

has weight b fi j v V g a fi j v V g

i i

Since E is semistable a nonnegative weight o ccurs in E This means

that there are v w such that hv w i E and such that b a where

i i i i

fi j v V g and fi j v V g

i i

Now hv w j v V i E hence dim E dimimV W

i i i

Since b a we have

dim E a dim V

dim E a b dim V

L

Let b e a parameter subgroup and write V V such that

i

iZ

P

i

i dim V b ecause G SLV t acts as t on V Then

i i

iZ

d d

Assume the stability criterion holds Let V W E then we

have to show that v for some v of weight What ab out weight

I think b oth should o ccur Take t t ck

L

Let F V V Then

i

i

X

V dim gr

F

Z

X

dim F V dim F V

Z

X X

dim F V dim F V

N N

X

N

dim F V N dim F V

z

N

V

for N big enough

We lo ok for fv w j i dg such that hv w i E and try to

i i i i

maximize the weight We take a elements v F V The b est strategy

i

obviously is to take a dim imF V W E This is p ossible Start

n

with F V W n big this gives a Then go down step by step

n

cho osing a a a a new basis elements The weight is then

n n n n

X X

w a a a N a

N

N

We know

dim F V a

d dim V

Hence

X X

d dim F V

N d w d dim F V N dim V

dim V dim V

N N

For stability one needs that there are at least two stages in the ltration

2

Remark This what never happ ens when dim W Ie for dim W

no E is semistable ck

Application to vector bundles on curves

Let C b e a smo oth pro jective connected curve with genus g g C For

a vector bundle E on C let

deg E

E g

rk E

ie h C E h C E rk E RiemannRoch Note that this is a

mo dication of the slop e used earlier

n

Let O b e an ample line bundle of degree and dene O n O

E n E O n Then E n E n

Mr will denote the mo duli space of semistable E with rk E r

and E

Since E E a induces an isomorphism Mr Mr a we may

without loss of generality assume that E

If g g will do then E is globally generated by C E

which has dimension N r So

SL

N

Q Mr

N

with a quotient scheme Q classifying quotients E of O with Hilbert

p olynomial n r n dim C E n n big see later lectures

We shall cho ose some by shifting a Once a is

chosen cho ose b a such that for any exact sequence

r a

K O E a

where E has the correct Hilbert p olynomial K b is generated by global

sections and H C K b

This gives an emb edding

Q Grass  SLr a

r a r a

E a O K C K b C O b C E a b

Claim For a

corresp onding p oint in Grass

E semistable vector bundle

is semistable

To test stability in Grass we have to consider

r a

V V k C E a

C F a V F a E a is generated by V

Idea For a F E nice subbundle with rk F r F

dim F a r a by RR

z

V

dim F a b r a b

z

imV W

Condition

a b a b

a a

For b

Argument Either F E or F E with some xed

May

As always C is a nice curve of genus g and we x the data r rk E and

deg E

g such that g Then H C E and by E

rkE

RR N dim C E r

By Grothendieck there is a quotient scheme Q parametrizing

N

quotients O E with Hilbert p olynomial n r n Ie for n

C

dim C E n r n Q is of nite typ e

N

Cho ose b Z so big that for all K kerO E H C K b

C

and H C K b generates K b For b xed the condition denes an op en

S

U Q since for every K there is some b satisfying subset U of Q We have

b b

b

the condition Because of quasicompactness and U U there is some

b b

b with U Q

b

There is a map

Q Grass

r b

N N

E C O b C E b O

C

N

Grass parametrizing quotients If b is chosen as ab ove C O b

N

E C E b is surjective since H C K b with K kerO

C

hence the map is welldened and we recover K b and hence K as the

N

subbundle of O b generated by the kernel of Hence the map is an

injection It is also equivariant with resp ect to the SL action on b oth

N

sides

N

Let V k and W V a subspace Consider the following situation

N

E V O O

C

C

K W O F

W C

W want to cho ose b in such a way that additionally H C K b for all

W

these K This is p ossible The pairs E W are parametrized by a quasi

W

compact scheme S having a constructible stratication S S such that

over every S E F is at Grothendieck Then use semicontinuity and

quasicompactness arguments as ab ove

Theorem Mumford semistability of the image of E in Grass for

b big is equivalent to semistability of E as a vector bund le

Proof We use the result of the example in the last lecture In our situation

this means that E is Mumford semistable i for all subspaces W V

dim imW C O b C E b dim C E b b

dim W dim V

Fix a surjection V O E We show rst that V C E

C

Let W b e the kernel of V C V O C E Then the image of

C

W C O b C E b is zero so by W must b e zero

Now we have to show that E is a vector bundle ie torsion free and that

it is semistable in the vector bundle sense with slop e For this let T b e

the torsion of E and take F in the HN ltration of E T which is a vector

bundle such that it contains exactly the p ortion of slop e Then E F

is the part of slop e Now E T is semistable of slop e i E T

and E T has no nontrivial quotients with slop e This is equivalent

to E F since E F is the largest p ossible quotient of slop e and

deg E T deg E Since equality in this relation holds only for T we

also see that E F implies that E is a semistable vector bundle of slop e

It suces to show that E F this implies E F Let

r rk E F and r rk F F

Let W kerV C E F Since everything in F has slop e g

H C F and F is globally generated Hence

C F C E C E F

is exact and W V C F We get an induced injection

C F W C E V

The dimension of C E V is dim H C E which injects into H C E F

H C F and we want to b ound the dimension of the latter

Since V O E F there are r sections which generically generate E F

C

r

ie we have a map O E F with nite cokernel Then

C

r

H C O H C im H C E F

C

whence h C E F dim H C E F r g

By denition of W

W C O b C F b C E b

hence the dimension of the image let us call it I is at most h C F b

r b By

b

dim W dim I r

b b

therefore using

b b

h r C E F r g r dim C F r

b b

If h C E F is zero then from b we see that or r ie

F By denition of F we must have hence as well lo ok at

F E E F and use E By denition of F again

this means E F ck

So assume H C E F Then in the HN ltration of E F there must

b e a quotient of slop e g should b e g rememb er we shifted

by g Then since the slop es in the HN ltration of E F are all we

get

deg E F r g g

hence

deg F r g g r r g

Plugging this into we get

g r r g

b

Since everything is globally generated can b e b ounded in terms

r

g g r r g Hence of sp ecically

r

by cho osing big a everything O a and b even bigger we get

a contradiction

We are given a semistable vector bundle E of slop e Let V

C E then we have V O E Let W V b e a subspace and F the

C

subbundle of E generated by W We have to check Let I b e the image

guring in the numerator on the right hand side of By our choice of b

I C F b and its dimension is the Euler characteristic of F b Let

as b efore r rk F and F We have to show

b

dim W r

b

Since W C F there is no harm in assuming W C F So we want

to show that

b

h C F r h C F r

b

If h C F this follows from r b and Otherwise we

have some step of slop e g in the HN ltration of F hence as b efore

r r g

and

r h C F r g g r r

if is suciently big But then for b big enough this is also r

b b

One should check that a single b works for all E in this pro of One should

also check that the pro of for stable instead of semistable do es go through

2

Now we have

i

Q Grass

ss ss

ss

fE semistableg Q i Grass Grass

s s

s

fE stableg Q i Grass Grass

where i is a closed immersion and we have a G SL action on b oth sides

N

The corresp ondence of semistable p oints on the left and right hand sides

comes from the fact that suitable p owers of Ginvariants can b e lifted

We can form the quo otient

ss

Q G M

s s

Q G M

ss

M is our mo duli space its p oints corresp ond to the closed Gorbits in Q

s

The p oints in M corresp ond to the isomorphism classes of stable bundles

their orbits are closed

L

Claim Orbit of E is closed E E is a direct sum of stable

i

bundles

Proof Sketch

If E is not semisimple ie not of the form on the right hand side we

have a nontrivial extension

E E E

with semistable bundles E E of the same slop e Let

V C E C E V V V

and dene a parameter subgroup of G by

b a

t G t t

V m V

where a dim V and b dim V t deforms E into E E hence

E E is in the closure of the orbit of E hence in the orbit and the extension

was trivial contradiction

L

Let E E There is a closed orbit within the closure of the orbit of

i

E let E b elong to this closed orbit Then by E splits into a direct sum

of stable bundles Now by semicontinuity the multiplicity of every sp ecic

E in E is at least that of E in E hence since ranks are equal they must

i i

coincide and E is in the orbit of E whence this is closed 2

May

ss

We have M Q SLN M is pro jective The notation XG in

dicates that we take the invariants of the homogeneous co ordinate ring to

construct the quotient

We get an ample line bundle on M from an ample line bundle on Q

N N

O E gives a quotient C O b C E b

and the determinant det C E b denes an ample line bundle on the

Grassmannian which can b e pulled back to Q We have to b e careful

however to get a vector bundle on the quotientwe have to take the SLN

action into account See b elow

Aside Natural line bundles

G

Recall our quotient construction R Pro j R We obtain coherent sheaves

G

on the quotient from mo dules M where M is a graded Rmo dule with

Gaction

A p oint x in the quotient corresp onds to a closed semistable orbit Y

GG Then

x

G G

M generates M at x M j generates M j

Y Y

G

x

M x generates M x M k x

G op erates trivally on M x

x

The rst equivalence comes from the fact that the action of G commutes

with taking quotients mo dulo invariant ideals

In our case we have x E and G is the subgroup of Aut E consisting of

x

elements having determinant on C E Hence we have to check whether

det C E b has trivial SLN action We have

