Alg Solutions of Diff.eq

8/3/2019 Alg Solutions of Diff.eq

1/118


2/118


3/118

Algebraic Solutions of Differential Equations 3c o n n e c t e d f ib r e s a n d c o m p l e x f i b r e q / c, a n d a d if f e r e n ti a l e q u a t i o n ( M , 17)o n ql /R w h i c h i n d u c e s ( M , 1 7)c lq /c o n q /c . S u p p o s e t h e r e e x i s ts a n a f fi n eo p e n s e t C c q / s u c h t h at , f o r a n y m a x i m a l i d e a l p o f R , t h e p - c u r v a t u r e o f(M /o M I V, 17)v a n i s h e s . I f s u c h a C e x i s t s f o r o n e s e t o f c h o i c e s (o //, R , ( M , 17)),t h e n s u c h a C e x i s ts f o r a n y s e t o f c h o i c e s . It 's e x i s t e n c e i s t h u s a n i n tr i n s icp r o p e r t y o f t h e g e r m o f ( M , 17)c a t t h e g e n e r ic p o i n t o f S c , w h i c h w e c a l l" h a v i n g p - c u r v a t u r e z e r o fo r a l m o s t a ll p " .

B e c a u s e " p - c u r v a t u r e z e r o " is a p r o p e r t y w h i c h i s l o c a l f o r t h e 6 ta let o p o l o g y , i t f o l lo w s t h a t i f ( M , 1 7)c b e c o m e s t r iv i a l o n a f in i te 6 t a lec o v e r i n g o f S c , t h e n i t h a s p - c u r v a t u r e z e r o f o r a l m o s t a ll p in t h e a b o v es e n se . G r o t h e n d i e c k ' s q u e s t i o n i s w h e t h e r t h e c o n v e r s e is t r u e :( I q u a t ) I f ( M , 17)c on Sc has p-curvature zero for almost all p, does itbecome trivial on a f ini te ~tale covering of Sc?

O u r m a i n r e s u lt is t h a t (I b is ) a d m i t s a n a f f ir m a t i v e a n s w e r w h e n t h ed i ff e re n t ia l e q u a t i o n i n v o l v e d is a P i c a r d - F u c h s e q u a t i o n , o r a s u i t a b l ed i r e c t f a c t o r o f o n e . R e c a l l t h a t i f K / C i s a n y f u n c t i o n f i e l d , a n d U / Ka n y s m o o t h K - v a r i e ty , t h e f in i te - d im e n s i o n K - s p a c e s o f a l g e b r a ic d eR h a m c o h o m o l o g y HER(U/K) a r e e a c h e n d o w e d w i t h a c a n o n i c a l i n te -g r a b le c o n n e c t i o n V, t h a t o f G a u s s - M a n i n ( " d if fe r e n t ia t i o n o f c o h o m o l o g yc l a ss e s w i t h r e s p e c t t o p a r a m e t e r s " ) . T h e r e s u l t in g d i f f e r e n ti a l e q u a t i o n s(H~R(U/K), 17)a r e c a l le d t h e P i c a r d - F u c h s e q u a t i o n s .

T h e s u i t a b le d i re c t f a c t o r s a r e t h e fo l lo w i n g . S u p p o s e a f in i t e g r o u p Ga c ts a s K - a u t o m o r p h i s m s o f U . T h e n i t a c ts o n th e de R h a m c o h o m o l o g yH g R ( U / K ) i n a h o r i z o n t a l w a y ( i. e., it r e s p e c t s 17). F o r a n y i r r e d u c i b l eC - r e p r e s e n t a t io n Z o f G a n d a n y a u t o m o r p h i s m a o f C , l e t Z " d e n o t e t h er e p r e s e n t a t i o n d e d u c e d f r o m X b y a p p l y i n g tr t o i ts m a t r i x c oe ffW e s a y t h a t X a n d X~ a r e Q - c o n j u g a t e . L e t Z ~, . . . , X , b e t h e n o n - i s o m o r p h i ci r re d u c i b l e r e p r e s e n t a t i o n o f G w h i c h a r e Q - c o n j u g a t e t o x. L e tP(X,)(HgR(U/K), V) d e n o t e t h e z , - is o ty p ic a l c o m p o n e n t o f ( Hg R ( U/ K ) , 17),i . e . t h e p a r t o f HER (U/K ) w h i c h t r a n s f o r m s b y X i, w i t h i ts i n d u c e d G a u s s -

rM a n i n c o n n e c t i o n . T h e n 0 ) P(Xi)(HgR(U/K), 17) i s w h a t w e m e a n b y ai = ls u i t a b l e d i r e c t f a c t o r o f (HER(U/K) , 17).

T h e p r o o f is b a s e d u p o n t h e s o m e w h a t s t r ik i ng f ac t t h a t i n c h ar a c -t e ri s t ic p , a s u i ta b l e a s s o c i a t e d g r a d e d f o r m o f t h e p - c u r v a t u r e o f t h eG a u s s - M a n i n c o n n e c t i o n i s a " tw i s t e d " f o r m o f t h e m a p p i n g " c u p -p r o d u c t w i t h t h e K o d a i r a - S p e n c e r c l a ss " . T h e p r o o f o f t h is f a ct isu n f o r t u n a t e l y c o m p u t a t i o n a l , a n d r a t h e r l o n g . I t d e p e n d s e s s e n t i a ll yu p o n t h e i d e n t i t y o f H o c h s c h i l d , w h i c h a s s e r ts t h a t f o r a n y d e r i v a t i o n Do f a c o m m u t a t i v e F p - a l ge b ra , a n d f o r a n y e l e m e n t X o f t h a t a l g e b r a ,w e h a v e D P - I( X p - t D X ) + ( D ( X )) P = X P - t D P ( X ) . I n d e e d , i n t h e c a s e o fa n H ~, t h i s i d e n t i t y i s t h e p r o o f .1"