C O b C E C E b

N

where GLN acts on C E k Try to nd a linear combination

det C E b det C E

such that the scalars op erate trivially We may cho ose

dim C E r and dim C E b r b

In this way we get an ample line bundle

r r b

det C E b det C E

z z

ample trivial line bundle

with nontrivial action

with a PGLN action which gives an ample line bundle on the quotient

If E is stable then Aut E scalars whence we have trivial action of G

x

and probably get a line bundle on the quotient

The Theorem of NarasimhanSeshadri

We are now in the compleyanalytic case k C and C is a compact

Riemann surface of genus g Then the fundamental group is

C hA A B B j A B A B i

g g g g

Let C b e the universal cover of C Given a representation

C GL C

r

r

we get a lo cally constant sheaf E on C as E C C C diagonal

action Hence we have a holomorphic therefore algebraic vector bundle

E E O

C

The representations are parametrized by an algebraic variety

Rep C GL

r

take co ordinates for A B the relation b etween the A and B gives

i i i i

equations for the co ecients we get an analytic at family of vector bundles

over C on Rep C GL

r

A representation is unitary if its image C is contained in Ur

the subgroup of unitary matrices in GL C E then gets a at hermitian

r

metric

Claim unitary E is semistable of degree

Proof The degree is essentially the integral over C of the curvature which

is zero here If F E is a subbundle it gets an induced metric that has

nonp ositive curvature Hence deg F deg E and E is semistable

To make this reasoning a bit more precise We have the subbundle

rkF rkF rkF

F E where E is asso ciated to the unitary representation

rkF rkF r

on C Hence we may assume that F is a line bundle rememb er

S

rkF

deg F deg F Take an op en cover C U such that we have

lo cal generators f of F on U In a unitary basis of E which is lo cal on

U f is given by holomorphic co ordinate functions f f Con

r

r

P

jf j This do es not dep end on the unitary basis and sider kf k

j

j

log kf k is either sub or sup erharmonic ie log kf k this form

should glue over C The degree of F is essentially the integral over C

of log kf k hence 2

C

r r

Claim C E C the C invariants in C

Proof Let f C E Then kf k dened lo cally with resp ect to some

unitary basis has a maximum somewhere on C since C is compact Lo cally

P

kf k jf j with holomorphic f which then have to b e constant

j j

log kf k is sup erharmonic ck so f is a constant section which has to

r

come from E and then from an invariant in C 2

If we have two representations with spaces E and E and asso ciated

vector bundles E and E then the vector bundle corresp onding to the rep

resentation on Hom E E is Hom E E and

Hom E E C Hom E E Hom E E

C

Hence the functor E E from unitary representations of C to vector

bundles over C is fully faithful

Theorem The functor we have constructed is an equivalence of categories

unitary representations

semisimple semistable

vector bundles of degree

of C

E E

Proof We have already shown that the image consists of semistable vector

bundles of degree zero and that the functor is fully faithful We have to show

that E in the image is semisimple ie direct sum of stable bundles Since

any unitary representation is semisimple use orthogonal decomp osition we

can assume E to b e irreducible We must then show E is stable We induct

on the rank of E If the rank is zero nothing has to b e shown So assume

rk E and E not stable Then there is a prop er subbudle F E of degree

which we can assume to b e simple ie stable take one of smallest rank

By our induction hyp othesis F comes from a representation F I dont see

that which by full faithfulness has to inject into E which therefore cannot

b e irreducible contradiction Mayb e one can argue as follows If F E

has degree zero its induced metric must b e at Then F O F where

F E is a subsheaf of the sheaf of constant sections of E This should then

pull back to a subrepresentation F E

Remark We see that irreducible representations corresp ond to stable bun

dles

It remains to show that we get all stable vector bundles of degree zero

up to isomorphism from irreducible unitary representations

Claim We get a continuous or even real analytic map

Rep C Ur Mr

C

Mr is the space for vector bundles of rank r and degree as a

C

complex analytic space

Remark This would b e clear if we knew already that Mr is a mo duli

space in some sense We have the universal bundle on C Rep C Ur

whose b er ab ove C E is E as constructed ab ove

There is an op en subset Rep C Ur U Rep C GL such

r

that for E and E parametrized by U H C E a This is b ecause

of semicontinuity of dim H C E a H C E a is a closed condi

tion which is never true on Rep C Ur Furthermore we want that

the kernel of C E a O b E a b is globally generated Similar

argument

S

N

C Cho ose an op en cover U U and over U a basis of C E a

This denes an analytic map U Q Q is the same as the algebraically

C C

dened Q since all quotients are algebraic We may assume that the image

ss

is contained in Q the complement is analytic hence closedor recall that

ss

X is op en in X quite generally Then these maps glue to give a welldened

analytic map

ss

U U Q Mr

C

C

Now Rep C Ur is compact can b e identied with a closed subset

g

of Ur and Ur is compact hence by restricting we get a realanalytic

hence continuous closed and prop er map

Rep C Ur Mr

C

The preimage of the stable bundles consists exactly of the irreducible rep

resentations hence we get a prop er map

s

IrrRep C Ur Mr

C

with closed image

The image is also op en Both sides are manifolds We show the map

induces a surjection on tangent spaces then the map has to b e op en by the

implicit function Thm But rst show that IrrRep is a manifold

g

SL A A B B A B A B is Claim GL

r g g g g

r

smo oth at p oints coming from irreducible representations of C

It is p erfectly clear that this map is smo oth everywhere Do es Faltings

think of representations mo dulo conjugation He said something of cheating

around here Anyway I dont quite understand the following lines

P P

Tangent map is approx A B A B

i i i i

Adjoint map comp onents of commutators with A and B only scalars

i i

b ecause of irreducibility

The real dimension of Rep C Ur is g r r

s

As to the dimension of M we have for a stable vector bundle E that

s

Aut E G so that we have only trivial action which implies M is

m

smo oth at E The tangent space is given by H C E nd E We know the

Euler characteristic is r g Since H C E nd E End E C we

s

get for the complex dimension of M

s

dim Mr dim H C E nd E r g

C

Now lo ok at the tangent map of

End E H C E nd E H C suE H C

R

z

was C

It should b e uE instead of suE In general if is a discrete group and

G is a Lie group with Lie algebra g then H g is the tangent space of

Rep G at where the action of on g is given by Ad Deforming

gives a co cycle g cob oundaries corre

sp ond to conjugation by elements of G NB This holds for representations

mo dulo conjugation

By Hodge Theory

suE C C E nd E C E nd E H C

z z

was C

H C E nd E

The real p oints cannot go into the rst summand they have to map diago

nally since they are invariant under complex conjugation Hence they surject

onto H C E nd E ie the tangent map is a surjection our map is a

submersion and the image is op en

One really should lo ok at representations mo dulo conjugation then the

real dimension of IrrRep C Ur is g r r which is the same

s

as the real dimension of Mr Hence we have an isomorphism on tangent

C

spaces

Since the image is op en and closed it must consist of connected comp o

s

is irreducible hence connected we p ostp one the nents Because Mr

C

pro of of this fact and the image is nonempty there are irreducible unitary

representations of dimension r it must b e everything

Hence every stable vector bundle of degree zero is obtained up to iso

morphism from an irreducible unitary representation of C Of course

we then also get every semisimple semistable vector bundle of degree zero

from a unitary representation 2

Corollary If E and E are semistable vector bund les of degree zero on C

then the same holds for E E

Proof First assume E and E are stable Then E and E come from

irreducible unitary representations E and E of C and E E is

obtained from E E which is again unitary Hence E E is semisimple

semistable of degree zero

Now let E and E b e arbitrary semistable vector bundles of degree Then

j j j

there are ltrations E of E i f g with E E stable of degree The

i

i i i

j

pro of for E stable now shows that E E E has a similar ltration E If

i

E were not semistable there would b e some F E of p ositive degree Then

j j

in the ltration F F E there would o ccur a quotient of p ositive degree

j j j j

Since F F injects into E E we have a contradiction ck 2

Remark Faltings says he do esnt know a purely algebraic pro of of this

Mo duli space as stack

Supp ose M is a mo duli space and supp ose further that we have a vector

bundle H on M If we then have a vector bundle family E on C S we get

a classifying map S M which we can use to pull back H to a vector

E

bundle HE H on S

E

If f S S is a morphism and E is a vector bundle on C S we

f HE and all sorts of further nice compatibilities have H f E

C

We also get an action of Aut E on HE How

So even if a mo duli space do es not exist we can study vector bundles on

it by dening a vector bundle H on M to b e an asso ciation E HE with

suitable prop erties

Example

a HE O This should corresp ond to O

S M

b Let x C Then we may take HE E j orif we want a line

fxgS

bundle HE det E j

fxgS

c An imp ortant example is the determinant of cohomology If C S

S then lo cally in S we can nd a complex of sheaves

d

H H

such that E ker d and R E coker d EGA I I I For example take

x C and N such that R E N x Then we can take

H E N x E N xE H