4/118


5/118


6/118

6 N.M. Katz:

1 . G e n e r a l i t i e s o n t h e K o d a i r a - S p e n c e r C l a s sa n d t h e G a u s s - M a n i n C o n n e c t io n1 .0 . Th e Geom etr ic Set t ing

T h r o u g h o u t t h is S e c t i o n 1 , w e w i ll c o n s i d e r t h e s i tu a t i o nD ~ i ~ X , J ~ U = X - D

( 1 . 0 . 1 ) s

Ti n w h i c h T i s a n a r b i t r a r y b a s e s c h e m e , S i s a s m o o t h T - s c h e m e ( v ia g ),w h i c h w ill p l a y th e r o l e o f a p a r a m e t e r s p a c e, X is a s m o o t h S - s c h e m e( vi a f ) , w h o s e f ib r e s o v e r S a r e " p a r a m e t e r i z e d " b y S , a n d D i s ( via /)a u n i o n o f d i v is o r s D i i n X , e a c h o f w h i c h i s s m o o t h o v e r S ( h e n c e a ls oo v e r T ) , a n d w h i c h h a v e n o r m a l c r o s si n g s r e la t iv e t o S (h e n c e a l s o re l a ti v et o T ) . T h i s s it u a t i o n p e r s i s ts a f t e r a r b i t r a r y c h a n g e o f b a s e T ' -~ T .

I n practice, X is u s u a l l y p r o p e r o v e r S , a n d s h o u l d b e t h o u g h t o f a sa p a rt ic u l a r ly n ic e c o m p a c t i f i c a t i o n o f t h e s m o o t h " o p e n " S - s c h e m eU = X - D , w h i c h is p s y c h o l o g i c a l l y p r i o r to X . W e a ll o w D to b e t h ee m p t y d iv i s o r, c o r r e s p o n d i n g t o U b e i n g p r o p e r o v e r S. W e d o n o tassume X p r o p e r o v e r S e x c e p t w h e n w e e x p l ic i tl y s o st at e.(1 .0 .2) Let D e r o ( X / T ) ( r e s p . D e r o ( X / S ) ) d e n o t e t h e l o c a l l y f r e e s h e a fo n X o f g e r m s o f T - li n e a r ( r e sp . S - li n e a r) d e r i v a t i o n s o f (9x t o (9x w h i c hp r e s e r v e t h e i d e a l s h e a f o f e a c h b r a n c h D i o f D . W e m a y n o w d e f in e t h es h e a f o f g e r m s o f r e l a t iv e ( t o T , r e s p . t o S ) K a h l e r d i f fe r e n t ia l s o n X w i t hl o g a r i t h m i c s i n g u l a r i ti e s a l o n g D , b y

12Jc/r ( lo g D ) = H om ~x (D ero ( X / T ) , C x )(1.0.2.1) ~ / s ( lo g D ) = Ho m~, , (Oero (X /S) , (gx) .F o r e v e r y i n t e g e r p - > 0 , w e d e f in e

f2xP/r (lo g D )= Ae~x 2~/T l o g D )(1.0.2.2) a~/s ( l o g D ) = A ~ a lx /s ( l o g D ) .(1.0.3 ) In f ac t , O ] / r ( I o g D) ( r e sp . 12~ /s (l og D) ) i s a subcom plex o f j . 0 ~ / r( r e sp . j , Q~/s) , w h e r e j : U c--*X d e n o t e s t h e i n c l u s io n . T o fix i d e a s, l e t u se x p l ic a t e t h e s e s h e a v e s in t e r m s o f l o c a l c o o r d i n a t e s . W e m a y c o v e r S b y


7/118

Algeb raic Solut ions of Differential Equ at ions 7a f f in e o p en se t s q /i , an d co v e r X b y a f f in e o p en se t s V~ su ch t h a t

(1.0.3.1)e a c h q / = q /i is 6 t al e o v e r A t ( r d e p e n d i n g o n i)v i a lo c a l c o o r d i n a t e s s l , . . . , s ,e a c h V = V is 6 t al e o v e r A t , ( n d e p e n d i n g o n i )

v i a lo c a l c o o r d i n a t e s X l , . . . , x ,t h e b r a n c h e s o f D w h i c h m e e t V~ a r e d e f i n e d b y t h e e q u a t i o n

x , = 0 , v = 1 . . . , c~ (a d ep en di ng o n i ).T h e n , o v e r , V t h e s h e a f D e r o ( X / T ) i s a f ree (~ v -mo d u le w i th b a s i s(1.0.3.2) x~ - - (v = 1, ~), (j = ct + 1, n), a~ x~ . . . , ~ - ~ f . . . . ~ ( # = l , . . . , r)a n d D e r o (X / S ) i s a f r ee (~ v -mo d u le wi th b a s i s(1 .0 .3 .3) x ~ - - (v = 1 , c t) ,a x , " " ' 0 x j ( j = 0 ~ + 1 . . . . n ).T hu s, ov er V, f2~/T(1Og D) is f ree o n (~v w ith ba sis(1.0.3.4) d x ~ ( v = l , . . . , ~ ) , d x j ( j = ~ + l , . . . , r ) , d s , ( # = l , . . . , r)Xvw hile f21 /s( log D) is f ree on (_9 w ith ba sis

d x v(1.0.3.5) ( v = 1, . . . , c0, d x j ( j = ~ + 1, . . ., n ).Xv(1 .0 .3 .6 ) C l e a r l y t h e f o r m a t i o n o f f 2 ] / s ( lo g D ) c o m m u t e s w i t h a r b i t r a r yc h a n g e o f b a s e S ' - ~ S.(1.0.3.7) R e m a r k . ( W h e n S i s o f c h a r a c t e r i s t i c z er o , t h e c o m p l e xI 2 ]/ s( lo g D ) i s q u a s i - i s o m o r p h i c t o j . t 2 ~ : / s . T h i s i s n o t t h e c a s e i n g e n e r a l,b e c a u se j . f2g. /s h a s " t o o m u c h " c o h o m o l o g y , w h il e I2 ]/s (l og D ) h a s o n l y" g e o m e t r i c a ll y m e a n i n g f u l " c o h o m o l o g y , w h e n c e o u r " p r e f e r e n c e " f ort 2 ] / s ( log O).)

1.1 . The Kodaira-Spencer ClassF r o m t h e d e f i n i ti o n s , i t f o l lo w s t h a t w e h a v e a n e x a c t s e q u e n c e o f

l o c a l l y fr e e sh e a v e s o n X(1.1.1) 0 - ~ f * (f2~/r) --~ t2~/r (lo g D) -~ f2~c/s log D) -~ 0


8/118


9/118


10/118

10 N.M. Katz:

w h i c h i s t h e negative o f t h e i s o m o r p h i s m ( 1.2 .1 .8 ) e x p l i c a t e d i n ( 1 .2 .1 .9 ),a n d w h i c h w e w i ll not u s e .

I n t h e s e t ti n g o f (1 .2 .1 .9 ), w e m a y c h o o s e o v e r e a c h V~ a m o r p h i s m~ : ~ 1 V i ~ K e r([3 )[ V / ~ - fg [ Vi w h i c h i s a s e c t i o n o f ~ : f q~ --~ ou f(i .e . , suchth a t ~k i. ~ = id , lv , ) .( 1.2 .1 .1 0 .2 ) I n f a c t , w e m a y s i m p l y c h o o s e ~bi s o t h a t ~ o ~bi = id~rlv, - q~i~[3,t h i s b e i n g p o s s i b l e b e c a u s e , o v e r V~, id ~e -C p ~ o[3 i s a p r o j e c t i o n o n t ok e r([ 3) . T h e n t h e d i f f e r e n c e ~ b i - ~b~ d e f i n e s a m o r p h i s m f r o m ~ r V / ~ V~ t of~l V~n ~ w h i c h v a n i s h e s o n t h e i m a g e o f ct, t h u s d e f i n i n g b y p a s s a g e t oq u o t i e n t s a m o r p h i s m f r o m ~ 1 V /c~ V~ t o fq [ V /c~ V~, s ti ll d e n o t e d ~ i - ~b~.T h e c o h o m o l o g y c l as s o f {~ k~ -~b~} i n H a ( X , H o m ( ~ , i f ) ) i s t h e e l e m e n tc o r r e s p o n d i n g t o t h e e x t e n s i o n c l a s s o f (1 .2 .1 .1 ) v ia t h e i s o m o r p h i s m(1 .2 .1 .10 .1 ) . To s ee tha t this i s o m o r p h i s m i s t h e n e g a t i v e o f (1 .2 .1 .9 ), its u f fi c e s t o r e c a l l t h a t(1.2.1.10.3) 0~o ~ i = id~elv, - ~o~o 3w h e n c e(1.2.1.10.4) ~ o (~ k, - ~s) = - (~ o, - ~os)o [3w h i c h g i v e s t h e e q u a l i t y o f t h e t w o c o c y c l e s {~b - ~bs} a n d { - (q ~ -c p~ )} .(1 .2 .1 .1 1) T h e " s e c o n d " i s o m o r p h i s m ( 1.2 .1 .1 0) is t h e dual o f t h e f i r s t(1 .2 .1 .8 ), i n t h e s e n s e t h a t t h e d i a g r a m

E x t a ] ~ , c~ )

E x t a ( c ~ , j , ~ ) ~a.2~.~o~ H a(X , n o m ( ~ , ~ ) )i n w h i c h t h e l ef t v e r t ic a l m a p a s s o c ia t e s t o t h e c la ss o f 0 - ~ f g - , ~ r 0t h e c la s s o f t h e d u a l e x t e n s i o n 0 ~ ~ - ~ ~ fq --* 0 a n d in w h i c h t h er ig h t v e r t ic a l a r r o w is d e d u c e d b y p a s sa g e t o c o h o m o l o g y f r o m t h ec a n o n i c a l i s o m o r p h i s m o f d u a l i ty Horn (~ , f~ )~ Horn (f~ , ~ ) , is c o m m u t a -t iv e . T o s e e t h is l e t u s r e m a r k s i m p l y t h a t i f { cp ~ - cpj} i s a c o c y c l e r e p r e s e n t -i n g t h e c l as s o f 0 ~ f f ~ ~ - - ~ 0 v i a (1 .2 .1 .8 ), w h i c h i s t o s a y , v i as p l i t ti n g ~ ~ ,, t h e n { ~ bi-c ~i } i s a c o c y c l e a r i s i n g f r o m t h e c l as s o f0 - ~ ~ - ~ 0 b y s p l i t t in g ~ - ~ , w h i c h is t o s a y , v ia (1 .2 .1 .10).K e a p i n g i n m i n d t h a t t h e i s o m o r p h i s m s ( 1 . 2 . 1 . 8 ) a n d ( 1 . 2 . 1 . 1 0 ) a r e t h en e g a t i v e o f e a c h o t h e r , w e f i n d :(1 .2 .1 .12 ) P ropos i t ion . L e t ~ eH l ( X , H o m ( ~ , f f ) ) b e th e cla ss o f th eextension O - - * f f - - - ~ - - * ~ - - ~ 0 via (1.2.1.8), and let ~e H I( X , H orn(if , o~))be the class of the dual extension O---~-~J/~-~f~--~O, via (1.2.1.8). Thenvia the canonical i somorphism H o m (~ , i f ) ~ H o m ((~ , ~ ) , ~ = - ~ .