The assertion follows from the long exact cohomolgy sequence for

This complex is unique up to a quasiisomorphism which in turn is de

termined up to homotopy equivalence

H H

y y

H H

quasiisomorphism means inducing isomorphism on cohomology The de

terminant of cohomology is now

det H C E det H det H

this is a welldened line bundle indep endent of all choices made

For a xed vector bundle F on C det H C E F is an example of a

vector bundle on mo duli space

Remark An exact sequence of vector bundles

F F F

induces an isomorphism

det H C E F det H C E F det H C E F

Hence it suces to consider line bundles L instead of general vector bundles

F For a line bundle L and a p oint x C consider

Lx L k

x

This induces an isomorphism

det H C E Lx det E det H C E L

x

z

this is of typ e b

This means that there is basically only one new line bundle det H C E

on M

P

If D n x div f is a principal divisor on C then O D is trivial

i i

det H C E O D and therefore using the consid hence det H C E

N

n

i

erations ab ove det E x is trivial So there is essentially only one

i

i

line bundle of typ e b This should b e made clearer At this p oint we get

a map from divisors on C to line bundles on M factoring over the principal

divisors Hence we get a map from isomorphism classes of line bundles on

C to line bundles on M Faltings said something ab out Jacobians and

algebraic equivalence

Example Consider our ample line bundle

det C E a b det C E a

Since H we may replace by H Then we get something of the form

det H C E a b det H C E a

something of typ e b det H C E

where

This means roughly that we can exp ect some kind of ampleness for

det H C E

May

Consider the following moduli functor

S vector bundles on C S isomorphisms

The pair on the right hand side are the ob jects and morphisms of a category

which is a groupoid ie all morphisms are isomorphisms So we have here

a group oid functor

The isomorphism classes of vector bundles which is what we are inter

ested in constitute the set of connected comp onents of the group oid

The corresp onding functor is not representable mo duli space do es not ex

ist

But we may consider it represented by a socalled mo duli stack

Msee b elow what a stack is

I dont know what the signicance of the following remark is at this

place

S

We can glue vector bundles If S U is an op en cover and we are given

vector bundles on C U for all with isomorphisms over C U U

which are compatible on triple intersections the data glue to give a vector

bundle on C S

When constructing a representing ob ject we have to face two problems

a vector bundle can have automorphisms other than G the action

m

of SLN or PGLN is not free This will b e OKthis seems to b e

what stacks are made for after all

the ob ject will not b e of nite typ e ie not quasipro jective HN

ltration can b e wild

To overcome the second problem we construct op en substacks M

n

Fix some n We want to restrict to those E with H C E n mayb e

this isnt necessary but it wont hurt and such that H C E n generates

E n

Given E on C S let

S fs S j H C E n and H C E n generates E n g

n s s s

C fsg C S Then S is open where E n i E n with i C

n s

in S By semicontinuity Grauert H is an op en condition hence

satised on an op en subset U of S Then E nj is a vector bundle on

C U

U by some Base Change Thm which generates a subsheaf of E nj

C U

b erwise E n corresp onds to global sections of E n hence generates

s

a subbundle in each b er The supp ort of the quotient is closed and is

prop er hence the image of the supp ort in U is closed and the p oints where

E n is generated by global sections constitute an op en subset again

s

The substack M should probably b e viewed as consisting of all the im

n

ages of S under the classifying maps S M Since S is always op en M

n n n

should b e kindofop en in M

S S

M S and so M Clearly S

n n

n n

These substacks will b e quasicompact whatever this will mean

Let T b e the mo duli space of vector bundles E on C such that H C E n

n

generates E n and H C E n together with a basis of C E n

Then T has an op en emb edding into the quotient scheme Q classifying all

n

N

where N r n is the dimension of C E n The quotients of O

C

conditions dening T are all op en quotient lo cally free of rank r H

n

H generates So T is a nice scheme representing the corresp onding functor

n

and we will identify the two Then we have a GL torsor

N

T M

n n

ie a kind of bundle with b er GL but b eware of automorphisms of E

N

On C M there is the universal vector bundle E and E is a lo cally

n

free sheaf on M of rank n I guess it is E n and has rank N

n

The stack M should therefore b e the quotient T GL This quotient

n n N

wont exist as a scheme since the GL action is not free the stabilizer of

N

t T corresp onding to E basis is given by Aut E For example we

n

always have G Aut E

m

The idea now is not to worry ab out the quotient but to work with T

n

and its GL action instead We represent the stack by T GL and argue

N n N

as if GL would op erate freely

N

Example A coherent sheaf on T GL should corresp ond to a coherent

n N

sheaf F on T with GL action Let GL T T b e the action of

n N N n n

GL on T Then we require that for any two maps

N n

GL S T

N n

we have an isomorphism of sheaves on S

F F

where which is transitive under multiplication of maps

I take this to mean the following

where x x x multiplication in GL

N

pr pr

Using GL GL T T one can also write down explicit

N N n n

conditions on F

These sheaves are the same as sheaves on the mo duli functor what

ever this is restricted to E with H C E n and globally generating

H C E n

Given a family of vector bundles E on C S and such that S S

n

S

E n is lo cally free of rank N Hence we can cho ose an op en cover S U

N

O such that E nj Fixing this isomorphism we get a classifying

U

U

map U T Q Further we have transition maps U

n

U U GL given by the matrices changing the one basis into the

N

other on U The transition maps are compatible on

triple intersections An equivariant sheaf F on T now induces sheaves F

n

U U on U by pullback Lo oking at GL T we see that

N n

we get isomorphisms F j F j compatible on triple intersections

U U

Hence the F glue to give a sheaf on S This shows that we get for each E on

C S a sheaf F E on S with some prop erties as discussed in the previous

lecture Is this meant by a sheaf on the mo duli functor

If we have two isomorphic families E and E on C S we get maps

U GL giving the isomorphism

N

E nj basis E nj basis

U U

F which are compatible with those This in turn yields isomorphisms F

induced by the resp ective transition maps Hence they glue to give an iso

F E In particular we get an action of Aut E on morphism F E

F E

It should b e easy now to see the existence of the sheaf E n on M It

n

certainly exists on T we have the universal sheaf E on C T T which

n n n

has H C E n and H C E n generating so E n will b e lo cally

t t

free of rank N dim H C E n so we have to check equivariance Let

t

S T and S GL b e maps Then E and E

n N

are isomorphic sheaves over C S we only change the basis of H C

hence

E E E E

Transitivity is equally clear

If we want to consider M as a limit of M we have a problem Increasing

n

n will also increase dim T and N so we dont get anything like op en im

n

mersions on the T level We must intro duce a suitable equivalence relation

n

of representations of stacks

Denition of stacks

Remark The notion of stack came up in the sixties But to swallow

schemes was already enough for one generation of mathematicians

Faltings

Denition An algebraic group oid is a pair of schemes S R together

with maps

dom ran

R S S

comp

diag

and R R R

S

id

S

where R R is constructed using ran R S for the left and dom R S

S

for the right factor such that

T Hom T S Hom T R

is a groupoid functor on schemes Hom T S are to be the objects of the

groupoid and Hom T R the morphisms Domain and range of a morphism

are given by dom ran the identity morphism is given by id and composi

tion of morphisms is given by comp

If S R is an algebraic group oid then SR is a stack

Example

a Let S b e a scheme and G a group acting on S via Then we can take R

G S with dom pr ran ids s and compg s g g s

g g s The resulting stack corresp onds to the quotient SG

b R S S closed immersion is an equivalence relation The stack

SR represents the quotient of S by this equivalence relation

Denition A sheaf on a stack SR is a sheaf F on S together with an

ran F isomorphism between the two pul lbacks of F to R ie dom F

satisfying the fol lowing condition Every map T S gives rise to a sheaf

F F on T For every map T R which should b e interpreted as

giving an isomorphism b etween the ob jects dom and ran

such that id F and such we get an isomorphism F

id F

that for with ran dom we get

comp

To put this into abstract nonsensical language A sheaf on SR is a

natural transformation from the group oid functor represented by S R to

the group oid functor T sheaves on T isomorphisms The corresp onding

sheaf on S is obtained as the image of id Hom S S in the sheaves on S

S

To get a nice ab elian category we will only consider smooth stacks

Denition A stack SR is smo oth if S and dom ran R S are

smooth

Since a stack is meant to represent isomorphism classes of something

which can b e obtained in a multitude of dierent ways lo oking at ob jects

with additional structure and mo dding the additional data out for example

we need a notion of when two stacks represent the same thing

Denition Two smo oth stacks S R and S R are equivalent vulgo

the same stack if there is a third stack S R such that we have smooth

and surjective maps

S S S

such that

R S S R R S S

S S S S

This means that we can check equivalence of objects in S by sending them

to S or equal ly wel l to S Heuristical ly this means that we get the same

equivalence classes

Another way to dene this is to say that we take the equivalence relation

R S S generated by pairs S R SR such that S S and R

S S

As an example we get the same coherent sheaves on equivalent stacks

S as in the denition ab ove Then F F F should induce Let S

an equivalence of categories b etween coherent sheaves on SR and coherent

sheaves on S R By descent a sheaf on S corresp onds to a sheaf on S such

that the two p ossible pullbacks to S S are isomorphic plus a transitivity

S

condition Now we have

R idS S S R S S S S S S S

S S S S S S S

S S S by the diagonal emb edding which implies that a sheaf on S R

descends to a sheaf on SR The converse is clear Exercise Write this out

in terms of group oid functors

Back to our mo duli problem

Let m n Then we have two smo oth stacks M T GL and M

m m M n

T GL which we want to relate

n N

We can assume that E M E M Then we get an op en subset

m n

T T dened by H C E m and H C E m generates E m T

n

n n

is stable under the GL action

N

Claim T GL and T GL dene equivalent stacks

m M N

n

Hence we get some kind of op en immersion T GL T GL

m M N

n

T GL

n N

Proof Let T T T represent isomorphisms pr E pr E I take

m m n

n

E of vector this to mean the following functor S isomorphisms E

n m

bundles on C S such that S S with resp ect to E and S S S with

m m m n

resp ect to E plus bases of E m and of E n this is representable

n m n

by the Hilbert scheme formalism The pro jections E p E E E

m m n n

are surjective by denition They are also smo oth We use the innitesimal

criterion Let T Sp ec A with a lo cal algebra A I A an ideal such that

I and let T Sp ec AI We must show that we can lift to

T T

T T

m

corresp onds to a vector bundle E on C T together and similarly for T

n

with a basis of E m Mo dulo I we have an isomorphism E F where F

F n is a vector bundle on C T for which we also have given a basis of

We use the isomorphism to identify E and F We lift the resulting identity

E to the identity on E The basis of E n can also b e lifted b ecause of on

lo cal freeness So we have dened as the classiying map of E E with the

two bases We still have to check that the pullbacks of R GL T

m M m

and R GL T are isomorphic This says that for two isomorphisms

N

n n

are isomorphic are isomorphic i E and E E and E E E E and E

n m m n

n m n m

This is of course clear

Hence by our denition the two stacks are equivalent and the E s corre

sp ond do es this mean the universal sheaves are the same 2

Naturally one wants to have a stack which is not bigger than necessary

ie a stack SR with minimal p ossible dimension of S

M We will get a lower b ound on dim S from the fact that So let SR

n

S will b e a versal deformation For s S we have a KodairaSpencer

map T H C E nd E which is surjective This should b e so since

Ss s

H is the tangent space at E of the mo duli spacesee rst lecture T

s S

is the tangent bundle of S Hence any E gives a lower b ound on dim S

dim T

Ss

Let s S b e a k valued p oint Let and consider k Then T is

Ss

s

given by all morphisms Sp ec k S extending Sp ec k S Each element

of T denes a vector bundle on C Sp ec k which can b e compared to

Ss

the constant bundle E obtained from k k k The dierence

S

lies in H C E nd E Cho ose an op en ane cover C U such that

r

O E j Write the transition maps into GL O as g g

U r U U

U

Then g denes a co cycle in H C E nd E This denes the Kodaira

Spencer map

This map is surjective We rst show that surjectivity is invariant under

going over to an equivalent stack Let S S induce an equivalence of stacks

Let s S map to s S Every map Sp ec k S extending s maps

to a map Sp ec k S extending s and their images in H C E nd E

coincide On the other hand every map Sp ec k S extending s can b e

lifted to a map Sp ec k S extending s b ecause of smo othness Hence

and T have the same image in H C E nd E T

Ss S s

From this we see that it suces to show surjectivity for T An element

n

of H C E nd E corresp onds to a deformation of E to C Sp ec k Since

we can lift the given basis of E n to a basis over k we see that our

deformation of E comes from a deformation in T

n

There is a kind of converse

Prop osition Given S smooth E on C S such that T H C E nd E

Ss s

is surjective for al l s S and such that for al l s S H C E n and

s

T such H C E n generates E n then there is an open subset T

n s s

n

that for s T E is isomorphic to a vector bund le parametrized by S and

s

n

E E pr GL SR as stacks where R S S represents Isompr T

N

n

Proof R is smo oth Take as usual a lo cal algebra A with an ideal I A

such that I and let T Sp ec A T Sp ec AI We have to lift

T R

T S

The lower map corresp onds to a vector bundle E on C T Since T

R S S we have another vector bundle E and an isomorphism E E

E to some E on C T Then E on C T Since S is smo oth we can lift

and E dier by a class in H C E nd E I and two extensions of T S

dier by something in T I Now

S

T I H C E nd E I H C E nd E I

S

the rst by versality the second b ecause H is right exact here This means

that we can adjust E to b ecome isomorphic to E whence the lift T R

Now let T T S classify isomorphisms Then the pro jections are

n

smo oth use a similar argument as ab ove hence the image T in T is op en

n

n

GL Thus we get an equivalence of stacks T SR 2

N

n

C There is a construction of Artin Given a vector bundle E on C

Sp ec k construct a smo oth scheme S of dimension dim S dim H C E nd E

and a vector bundle on C S extending E probably In this way one can

construct parts of the mo duli stack directly

May

S

We have our mo duli stacks M M where M S R with smo oth

n n n n

n

schemes S and group oid structures R on S The inclusions M M

n n n n n

can b e represented by op en immersions S S on suitable representa

n n

tives The S are versal deformations ie the tangent map

n

T R E nd E

S

n

is surjective

There is a metho d of M Artin to construct neighb orho o ds in S of a

n

given p oint E

Given E k we can construct a versal deformation of E over C Sp ec k t t

d

such that any other deformation can b e obtained by basechange from it we

can cho ose d dim H C E nd E in general it has to b e

By the Artin Approximation Theorem there is a subring k t t

d

A k t t etale and of nite typ e over k t t such that the

d d

versal deformation of E can b e dened over A and we get a versal family

over Sp ec A which constitutes a neighb orho o d of E

Cho ose S Sp ec A with various As we can take the union to b e

n

nite

nite b ecause of quasicompactness and R isomorphisms This gives a

n

kind of direct construction of M

n

Mo duli stack is connectd

As promised

Theorem The moduli stack for xed rank and degree is connected

ie each S for n can be chosen connected or alternatively there is

n

no open and closed nontrivial subset S S that is stable under R ie

n n

n

S random S

n n

Proof The idea is to use the known result for line bundles and reduce the

general case to it by ltering vector bundles by line bundles

Claim Let E have rank Then for N E N has a section s with

sx for all x C

Proof Let T fs x C E N C j sx g this is a closed sub

variety of C E N C Its b er over x C under pr is C E N x

If N is suciently big then H C E N x hence by RR applied to

E N and E N x

dimb er dim C E N rk E

and therefore

dim T dim C E N rk E dim C E N

Hence pr T is a prop er subset of C E N and an element in the com

plement is the section claimed to exist 2

We get E as an extension

O N E F

with deg F deg E N Since H is right exact on a curve we have

H C E M H C F M

and analogously for E M x and F M x for M Hence we can

rep eat the construction to obtain a ltration

E E E E

r

O N for i r and where r rk E such that E E

i i i

r

P

E E N Since we can always increase N in the con det E O

r r i

i

struction ab ove we may assume N N N

r

Claim Such extensions with xed r N and deg E are classied by an

i

irreducible variety

Proof For line bundles we get a copy of the Jacobian which is irre

ducible So supp ose we know the assertion for r and bundles F We

have to classify extensions

O N E F

We have Ext F O N H C Hom F O N Since

H C Hom F O N Hom F O N

induction on ltration of F use that degrees in there are N H

denes a vector bundle Let F b e the variety classifying F s and F the

universal bundle on C F F Then H Hom F O N is a bundle

on C F with H whence R H is lo cally free by RR Its total

space classies extensions E 2

So we have an irreducible variety T classifying ltrations as ab ove The

conditions for S dene an op en subvariety T T which maps onto S R

n n n

Since T is irreducible S R is connected or also irreducible if this notion

n n

were dened By cho osing a connected comp onent of S we see that we

n

may take S connected 2

n

Remark This argument also works for semistable bundles where we have

used the result

Theorem Assume g

a The closed subset of nonsemistable E has codimension in S

n

b Suppose char k Then the closed subset of nonstable E has al

ways codimension in S Unless g and the rank r the

n

codimension is

In particular the stable bund les are dense hence they exist for any degree

and rank

Proof To estimate the dimension we use versality T R E nd E

S

n

But in case of singularities the tangent space may b e to o big To overcome

this we lo ok at a generic p oint

Let S S and T fE j E not semistableg and let b e the generic

n

p oint of an irreducible comp onent of T Then over C k E has a non

trivial HN ltration This HN ltration is already dened over k It is

dened over a nite extension M of K k which is a purely inseparable

extension of a Galois extension L of K Since the HN ltration is invariant

under the Galois group it will b e dened over K if it is dened over L

If char k we are done Otherwise let p char k Consider a purely

p

p

t There exists a derivation on K inseparable extension K K K

p

p