11/118

Algebraic So lutionsof D ifferentialEquations 11(1 .2.2 ) I n t e r i o r P r o d u c t a n d C u p P r o d u c t .

L e t u s n o w r e c a ll t h e m o r p h i s m I o f in ter ior produc t f o r e a c h i n t e g e rv=>l .(1.2.2.1) I : H o m ~ x ( ~ , cg) _ , H o m ~ , , ( A ~ , C ~| ~ ) .

In t e rm s o f l oca l s ec t i ons f l , . . . , f v o f ~ an d q~ o f Hom(o~, ~) ,v(1 .2 .2 .2 ) I ((p )( fx A . . . A f~ ) = ~ ( - 1) -1 q~ ( f j)Q fx ^ " " ^ f j ^ " " ^ f~ .j = l

(1.2.2.3) Pr op os i t ion . The d iagramE x tl ( .~ ,, f f ) a v E x f l ( A ~ , f f |

14~ x , H o r n ( ~ , ~ ) ) - ~ -' , ~ 1 ( X , H o r n (A " ~ , . ~ | A v-1 ~ ) )in wh ich the ver t ica l i somorph isms are (1.2.1.8) a n d (1.2.1.8 bis), i s com-mu ta t i v e .

Pr o o f . Ju s t as in (1.2 .1 .9), l e t us c ho os e a c ov er in g V~ of X an d sec t ionsq h: ~ 1 V ~ - , o~1V~ o f ~: ~ - , o ~ A b u s i n g n o t a t i o n , w e d e n o t e b y A v qh t h ec o mp o s i t i o n(1.2.2,3.1) A ~ I V ~ -a v (~ ~ A ~ I V ~ P r~ , K ~(i .e., A v ~oia~ " AV(q~i) m o d u lo K E ) . Cl e a r l y t he A ~ ~o g i ve l oca l s ec t i ons o fA ~ : K ~ a n d h e n c e 1 - c o c y c l e { A V q g i - A ~ o j } w i t hcoeff i c i en t s in H o m ~ x ( A ~ , f # | h a s a s c o h o m o l o g y cl as s i nH i ( X , H o m ~ x ( A ~~ , C~| ~ ) ) ~ E x t ~ AV o , f r 1 7 4 ~-1 ~ ) t h e c la s s o fA ~ ( 0 - ~ f r T o c o n c l u d e t h e p r o o f, i t r e m a i n s o n l y t on o t i c e t h a t(1.2.2.3.2) A ~ cp g- A ~ cp~ = I(~ 0~ - tp~).

T o s e e th i s , w e c a l c u l a te , f o r l o c a l s e c t io n s f l . . . . . f ~ o f(A ~q9 - A vq~j) f~ ^ . . . ^ f~ )= q0i(fx ^ . . . ^ q0i(f~ - ~oj(f~) . . - ^ ~o~(f~)

= ( q ~ ( f l ) + ( ~ o i - ~ o ~ ) ( f , ) ) ^ . . . ^ ( q ~ i ( f ~ ) + ( ~ o , - q ~ i ) ( f ~ ) )- ~ o ~ (f ~ ) ^ . . . ^ ~ o A L )

= i (-- 1)a-l((Pi--~O J)(fa)| A " '" ^ f- - - ^ . . . ^ f ,a=l

+ t e r m s i n K z ( A ~~ )= I ( ~ o i - q ~ ) ( f ^ . . . ^ f ~ ) . Q . E . D .


12/118


13/118


14/118


15/118

Algebraic Solutions of DifferentialEquations 15A s a c o r o l l a r y o f (1 .3 .2 ) a n d t h e def in i t ion o f t h e G a u s s - M a n i n c o n -n e c t i o n , w e h a v e

(1 .4 .1 .7 ) P ropos i t ion . S u p p o s e t h a t t h e H o d g e = ~ D e R h a m s p e c t r a lsequence(1.4.1.7.1) EP ' ~= R q f , ( f 2 ~ l s ( l o g D )) =~ RP +qf , ( I2~/s ( log D))i s degenera te a t E l , i . e . , tha t gr~ R p + q , (0~1s ( l o g D ) ) = R q f , (O~/s) . Th e nth e a s s o c ia t e d g r a d e d m a p p in g i n d u c e d b y t h e G a u s s - M a n in c o n n e c t i o nis t h e cu p - p r o d u c t w i th t h e Ko d a i r a - S p e n c e r m a p p in g (1.3.2.2)

p ~ H ~ ( S , a ~ lT | R i f , ( O ero (X/S ) ) ) ;i .e . , the d iagram

g rp RP + qf , (Q~c/~ log D))(1.4.1.7.2)

g q f , (f2 P/s( l o g D ) ) -c o m m u te s .