such that t and K K ie K fu K j u g acts

O K via its action on the second factor We claim that on O

C K C Sp ecK

the HN ltration on E E K is invariant under and hence dened

K

over K Let F E b e the rst step in the HN ltration and consider

F E F

how is this map dened It seems to require F F K which probably

K

is what we want to prove This map is O K linear and must b e

C K

b ecause of the dierent slop es

We can now replace T by a nonempty op en T T and nd a ltration

by vector bundles E which induce the HN ltration on each E on E j

C T

b er Consider the following diagram

R E nd E T

HN T

y

T R E nd E

S

y

y

R E nd E E nd E

HN

where E nd denotes endomorphisms resp ecting the HN ltration The

HN

map on the lower right is surjective b ecause R is right exact The map

R E nd E E nd E is zero hence at a p oint E T we have T

HN T

co dim T dim H C E nd E E nd E

T T E HN

SE

Now E nd E E nd E has a ltration with sub quotients Hom g r E g r E

HN

where g r E has slop e Because of the slop es and semistability

H C E nd E E nd E Hence the dimension of H is given by

HN

RiemannRoch

dim H E nd E E nd E

HN

X

g rk E nd E E nd E deg Hom g r E g r E

HN

since g rk and

deg Hom g r E g r E rk g r E rk g r E

with the old denition of slop e deg rk and at least one of these o ccurs

since the ltration is nontrivial This proves part a

To prove part b we show that the nonstable bundles have the claimed

co dimension in the semistable ones Let now T b e the set of nonstable

semistable bundles and let again denote a generic p oint of a comp onent of

T Let E on C b e semistable but not stable We get a ltration of E whose

P

sub quotients are direct sums of stable bundles E F E where the

sum is over stable subbundles with the same slop e as E continue with E E

By analogous arguments for the Galois case as used for part a we see that

this ltration is dened over k since we assume char k we need not

worry ab out inseparable extensions By the same reasoning as ab ove we get

the estimate g rk for the co dimension Why is in this case H

We might have isomorphic stable bundles in dierent sub quotients Mayb e

this is b ecause the ltration is canonical image of stable bundle is stable

hence has to b e resp ected by global endomorphisms If the ltration has at

least two steps the rank is so the comdimension must b e and can

b e only if g and the rank is which implies that the ltration has two

steps with line bundles as sub quotients ie E has rank

We therefore may assume that the rank is zero ie that E k is a direct

sum of stable bundles This splitting is dened over a separable extension L

of K k this is true even in characteristic p Sp ec L Sp ec K is an

etale cover which extends to

op en

etale

T T T

hence T T T and we can estimate the co dimension by the

T E T E SE

to R E nd E E nd E dimension of the cokernel of the map from T

T

where E nd means endomorphisms resp ecting the direct sum Then our old

argument nishes the pro of Really It seems there are diculties with

n

vector bundles of the form E nfold direct sum of a stable bundle E how

can the global endomorphisms b e restricted to make the argument work 2

Example For g and rk co dimension do es indeed o ccur Consider

nontrivial extensions with all degrees zero

L E L

with line bundles L L Then H C E nd E k S can b e chosen

n

to have dimension g L and L dep end on prame

ters each which gives a subspace of deimension hence co dimension

dim H C Hom L L since H generically ie the nontrivial

extension is unique

The GITquotient again

We try to construct again the GITquotient GIT Geometric Invariant

ss ss

Theory Let S S the op en subset of semistable bundles and R the

n

n n

induced group oid structure We want a pro jective quotient

The determinant of cohomology denes a line bundle L on S and on

n

ss ss ss

invariant global sections on S Supp ose L is generated by its R S

n n n

f

N ss ss

P which is R invariant meaning f dom Then we get a map S

n n

ss N

f ran R P Is the image closed whence pro jective Supp ose so

n

nite

N ss ss

M P and invariant normalization S Then take the R

n n

check the b ers and analyze curves in them to see if we have the quotient

we want

By Langtons PhD Thesis Harvard Mumford the image is closed

Theorem Let V be a discrete valuation ring with fraction eld K and let

E be a semistable bund le on C Sp ec K

K

Then E extends to a bund le E on C Sp ec V with semistable special ber

Proof We may assume V to b e complete Given E on C Sp ec V V the

completion of V cho ose a xed extension E on C Sp ec V Let b e a

uniformizer of V Then for some n we have

n n

E E E

n n

E are the same over V and over V E can E Since the subbundles of

b e dened over V

Now let E b e any extension of E and E its sp ecial b er Let E b e

K s s

the HN ltration of E and let F b e its rst step If E is not semistable

s s

we have F E Under the assumption that the Thm is false cho ose

s

E such that F is minimal and among those such that rk F is minimal

F lives on the sp ecial b er Extend it by zero to all of C Sp ec V Dene

E E E such that E E is the image of F E E I have changed

this from my notes which didnt quite make sense Then we have

E E F E E F

F and E F change places Let F b e the rst step in the HN ltration

s

of E E Then F F by the exact sequence ab ove By minimality

of F F F note E E We get a strict map F F whose

K

K

kernel must b e zero since rk F is minimal this do esnt seem to b e the right

F b ecause rk F was minimal justication Hence F F and then F

here this is OK So the sequence splits by F F

We have found a new extension E E such that

E E

s

s

F F

Continuing in this way we obtain a decreasing sequence

n n

E E E E E

n n n

such that the rst step F in the HN ltration of E is the image of E

s s

n

and is F

T

n

E E On the formal scheme C V consider E

n

Let Sp ec A C b e an op en ane subscheme then over A V we have

k

n n

a decreasing sequence of pro jective mo dules E with E E let E b e

their intersection

Claim rk E rk F and we have a surjection E F F corresp ond

ing to F

Proof Let p A V b e the prime ideal induced by the maximal ideal

k

of V Then

n

length E E nrk E rk F

p s

By commutative algebra there is a basis e e e e of E such

a a ab

n

that a rk F and E is generated by e e which generate F and

a

n n

e e We can cho ose this basis to b e indep endent of n use

a ab

completeness of V Then we see that E is generated by a e hence

a

equals F 2

This gives us a formal subbundle G E with G F By Grothendieck

s

G is algebraic hence comes from a subbundle G E Now

G G F E E

K s s K

contradicting the semistability of E 2

K

Remark We see that the construction in the pro of when starting with an

arbitrary extension E of E eventually must lead to a semistable bundle

K

June

Raw notes by Bernd Steinert

Recall S Langstons Theorem from the last lecture

Theorem Let V be a discrete valuation ring with fraction eld K let C be

dened over V and let E be a semistable bund le on C K

K V

Then E extends to a bund le E on C with semistable special ber

K

The argument fot its pro of was as follows

Take E b est p ossible and lo ok at the HN ltration

E E E E

n n n n

F where The inclusions E E give rise to isomorphisms E E

n n

F is the rst step in the HN ltration of E E

T

n

E this is a formal vector bundle Let E

n

n n

Claim E E E

S

Proof Write C Sp ec R as a union of op en ane subschemes E E

R

and F corresp ond to free mo dules over R V V

V

n

E E E E

F

n

Cho ose a basis e e e e of E such that e e are in E

a a ab a

n n n

and induce a basis of F Then e e e e E form a

a a ab

n

basis of E lo ok at index

n

n n

This can b e done indenp endently of n Chosen e e go o d for E

a

n n

mo dify them by something in E to get elements go o d for E this is

n

n n n

d

p ossible b ecause E E E Then e lim e exists in E V

i

i

n

and is go o d for all n 2

We had the following picture

Let V b e a complete discrete valuation ring with eld of fractions K

Let K L and W the normalization of V in L Then

N ss

P S S

x

Sp ec V Sp ec K

Sp ec W Sp ec L

N

where the map is dened via global sections of det H this will b e

shown to day

Using Langtons Theorem we see that E extends to E on C W

L R

hence

pr

Sp ec L Isom S original map

pr

S map extends to Sp ec V

Replace lifting by equivalent one can extend

In this way one sees that the image is closed

 M

Construction of global sections of det H on S

that are invariant under Isom

Remark If E is not semistable any such section vanishes at E

Proof Assume E g for general E see b elow and let F E

b e a subbundle violating the semistability prop erty ie F E

Consider the following family over P of vector bundles E E F F

F E E F

What is the determinant of cohomology Let C P P b e the pro jection

Then

detR E detR F O H C F detR E F

z

O something

Hence deg det and all global sections must vanish 2

How to construct sections

Cho ose some xed F such that F E g det H C E F

has global section H C E F

Observation If rk F rk F and det F det F then

det H C E F det H C E F

as line bundles on the mo duli stack of E s

Proof This is OK if rk F Otherwise use induction Write

i

O N F G

i i

then det G det F N and

i i

det H C E O N det H C E G det H C E F

i i

2

N

det F Fix some F Consider F with rk F N rk F and det F

Then

N N

det H C E F det H C E F det H C E F

Theorem If E is semistable and N is suciently big then there exists

an F as above such that H C E F

Proof This condition denes an op en subset in the mo duli stack of F s

We may assume that E is stable Write

E E E E E

d

with E E stable If all E E F have no cohomology then E F has

i i i i

no cohomology as well

Lo ok at S lo cal deformation space of F

Remark Replace F by F F L H E F L is dual to

H F Hom E L hence we need not worry ab out det F

C

S is smo oth F extends to a bundle F on C S and there is an epimor