v > f2 ~ i T | g r F - ' R P + q f , (f2 ~ cis lo g D ) )7

" > | + % D ) )

(1.4.1.8) R e m a r k . I n c a s e X / S i s p r o p e r , i t fo l l o w s f r o m D e l i g n e ' s m i x e dH o d g e t h e o r y [ 8 ] a n d a s li gh t m o d i f i c a t io n o f h is a r g u m e n t ( [ 6 ], T h e o -r e m 5 . 5) t h a t(1 .4 .1 .8 .1 ) I f S i s a n y s c h e m e o f c h a r a c t e r i s t i c z e r o , t h e s p e c t r a l s e q u e n c e(1 .4 .1 .7 .1 ) i s d e g e n e r a t e a t E l , a l l o f it s t e r m s E ( q , E ~ ;~ a r e l o c a l l y f re e ,a n d i ts f o r m a t i o n c o m m u t e s w i th a r b i t ra r y c h a n g e o f b a s e S ' - ~ S .(1 .4 .1 .8 .2 ) I f S is a n y r e d u c e d a n d i r r e d u c i b l e s c h e m e w h o s e g e n e r i cp o i n t i s o f c h a r a c t e r i s t i c z e r o , t h e r e e x i s ts a n o n - v o i d Z a r i s k i o p e n s e tq / i n S o v e r w h i c h t h e a s s e r t i o n s o f (1 .4 .1 .8 .1 ) a r e v a li d . W e c o n c l u d e t h i ss e c t i o n b y s t a t i n g e x p l i c i t l y a v e r y u s e f u l c o r o l l a r y o f (1 .4 .1 .7 ).(1 .4 .1 .9 ) Coro l la ry . H y p o t h e s e s a s i n (1.4.1.7), f i x an in teger n>=O, andsuppose tha t M ~= R"f , (O~c ls ( log D)) i s an O s-submodu le s tab le un der theGauss -Manin connec t ion , i . e . , tha t(1.4.1.9.1) V ( M ) c 0 ~1T M .( W e th e n s a y t h a t M i s h o r i z o n ta l . )

L e t u s d ef in e t h e i n du c e d H o d g e f i l tr a t i o n o f M , F i ( M ) , b y(1.4.1.9 .2 ) F ' ( M ) = M c~ F i R " f , ( f2~ ls( l o g D ) ) .


16/118

16 N.M . Katz :

T h e n g r~ (M ) = F p (M ) / F p +a M ) ~ g r~ R p +q , ( f2]/s ( log D))(1.4.1.9.3) l[

Rq f , (~2P/s log O )).B e c a u s e M i s h o r i zo n t a l, i t f o l l o w s f r o m (1.4.1.7) t h a t w e h a v e a c o m m u t a t i v ed i a g r a m a f , ( C , , s ( lo g D ) ) - , | ( O , / s ( lo g O ) )

JV (M ) v ~ / r |(1.4.1.9.4)

grV (M ) v __~ f2~/r gr P - 1 M )

1 ]R ' f , ( ~ / s ( l o g D )) P , ~ / r | R q+ ~ , (~xV})~(logD))W e d e d u c e f r o m i t t h a t t h e H o d g e f i l t r a t i o n F i ( M ) o f M i s h o r i z o n ta l (i. e .,e a c h F i ( M ) is h o r iz o n t a l) i f a n d o n l y i f t h e r e s tr i c ti o n t o O p g r ~ ( M ) o f t h em a p p i n g " c u p - p r o d u c t w i t h t h e K o d a i r a - S p e n c e r c l a s s "(1.4 .1 .9 .5 ) p : @ g r~ (M )- , @ f2s~/r - I (M )p pvan i shes .(1.4.1.10) R e m a r k . In p rac t ice , the M in (1 .4 .1 .9 ) wi l l be e i the r a l l o fR" f , ( f2 ~ / s ( lo g D) ) , o r t h e p r i m i t i v e p a r t o f R" f , ( f2 ] / s ) i n c a se O i s v o ida n d X / S i s p r o j e c t i v e a n d s m o o t h , o r t h e p a r t o f R " f , (f2 ~c/s(lo gD )) w h i c ht r a n s f o r m s a c c o r d i n g t o a p r e c h o s e n i r r e d u c ib l e r e p r e s e n t a t i o n o f af in i te g r o u p ( o f o r d e r p r i m e t o a l l r e s i d u e c h a r a c t e r i s t ic s o f S ) w h i c h a c t sa s a g r o u p o f S - a u t o m o r p h i s m s o f X a n d p r e s e rv e s D . T h i s l a s t c a s e w i lla r i s e w h e n w e d i s c u s s S c h w a r t z ' s l is t (c f . 6 .0 ).