phism T R E nd F

S

i

Study R E F

Replace S by an op en subset such that E F is a vector bundle there

Claim The cup pro duct E F R E nd F R E F is

identically zero

Proof The tangent vectors are given by Sp ec k S

Let F b e the constant deformation F diers from F by a co cycle

where has values in E nd F

Let s C E F b e a global section then s s s extends for

some Cechco cycle s

We get s d s where d is the Cech dierential But this

is just what we get by the cup pro duct pairing 2

Cho ose a closed p oint s S There is a map

C E F Hom E nd F Hom E F

C C C C

and this map vanishes To get the map lo ok at

C E F Hom E F C E nd F

C C C

which is given by contracting E

Let G E b e the subbundle generically generated by the image of

C E F F E This is the smallest subbundle G such that all

sections lie in G F

C E F C G F

The vanishing of the pairing is equivalent to Hom E F

C C

Hom E G F

C C

Stability of E implies G E with some rk E

Replacing S by an op en subset we may assume that the G s dene a

subbundle of E on C S

Idea Estimate the variation of G

Deformations of subbundles G E have tangent space Hom G E G

C

This gives a map T Hom G E G which can b e describ ed in the fol

S C

lowing way comes from k with

Let G E b e constant and let g b e a lo cal section of G Let g e b e

a lo cal section of G

Upp er b ound for Hom G E G

C

rkG

F G is generically surjective enough to estimate Hom F E G

C

so we get a lower b ound for deg G may assume F semistable We then get

b ounds for h Hom G E G h Hom G E G Since

H E nd E H Hom G E G H is right exact we get a xed b ound

for dim Hom G E G

C

Replacing S by an op en subset we get a subbundle T T with

S

S

const such that rk T T

S

S

ker T Hom G E G T

S

S

The pairing

R G F G F T

S

y

R E nd F

vanishes Consider

C G F Hom G F C E nd F

The image is contained in a subspace of dimension c

Key observation

h G F h G F G F rk G rk F

rk G rk F

for rk F

g g

r

r

There exist r rk G sections g g of G such that F G is

r

generically surjective This gives an injection

r

Hom G F C E nd F

r

and the image is contained in ab ove subspace hence h G F r c a

contradiction 2

June

Raw notes by Bernd Steinert

In the last lecture we have seen that

E semistable F H C E F

In particular E F g One can prescrib e rk F and det F

to some extent

N

Corollary functions generate det H C E F over the semistable

locus

To day we will show the following

Assume given a complete curve B and a vector bundle E on B C such

that deg det R pr E F and such that E j is semistable for at least

bC

one b B

Claim All restrictions E E j are semistable and have the same

b bC

JordanHolder series

Remark All b ers E are semistable

b

M

Proof Find F with rk F M rk F and det F det F such that

is semistable Then F where E H C E

b b

M

detR pr E F detR pr E F

has degree zero

Let B detR pr E F with b Then b

for all b hence E F has trivial cohomology and E is semistable for all

b b

b B 2

Remark Replace E by E F which is semistable and has trivial

cohomology

Remark We may assume B P

Proof Cho ose a nite map B P and consider E on P C Let

x P Then

O m O E E j

B x B

fxgC

O m O E

B x B

M

E

b

b x

where means JordanHolder equivalent and the b are taken with

their multiplicities in the last line Since we are free to vary as we like the

claim for P implies that for B

Note that all b ers of E have trivial cohomology hence the function

vanishes nowhere and our assumptions are valid for P and E 2

Claim All restrictions E j are isomorphic

B fcg

O S Lo ok at Proof Cho ose nite subsets S S C such that O S

R O S pr E pr

M

E j O S pr E pr

B fsg

sS

This follows essentially from the exact sequence

M

O O S k

sS

Also

M M

pr E pr O S E j E j

B fsg B fs g

sS s S

There is much freedom to cho ose S and S Check multiplicity of indecom

p osables which is semicontinuous Claim 2

S

U such that E j Corollary There exists an open cover C

B U

E j pr

B fc g

i

E j E The Proof Use semicontinuity Consider R pr Hom pr

B fc g

restrictions to each b er B fcg are isomorphic By Grauerts semi

continuity these are vector bundles and commute with basechange

Apply to i and lift isomorphisms on b ers 2

Now lo ok at the HN ltration HN E slop es r They glue together

r

to form global subbundles HN E E on B C

r

pr O r pr G for vector bundles G on C Also HN E HN E

r r r r

This is as canonical as things can b e here

We know HN E j has slop e g so Euler characteristic

r

fbgC

Claim We have equality

Proof

O

det R pr E det R pr O r G

r

z

r

exterior tensor pro d

X

O r H C G

r

r

X

r HN HN O

r r

r r

X

O HN r HN

r r

z

r

E

X

O HN

r

r

P

Hence HN b ecause det R pr E has global sections and since

r

r

all HN they must b e Thus all HN are semistable subbundles

r r

of the same slop e as E ie g 2

L

G This implies that G is semistable and E

r r b

r

We now only have to give the argument for Remark

We have shown now E F is JHconstant if H C E F

b

Idea Let G b e stable We want to check that G o ccurs with the same

multiplicity in each E E j

b

fbgC

We get the JHseries for E F by taking the JHseries for E ie

b

P

E m G with G stable plus taking the JHseries for each G F

Let m G b e the generic multiplicity ie the multiplicity in the generic

and let m G b e the multiplicity in b B p oint of B E j

b

k C

Claim m m

b

Proof Consider

Sp ec V B

B

s b

Pull back on Sp ec V C

This is really dicult Why is this so dicult Mayb e I should give up

Take JHseries E on generic b er C E E extend to E E

such that E E is torsion free

The E are bundles E j have the same slop e as E

C fsg

Lo ok at E j E j Since E is semistable this map is strict and

C fsg C fsg

E E is a bundle

In some stage this quotient is my JHconstituent

G so is sp ecial b er E E If generic b er

So multiplicities can only jump up

And we have a matrix eq

Always m m Adding these up we get equality everywhere 2

b

New construction of mo duli space

ss

As stack M SR where S was a versal deformation and R a group oid

Let

N

S P

det H p O

b e the map dened by E F

nite

N

Factor this as S M P where M is the normalization in par

ticular M is normal M is dened by the normalization of k in

M

ss n

M det H

n

We have the scalar automorphisms G S R RG still acts on S

m m

s

The stable lo cus S S is op en and Rinvariant

s s s s s s

R G S S eqivalence relation M S R algebraic space

m

s

Claim M M is an op en immersion

Proof By Zariskis Main Theorem quasinite nite op en immer

sion Here nite do es not matter b ecause we have taken M to b e the

normalization

Hence we have to show that the map has nite ie dimensional b ers

If not there is a not necessarily complete curve B contained in some

b er whence a family E of stable vector bundles on B C By Langstons

Theorem this extends to a semistable family over B the completion of B

Ratios are constant on it hence det H is trivial and E is JHconstant

Then all b ers are isomorphic and stable hence B maps to a p oint in

s

M

Hence

op en imm

nite

s

M M M

then M M and we are done 2

Comparison with what we get from GIT

Finally

M M

n

M det H C E F A A

n

n n

is a graded ring

One has

norm

A k

G

A Q subring from GIT

ss

with M Q G G PGL

N

Claim All are equal

subring from GIT sections on mo duli space from GIT

M mo duli space

op en

s s

M M

Next time Higgs bundles

So far

GL vector bundles of rank r

r

SL vector bundles with trivial det

r

or det a xed line bundle

There are other reductive groups

Sp vector bundles E with an alternating bilinear map h i E E O

identifying E and E

O or SO vector bundles E with a symmetric bilinear map h i E E

r r

O identifying E and E

In these cases there is also a notion of semistability parab olic sub

groups or subalgebras Let

E E E

b e isotropic subbundles Semistable means the parab olic subalgebra has

degree

This notion coincides with the old notion

This is not true for stability one only excludes isotropic subbundles

June

Raw notes by Bernd Steinert

Higgs bundles

Let C b e a curve as b efore

Denition A Higgs bundle is a pair E such that E is a vector bund le

on C and C E nd E

C

Why do we care ab out these

Note C E nd E dual of H C E nd E

C

Thus

cotangent bundle to mo duli

Higgs bundles

space of vector bundles

Of course there are several diculties

Formal computation of the tangent space

We have to lo ok at isomorphism classes over k with Let E

b e a Higgs bundle over k We have to parametrize its extensions to k

S

E id mo d and U with op en anes U and let E Write C

on U

Gluing E gives rise to a co cycle g with g U U E nd E

and we have g commutator mo dulo trivial deforma

tions ie g g g with g

The quotient gives the tangent space It can b e written as the hyp erco

homology of the complex

Ad

E nd E E nd E

C

deg comutator deg

Ad

One can dene H C E nd E E nd E To calculate it one

C

has to lo ok at the Cech complex in degree

Ad

y y y

Cech complex

Dierently from vector bundles there can b e an H here

There is an exact sequence of complexes

E nd E E nd E

y y y y y

E nd E E nd E

C C

It gives rise to a long exact sequence

i i

H E nd E H E nd E E nd E

C C

Ad

i i

H E nd E H E nd E

C

K H E nd E E nd E has an inner pro duct that is non

C

degenerate

Hom K K ie degrees and

C

Taking Serre duality

H K dual to H Hom K H K

C

Check directly This pro duct is antisymmetric

It is also the same as the canonical symplectic form on the cotangent

bundle at least if restricted to stable bundles as usual

Invariants

r r r

detT T a T a

r

i

with a C In particular a tr and a det

i r

C

The characteristic variety is given by

r

M

i

C Char Sp ec S

C

i

S symmetric algebra A Higgs bundle over C S gives a map S

Char Lo ok at its b ers

i

Conversely Given a C we construct a nite at D C

i

C

r

a a D Sp ec S

r

O

C

S

ie in lo cal co ordinates If C U and is a generator of on U