2. The Cartier Operation and the Conjugate Spectral Sequence2 .0 . T h r o u g h o u t th i s s e c t io n , w e w i ll c o n s i d e r t h e s i t u a t i o n o f 1 .0 , w i t ht h e a d d i t i o n a l a s s u m p t i o n t h a t S i s a s c h e m e o f c h a r a c t e ri s ti c p (a p r i m en u m b e r ) , i .e . t h a t p . l s = 0 i n ~Ys.(2.0 .1 ) R e c a l l t h a t f o r a n y S - sc h e m e n : Y- --, S , t h e S - sc h e m e y ~v ) i s b yd e f in i t i o n t h e f i b r e p ro d u c t o f 7 ~ : Y- - - ,S a n d t h e a b s o l u t e F r o b e n i u s


17/118

A l g e b r a i c S o l u t i o n s o f D i f f e re n t i al E q u a t i o n s 1 7m o r p h i s m F a b ~ : S - ~ S ( s o o n t h e r i n g l e ve l , F ab s i s j u s t " r a i s i n g t o t h ep - t h p o w e r " ) . T h u s b y c o n s t r u c t i o n Y tP~ s it s i n a c a r t e s i a n d i a g r a m

(2.0.1.1)

T h e p a i r o f m o r p h i s m sI (P) = Fa*bs(~) [ ~S r~b~ ~ S

Fa b : y-- --~ y7z : Y---~ S

d e f i n e s a m o r p h i s m F : Y - - ~ Y ~P ~, t h e r e l a t i v e F r o b e n i u s , w h i c h f it s i n t o ac o m m u t a t i v e d i a g r a m

y f ~ y~p~ ~ ~ y r ~ y (p )~(P)

i n w h i c h F . ~ i s F , bs: Y(P) ~ Y(P) a n d a - F i s F~b : Y---~Y . I n t u i t i v e l y , Fr a is e s t h e " v e r t i c a l c o o r d i n a t e s " t o th e p - th p o w e r , a n d e r a i se s th e" S c o o r d i n a t e s " t o th e p - th p o w e r .(2.0 .2) C o n s i d e r n o w t h e s p e c ia l c a s e Y = X is a s m o o t h S - s c h e m e , s a yo f r e la t iv e d i m e n s i o n n. T h e n :(2.0.2.1) F , ( C x ) i s a l o c a l l y f r e e ( gx c,~ m o d u l e o f r a n k p n; i n d e e d i fx l . . . . , xn a re l o c a l c o o r d i n a t e s o n a n o p e n s e t V c X ( i . e . , an 6 t a l em o r p h i s m V ~ A ~ ), t h e n a b a s e o f F , (C v) a s (fv~p~ m o d u l e is g i v e n b y t h ep~ m o n o m i a l s x ~'~ .., x ~ " h a v i n g O < w i ~ p - 1 .(2 .0 .2 .2 ) F - l ( ( g x t , , ) i s p r e c i s e l y t h e s u b s h e a f o f (9x w h i c h i s 'k i l le d b y a l lo f D e r ( X / S ) .(2.0.2.3) F , ( f 2 ~ . / s ( l o g D ) ) is a n C x ~ M i n e a r c o m p l e x o f l o c a l ly f re ec o h e r e n t s h e a v e s o n X ~p~.

2.1 . The Cartier OperationT h e f u n d a m e n t a l f ac t a b o u t D e R h a m c o h o m o l o g y in c h a ra c te r is t i c p

i s t h e f o l lo w i n g t h e o r e m o f C a r t i e r w h o s e s t a t e m e n t w e r e c a ll (c f. [ 3 ] ,c h p t . 2 , a n d [ 2 4 ] , 7 . 2 ) .( 2.1 .1 ) T h e o r e m . T h e r e i s a u n i q u e i s o m o r p h i s m o f (PxC,~ m o d u l e s f o r e a c hi n t e g e r i > 0

2 Inventionesmath., Vol.18


18/118


19/118

A l g e b r a i c S o l u t i o n s o f D i f f e r en t i al E q u a t i o n s 1 9

W e a r e t o b e g i v e n a m o r p h i s m K (q ~) o f f il te r ed com plexes o f f ~ l ( t ~ z ) -m od ules on Yz~( 2 . 2 . 1 . 4 ) q ~ l ( K z ~ , F ) ~ ~ fz~X((-0z~) ~(~)-~ (Kz~ , F )~ " f z ~ ( rwhich sa t i s f i e s t he na tu ra l t r ans i t i v i t y cond i t i on fo r a compos i t i on o fm o r p h i s m s o f Z - s c he m e s .

Co ns ide r no w the spec t r a l s equ ence o f d~z~-modules(2.2.1.5) E ~' " ( Z ~ ) = R P + q f z , .( g r ~ ( K z ,) ) ~ R P + " f z , . ( K z ) ,o n t h e Z - s c h e m e Z 1 , w h o s e E ~ q t e r m w e d e n o t e E ~ ' q ( Z O . F r o m t h e giv e nfunc to r i a l i t y o f (Kz~, F) i n Z I , we deduce , fo r eve ry m orp h i sm q~: Z2- - , Z1o f Z - s c h e m e s , m o r p h i s m s o f (g z~ -m o du le s c a ll e d " c h a n g e o f b a s e m o r -ph i sms " ,(2.2.1.6) ~0" (E~' ~ ZI)) - - - ~ ) ~Er '" (Z2)w h i c h r e n d e r c o m m u t a t iv e a ll d i a g r a m s

(2.2.1.7)* E p , q, , ( z o )

q,*~d~' q) l

K(~o) E p . q ~ Z~, 2t

I a , qK ( o ) .~ E p + r , q + 1 - , ( Z 2 ) .T h e s e m o r p h i s m s a r e c o m p a t ib l e w i th t h e u s u al i s o m o r p h i s m o f E , + tw i t h t h e c o h o m o l o g y o f ( E , d ,), i n t h e s e ns e th a t t h e d i a g r a m

~o* (K e r d~ q in EP 'q(Z O) -K3~')!(2.2.1.8) ~,*~ . . . . i ca lp r o l ec t ion )

4,c o m m u t e s .

, K e r d~' q in E~' q Z2)canoni ca l p ro j ec t ion

F u r t h e r , t h e i n d u c e d m a p p i n g o n Eoo is t h e a s s o c i a te d g r a d e d o f t h ec h a n g e o f b a s e m o r p h i s m d e d u c e d f r o m (2 .2 .1 .4 ):(2.2.1.9) (P * R P z l , ( K z , ) r ~ ) , R p f z 2 , ( K z 2 ) .

F o r e a c h i n t e g e r r o ~ 1, w e s a y t h a t t h e f o r m a t i o n o f E,o c o m m u t e swi th base chan ge i f fo r eve ry Z - s che m e q~: Z t ~ Z , a nd a ll pa i r s (p , q ) o fin tegers, the m or ph ism (2.2 .1 .6)(2.2.1.10) ~o* (EP, q (Z1)) - r ~ , ) ~ EPgq( Z z )is an i s o m o r p h i s m . W e s a y t h a t t h e f o r m a t i o n o f t h e s p e c tr a l s e q u e n c ef rom E ,o on co m m ute s w i th base ch ang e if fo r al l r > t o, t he fo rm at ion o fE , c o m m u t e s w i th b a s e c h a n g e .2*


20/118


21/118


22/118

22 N.M. Katz:

P r o o f T h e o n l y p o i n t is t h a t u n d e r e i t h e r o f t h e h y p o t h e s e s, th ec a n o n i c a l m o r p h i s m o f b a se c h a n g e(2.3.1.2.2) F * s R " f , (bX/S (log D)) --~ R"f~,") (a* (~bx/s(log D )))wh ich com es f rom th e ca r t e s i an d i ag ram (2 .0 .1 .1 )

X X( 2 . 3 . 1 . 2 . 3 ) l f (P) l f

S Fa b ~ 'Sis a n i s o m o r p h i s m .(2.3.1.3) R e m a r k . Under t he i somorph i sm (2 .3 .1 .2 .1 ) , t he Gauss -Maninc o n n e c t io n d e d u c e d o n F* s R " f , (~2bX/S log D)) ann ihi la tes the image u n d e rF*s of R " f , ( ~ x / s ( l o g D )) ( co m p a r e [24], 5.1.1).(2.3.2) Proposit ion. In the geom etric s i tuation 1.0, suppose th at X is properover S (and that S is a scheme o f characteristic p, as i t has been throughoutSec t ion 2). Suppose fur ther tha t(2.3.2.1) Ea c h o f th e H o d g e c oh o mo lo g y s he av es R " f , ( ~ x / s ( l o g O ) ) is aloca lly f ree shea f o f f in i t e rank on S and (hence) that i ts form at ion com-mutes wi th arbi trary change o f base S ' -* S .(2.3.2.2) The H odg e =~ De R ha m spec tra l sequence

E ~ b = R b f , ( ~ x /s (log D)) ~ R a+b , (I?~c/S log D))is degenerate at E 1

Then the conjugate spectral sequence, w hich, than ks to (2.3.1.2.1) a n dthe hyp othesis (2.3.2.1), m a y b e w r i tt e n(2.3 .2.3) r 'b = F~*sR " f , ( ~ x/s ( log D)) ~ R" +b , (g2~/s(log D )),is degenerate at E 2 .

P r o o f By (2 .3 .1 .4 .1), i t fo llows tha t the co nju nc t ion o f the h yp oth ese s(2 .3 .2 .1 ) an d (2 .3 .2 .2 ) r em ains t rue a f t e r an a rb i t r a ry c han ge o f base S ' ~ S ,a n d i m p li es t h a t t h e f o r m a t io n o f t h e H o d g e ~ D e R h a m s p ec tr alseque nce com m utes w i th a rb i t r a ry cha ng e o f base S ' - - , S . F ro m (2 .2 .1 .2 )i t fo l low s a l so th a t the f orm at io n o f the conE~'b c o m m u t e s w i th a r b i t r a r ychange o f base , whi l e by genera l p r inc ip l es t he fo rmat ion o f t he en t i r ec o n j u g a t e s p e c tr a l s e q u e n c e c o m m u t e s w i t h a n y f i a t b a s e c h a n g e S ' -+ S .(2.3 .2 .4) W e m ay assu m e tha t S i s a f f ine , beca use the qu es t ion i s loca lon S . W e wi sh t o r ed uce t o t he case in w hich S is noe the r i an . So supposeS=Spec(A) . C lea r ly t he re ex i s t s a subr ing A o c A which is f ini telyg e n e r a t e d o v e r Z , a p r o p e r a n d s m o o t h A o - sc h e m e X o , a n d s m o o t h


23/118


24/118

24 N.M. Katz:o f f in i te r a n k i m p l i e s t h a t f o r a s u i t a b l e s u b r i n g A I o f A , A o c A 1 c A ,w h i c h i s f in i te ly g e n e r a t e d o v e r Z , t h e A l - m o d u l e E~'b (AO i s l o c a l l yf r e e o f f i n it e r a n k , o r w h a t i s t h e s a m e , f la t, s in c e i n a n y c a s e i t o f f in i t et y p e o v e r t h e n o e t h e r i a n r in g A 1 .