C

S

write a a Then D V where

i i

r r

O V O U T T a T a

D C r

The overlap rule on U U is T T

D is a curve but in general it need not b e nonsingular It is a complete

intersection

E is an O mo dule by T It is torsion free but not necessarily

D

lo cally free Ie

coherent torsion free sheaf on C

Higgs bundle

with some conditions

Denition A Higgs bund le E is cal led stable if for any stable

subbund le F E we have F E

ss

By GIT or other metho ds there exists a coarse mo duli space Higgs

s

which is pro jective over Char It has an op en subset Higgs which is a

manifold The pro ofs should b e similar to the ones b efore

Why is it a manifold

Lo ok at H C E nd E E nd E this is dual to H E nd E

C

E nd E k

C

Remark If E is a Higgs bundle then det E tr is also a Higgs

bundle

The obstruction for E is the same as the obstruction for det E Hence we

are reduced to rank In this case

Higgs PicC C

C

and this is smo oth

s

s

We have Higgs cotangent bundle to M as a dense op en subset by

estimating dimensions the complement has co dimension at least

A Theorem of Laumon

We now supp ose char k

s

Lo ok at the map Higgs Char and recall that we have a nondegenerate

s

inner pro duct on the tangent bundle of Higgs

Theorem Laumon If S is smooth and is mapped into a ber of the

map above then the image of the tangent bund le T is isotropic

S

Corollary dimb er dimHiggs

This is not really a theorem ab out stability

Proof Thm Let E b e the familiy of Higgs bundles on C S

corresp onding to S Higgs By assumption a a is constant on S

i i

The tangent bundle is mapp ed

T R pr E nd E E nd E

S C

E nd E has an inner pro duct Try to get sub complexes

We have the following rules

We may replace S by an op en subset or by an etale cover

We may replace C by a nite cover C C with C smo oth here the

inner pro duct gets multiplied by the degree of the covering

Now lets play the game

Let b e the generic p oint of C K k We have C the function

K

eld K C and E b ecomes a vector space with an endomorphism Let

b e the Jordan decomp osition of Then ker ker and so

s n n

n

on dene subvector spaces so that we get stable subbundles

E E E

over C

K

Everything holds over some op en subset of S hence without loss of gen

erality already over S Then we get

Ad

lt lt

T H E nd E E nd E

S C

y

L L

H E nd g r E E nd g r E

i i C

i i

The inner pro duct b ecomes the pullback

We may replace E by g r E hence we may assume

i n

Cho ose a covering C C such that all eigenvalues of lie in

s

K C we also have to take a covering of S

Then we get a complete stable ag of subspaces in the generic p oint

and subbundles

E E E E

r

This reduces us to the case rk E In this case

Higgs PicC C

C

z

Char

and the inner pro duct is given by E L is a line bundle

Ad

E nd L E nd L

C

O

C C

The complex splits whence the assertion 2

Why is this interesting

We have the following dimensions

vector bundles r g

r

P

i

dim C char variety

i

r

P

ig g

i

r g

Higgs r g

s

Corollary The map Higgs Char is at and the bers are maximal

isotropic ie Lagarangian

Fib ers over generic p oints of Char

Claim a generic D smo oth and irreducible

i

Corollary JacD irred b er line bund le on D of deg

Proof Claim Lo cally D C A is given by

r r

P T T a T a

r

P P

Lo ok at p oints in C A Char where P a lo cal

T

co ordinate in C

Check For a given p oint in C A this has co dimension in Char

For r Have sections with zeroorder and

P

P gives conditions

P

gives condition on a

r

T

For r Hyp erelliptic ones cause some problems Weierstrass p oints

r never o ccurs

D irreducible

Otherwise D D D is a disjoint union Then P P P with

i

p olynomials P P of degrees r and r resp and cocients in C

We get for the dimension

r g r g r r g

but r r r r r r r a contradiction 2

June

Last time we have obtained the Higgs bration

r

M

f

ss

i N

C A Higgs T Char

s

M

i

N is the dimension of mo duli space where the complement of the cotangent

ss

s

bundle of M in Higgs has co dimension

Higgs classies pairs E with C E nd E and semi

C

stability of Higgs bundles is dened with resp ect to invariant subbundles

For some xed degree the generic b er of f is a connected abelian variety

PicD where D C is some covering

From now on we will assume that det E O ie trivial so that we can

replace GL by SL which simplies things since there is no G factor

r r m

Corresp ondingly we restrict to C slE ie tr This

C

implies that we must drop the summand C in Char the dimension

go es down by g

Remark We may replace SL by any semisimple group G eg SO or Sp

r r

r

bundles with a nondegenerate symmetric or symplectic inner pro duct We

then have to take C gE where g is the Lie algebra of G

C

The pro of that D is irreducible can b e dicult for G SL

r

ss

O so that we get the same global f is prop er hence f O

Char Higgs

functions

r

M M M

ss

i n s n

ss

s

C S Char O Higgs O M S T

Char Higgs M

C

i n n

i

This is an isomorphism of graded vector spaces if we give C degree

C

i

s

s

Corollary M O

M

It is dicult to determine the higher cohomology b ecause we dont know

much ab out R in this situation But we can show that certain classes in H

are nonzero

How do we get classes in H

We have an ample line bundle

L det H C E F

on M and on Higgs This dep ends only on rk F since det E is trivial

L has a rst Chern class c L H M and we get a map

M

n n n

c L M S T H M S T H M S T

M M M

M

where the second arrow is some kind of derivative use that is dual to T

and contract once

Claim If n this map is injective

Proof The idea is to restrict to the generic b er

Recall the Higgs bration

f

Higgs Char T

M

y

M

We lift L from M to T and push it to Higgs Since the complement has

M

co dimension we get an injection on H We claim that there is a map

such that the following diagram commutes

L

n

H Higgs O H M S T

Higgs M

n

x x

c L

L

n

M S T Higgs O

M Higgs

n

Let Char O b e of p ositive degree Then c L is some class

Char

in H M S T To show that this is nonzero restrict to generic b ers of

M

f

Restrict L to generic b er Pic D There det H C E

det H D E is known to give an ample line bundle divisor on

Pic D

Ad

Aside Higgs is a symplectic manifold T H slE slE

Higgs C

and the form induced by this restricts to the canonical symplectic form on

T This gives a canonical identication T Thus for a lo cal

Higgs

M Higgs

T ie we get a vector eld function O we have d

Higgs Higgs

Higgs

H from the Hamiltonian vector eld of

If is induced from O we have d on the b ers since is

Char

constant along b ers and H is tangential to the generic b ers since the

tangent spaces of the b ers are maximal isotropic

Claim The map is given by c L H H Higgs O

Higgs

Proof Later 2

To see that Take generic b er f A an abelian

variety Since L is ample on A one knows that

H A O t c L H A

A

A A

induces a p erfect duality b etween H A O and t Is t A T

A A A A

Hence if c L H then H on f Since H is tangential to

f this means that H everywhere hence d and is constant

contradicting our choice of

Remark Lo oking at functions and H do esnt note removal of bits of co di

ss

mension Hence it do esnt matter whether we lo ok at Higgs Higgs

s

Higgs T T or T

s s

M M s M

2

What is c L

Facts on Chern classes and the Chern character can b e found for example

in chapter of Hirzebruch Berger Jung Manifolds and Mo dular

Forms

We take a base space or stack S and let b e the pro jection C S

S Let E b e a vector bundle on C S Eg S M or S Higgs with

universal bundle Then L det R E is a line bundle on S and we have

c L c R E HBJ p

If E is an SL bundle which we will assume then c E

r

Recall the Chern character of a virtual vector bundle F

c F c F chF rk F c F

and the total Todd class of F

T dF c F c F c F

By GrothendieckHirzebruchRiemannRoch we have using c E

c L chR E

chE T dC

c T c E

C

c E

hence since shifts degrees down by one

c L c E

In what cohomology theory do the Chern classes live

p p

c can b e dened in H hyp ercohomology where

p

d d

p p p

is the truncated de Rhamcomplex

d

For example c L H is the obstruction to

have an integrable connection on L

A connection on L is a map r L L such that r l

rl l d for O l L ie a prescription how vector elds act on

L

r is called integrable if the curvature r vanishes or equivalently if the

action of vector elds preserves commutators

Connections always exist lo cally Cover your space by op en anes U

i

such that there is a connection r on L over U Then r r is a

i i i j ij

form on U U and r R for some R also called curvature with

i j i i

i

dR On U U we then have R R r r d one of

i i j i j ij j j ij ij

the signs is correct This gives us a Cechco cycle in the Cech complex

d

vanishing i there is an integrable asso ciated to

connection on L globally

We have a sequence

d

H H H

where the quotient measures the existence of a global connection the image

of c L in H is called the Hodge class of L and where the subspace

measures if the global connection can b e made integrable

An imp ortant prop erty of the rst Chern class of line bundles is that

c L L c L c L This means that in chracteristic zero it

n

do esnt matter if we lo ok at L or at L with n when we want to know

if c L vanishes

Knowing c for line bundles we can dene c for vector bundles by the

i

splitting principle

For c E Lo cally dene connections r then r r E nd E

i i j ij

and so on A similar construction as for line bundles gives us a class

Atiyah class in H E nd E that is the obstruction to having a

global connection on E

E nd E O

The b er over O is a homogeneous space under E nd E it is the

space of connections on E Lo ok at the long exact cohomology sequence

E nd E k H E nd E

The Atiyah class E is the image of k in H E nd E Hence

has an element mapping to ie a global connection i this class is zero

p p

In general we get the pro jection of c E to H by applying sym

p

metric p olynomials ie the co ecients of the characteristic p olynomial

E nd E O X

Let E b e an SL bundle Then

r

tr

slE E nd E O

We have c E and c E c E cf HBJ p hence

c slE c E nd E c E E r c E

Lo ok at the Chern character chE r c E and use chE E

chE chE cf HBJ p For the rst equality note that the sequence

ab ove splits at least if r is prime to the characteristic by O E nd E x

r x id

E

This means that for many purp oses we can use slE in place of E This

is related to the fact that dierent representations of SL give Chern classes

r

diering by a constant factor

Example Consider a stable vector bundle E on a curve C Then its Atiyah

class lives in

H C E nd E H C E nd E k

C

and is prop ortional to deg E Hence stable SL bundles always have a

r

global connection Over C the Theorem of NarasimhanSeshadri tells

us that there is even a unitary one

f

Now lo ok at the universal bundle E on C M M and its Atiyah

class

H C M slE

C M C M

Since restricted to a b er of f is trivial we see that

C

H M f slE H M

C C M

Lo ok at R f As to we have

M

H C M slE

M M

M R f slE

M

Hom T R f slE

M

Hom T T

M M

Claim id Hom T T with some choice of sign

M M M

S

Proof Cho ose an op en cover C U such that E j is trivial On

i U

i

U U U we have transition functions g parametrized by M Take

ij i j ij

the trivial connection r on U M Then taking the derivative in M

i i

direction

r r dg g

i j ij

ij

U M E nd E

ij

M

Hom T E nd E j

M U

ij

Now follow the denitions 2

Claim H M is prop ortional to c L

C M

L is the determinant of cohomology or its inverse

Proof c E tr tr mo d in

C M C M C M

M

H C M mayb e its H which maps to H M Serre

C M M

duality H C H C O k

C C

Hence by RiemannRoch c L f c E I cannot say that

C

this is particularly clear to me 2

This class in H M classies an extension of O by which

M M M

dually is given by

O D i L T

M M

where D i L are the dierential op erators of order on L and where O

M

is emb edded in D i L as the dierential op erators of order Dierential

op erators of order corresp ond to connections

This is equal to the obstruction to dening a connection on E on C M

relative M ie r E E By a Theorem of A Weil r exists

C

lo cally in M and two such rs dier by an element of f slE

C M

r

Another formulation is the following Consider the mo duli space M of

stable SL bundles E together with a connection r E E Then

r C

r

the canonical forgetful map M M gives a homogeneous space under

with class c L Hence

M

r

M Sp ec S Di L

where the relation means that one has to identify S Di L k

with S Di L Di L identity op erator

A reference for these things is

G Faltings Stable Gbund les J Alg Geom

July

First a correctionexplanation of some arguments from the last lecture

f

T Higgs Char

M

y

M

Higgs is symplectic For O we have the Poisson bracket

Higgs

f g hd d i H where h i is the symplectic form on T It

Higgs

satises the Jacobi identity If and are pullbacks from Char we have

f g

We have the map

Char O Higgs O Higgs T H Higgs O

Char Higgs Higgs Higgs

f H c L H

and we have shown that its kernel are the constants

r

Let M b e the mo duli space of stable SL bundles with connection intro

r

r

duced at the end of last lecture We have the canonical map f M M

r

M Conn Sp ec S Di L classies connections on L

L

M

r

Claim M O r k

M

r

We have M O r M f O r and f O r is a twisted version

M M M

n

of S T It is still ltered F with graded pieces g r F F S T

M n n n M

Try to lift and lo ok at obstructions

F F F F F F

n n n n n n

Show that

n n

S T F F H F F H S T

M n n n n M

is injective for n Then a section of F F must already come from

n n

a section of F F Induction then shows that all sections must come

n n

from F k

Claim This map is given by

L L

c L

n n

H S T M S T Higgs O

M M Higgs

n n

c L

H Higgs O

Higgs

The injection on the far right comes from the fact that the complement

of T in Higgs has co dimension

M

Example n

O D i L T

M M

Lift sections of T lo cally to D i L dierences on overlaps in O

M

determine a class in H O

of degree in T corresp onds to M T Form d and use

M

M

duality to obtain a vector eld on T total space of cotangent bundle

M

S

Let M U b e a lo cal cover trivializing L with transition functions

i

We have g Then c L is represented by d log g dg g

ij ij ij

ij

hd dg g i g f g g g g

ij ij ij

ij ij ij

This is where g is to b e considered as a funtion upstairs ie on T

ij

M

the obstruction to lifting to D i L

P

a By similar computa For n we can write lo cally

I i in

I

tions using the appropriate variant of Leibnizs rule and the case n we

get the result

Varying the curve

We want to vary the curve C The curves of genus g are parametrized by a

mo duli space of dimension g

Assume we have a family C B of curves of genus g over a base B

f

Then we get a family M B of corresp onding mo duli spaces or stacks

and a line bundle L det H on M everything considered is assumed to b e

smo oth

n

We want a pro jective connection on f L

Given a vector eld Z on B nd a secondorder dierential op erator D

Z

on M acting on L D should b e rstorder along B and D Z should

Z Z

b e O linear meaning we get the same result from commuting functions

B

in O with D or Z

B Z

A dierential op erator has order zero if it commutes with functions it

has order n if commuting with functions gives dierential op erators of order

n

n

Our construction will work for L K with K is the canonical

MB

bundle

First we want to solve the related problem obtained by replacing D i L

by S D i L here the exceptional value is

r r

We have a family of mo duli spaces M B as well and M is in

nitesimally constant Let Z b e a vector eld on B

r

Claim Z lifts canonically to L on M resp ecting commutators

Z

Argument We assume that C is over R k t t C is a formal

d

scheme over R with a formal

Claim Restriction induces an eqivalence

E integrable r on C E integrable r E E on C

Ck

C is the sp ecial b er t of C

i

Proof We show that this even holds on any op en ane U C where

U U Spf R The claim is then obtained by gluing

k

U ab ove is the restriction for we use the trivialization U

k

Spf R to lift

Let E r on U b e given Then

r

t

i

U E j U E

U

and E is generated by the right hand side

Eg for d Start with some e If e is constructed and is ok mo d

n

t

n n

t then r e t Replace e by e we are in characteristic

t n

n

zero obviously then e is ok mo d t

For d use a similar argument commute

t

i

f

d

r r r

That is Let b B M Then M the formal completion of M

along f b is canonically isomorphic to f b B hence tangent vectors

r

of B lift to M ; action

d

Or b etter Consider B B B the diagonal emb edding B B the

d d

r r

pr M 2 formal completion along the diagonal Then pr M

r

We have the map M M where the latter varies with C The map

r

T r H slE H slE slE H slE T

MB

M B dR

is its tangent map b erwise

A vector eld Z T H C T the latter is the tangent space of

B C

the mo duli space of curves at C the map is the tangent map of B mo duli

space now induces

L

Z

H slE H C slE

dR

Claim This is the cup pro duct

We get the image of Z in H C T as follows

C

S

Write C U and take Ck with U U k The gluings of the

i i i

constant deformation and of the deformation given by Z dier by with

ij

U U T this gives a co cycle representing the image of Z

ij i j C

r r r

We work in B ie over k t t Then M M B M is the

d

b er over and we can extend by a constant r on B

S

Write C U with U U B and where U U is U U

i i i i j i j

P

t Additonally the universal bundle E on M is trivialized E j

i i U

i

B E j

U

i

Then

T r mapping to T

M

M

family of vector bundles r on C

family on C constant wrt E extension

given by E r with C slE

Now apply vector eld Z

Compute the commutator L The original E is given by a co cycle

Z

g on U U is given by on U such that d log g Ex

ij i j i i i j ij

tend to a parameter family over k s t where corresp onds to and

s

Z corresp onds to E rs t is a vector bundle with integrable connec

t

tion Take the formal completion in the origin then E b ecomes constant

E k s t For the transition functions g s t on U U this means

ij i j

log g s t Z g g g s g sg s Take the tderivative

ij ij ij ij i j

ij

t

sderivative change connection change the trivialization of E

A plausibility argument is as follows

r

M parametrizes lo cally constant sheaves sheaves that are parametrized

by lo cally constant transition fucntions g on U U which are therefore

ij i j

insensitive to changes of the gluing If we change the connection by the

transition fucntions now satisfy g rg To compute the commutator

L we compare Changing the gluing rst then the connection do es

Z

nothing but changing the connection rst and then the gluing gives us h Z i

2

We want to apply RiemannRoch to the universal bundle E on

f

r r

C M M

B

Lo cally E has an integrable connection r lifting its universal connec

i

We have tion mapping to slE

C

r r slE slE f

r

i j

M

mo d f r slE f

r r

C

i

M M

This denes a class in

r

T r f f H slE slE

r r

M B C

M M

Claim This is the canonical map T r T T r

B

M M B

We have a splitting T r T T r

B

M M B

Complex top ology argument

r

M parametrizes lo cally constant in C direction transition functions

g The obstruction for a global connection on L is given by the derva

ij

r

tives of g in M direction By lo cally picking canonical connections with

ij

r we can represent this ob ject by d log g giving a class in

ij

H C sl C sl C is constant

r r

Then c E is represented by a class which mo d f is represented

r

M

r

by the square of this class Hence f c E M is a class

r

M

corresp onding to the inner pro duct on T r

M B

r

Result The pullback of L to M has a connection with curvature given by

the inner pro duct on T r induced by T r T r and the inner pro duct

M M M B

on T r

M B