T h e t r u t h o f t h is l a s t a s s e r t i o n is a p a r t i c u l a r l y s i m p l e c a s e o f ( E G A IV ,1 1.2 .6 .1 ), in t h e n o t a t i o n o f w h i c h B o = A o , M o = E ~ ' b ( A o ) , a n d t h e A ~a r e a ll t h e a b s o l u t e l y f in i te l y g e n e r a t e d s u b r i n g s o f A w i th A o c A ~ ~ A .( 2.3 .2 .5 ) H a v i n g r e d u c e d t o th e c a s e i n w h i c h S is t h e s p e c t r u m o f an o e t h e r i a n r in g , w e m a y f u r t h e r a s s u m e t h a t S is t h e s p e c t r u m o f an o e t h e r i a n l o c a l r i n g ( a g a in b e c a u s e th e q u e s t i o n i s l o c a l o n S ). W e m a yn e x t s u p p o s e S to b e th e s p e c t r u m o f a c o m p l e t e n o e t h e r i a n l o ca l r in g( b y f a i th f u l f l a t n e s s o f t h e c o m p l e t i o n ) , a n d f i n a ll y t h a t S i s t h e s p e c t r u mo f a n a r t i n i a n l o c a l r in g . L e t u s f ir s t e x p l a i n t h i s la s t r e d u c t i o n s t ep .

S u p p o s e S = S p ec (A ) , A a c o m p l e t e n o e t h e r i a n l o c al r in g w i th m a x i m a li d e a l m , a n d s u p p o s e t h a t f o r e a c h i n t e g e r n_>_0, t h e c o n j u g a t e s p e c t r a ls e q u e n c e o v e r S , = S p e c ( A / m ~+ 1) i s d e g e n e r a t e a t E 2 . L e t ' s d e n o t e b y(c,,.E'~'b(n) , d~(n)) t h e c o n j u g a t e s p e c t r a l o v e r S . ( in c l u d in g n = o % p u t t i n gS ~ = S ). T h e n a s r e m a r k e d a b o v e , w e h a v e(2.3.2.6)w h e n c e(2 .3 .2 .7)

e~ b ~ ~ c~

d 2 (00) --- ~ d2 (n)-- 0s u p p o s e d 2 ( o o) . . . . . d , ( oo ) = O. T h e n(2.3.2.8) . ,b _ . ,b 9onEr+l(OO)-conE2 (oo)= ]lnm o n E a 2 ' b (n )= l im rw h e n c e(2 .3 .2 .9 ) d r + , ( o o ) = l im d ~ + l ( n ) = 0 .T h i s s h o w s i n d u c t i v e l y t h a t (c on E ~'b (o o), d , ( o o )) is d e g e n e r a t e a t E 2 , a n dc o m p l e t e s t h e p r o o f o f v a l id i ty o f t h e r e d u c t i o n t o t h e c a s e in w h i c h S ist h e s p e c t r u m o f a n a r t i n i a n l o c a l r in g .( 2.3 .2 .1 0 ) S u p p o s e n o w t h a t S = S p e c ( A ) w i t h A a r ti n l o c a l, a n d d e n o t eb y l n g a (M ) t h e leng th o f a n A - m o d u l e M . T h e n a n e c e ss a r y a n d s u f fi ci en tc o n d i t i o n t h a t t ile c o n j u g a t e s p e c t r a l s e q u e n c e d e g e n e r a t e a t E 2 i s t h a t(2 .3 .2 .11) ~ lnga (~o .E ~ , b )= Z lng a ( ton E'b )"a,b a,bT h e s e c o n d t e r m i n ( 2 . 3 . 2 . 1 1 ) i s(2 .3 .2 .12) ~ lngA (~o,E~;b) = Z lngA ( R " f , ( t2}/s ( l og D ) ))

a , b I I


25/118

Al gebra i c So l u t i ons o f D i f f e ren t ia l E qu a t i ons 25w hi le th e f i r s t t e r m i s, by (2 .3 .1 .2 .1 ) ,(2.3.2.13) ~ lngA(co ,E~ 'b ) = ~ lngA (F~s Raf , ( f2bx /s ( log D))) .

a ,b a ,bN o w b y h y p o t h e s i s ( 2 . 3 . 2 . 1 ) , e a c h o f t h e A - m o d u l e s Raf , ( f2bx~s( log S)) isf re e o f f in i te r a n k , a n d h e n c e e a c h o f t h e A - m o d u l e s F~*sR a f , ( ~ / s ( l o g D ))i s free o f the s a m e f i n it e r a n k . I n p a r t i c u l a r , w e h a v e , f o r e a c h a , b(2.3.2.14) lngA (F* s R a f , (f2bx/s log D)) ) = lng a ( R a f , (f2bX/S log D ))) .P u t t i n g to g e th e r (2 .3 .2 .1 2 -1 4 ), t h e c r i t e r io n (2.3 .2 .1 1 ) fo r d e g e n e ra t i o na t E 2 m a y b e w r i t te n(2.3.2.15) ~ ln g a ( R a f,( f2 b x /s (lo g D ) ) ) = ~ l n g A ( R " f , ( f2 ~ / s ( lo g D ) ) ).

a,b nT h i s l a s t e q u a l i t y h o l d s , i n v i r t u e o f t h e h y p o t h e s i s (2 .3 .2 .2 ) t h a t t h e H o d g e=> D e R h a m s p e c tr a l s e q u e n c e i s d e g e n e r a t e a t E ~ . T h i s c o n c l u d e s t h ep r o o f o f d e g e n e r a ti o n . Q . E . D .(2 .3 .2 .16) Corol lary . U n d e r t h e h y p o t h e s e s o f p r o p o s it io n (2 .3 .2 ) , th ef o r m a t i o n o f t h e c o n j u g a t e s p e c t r a l s e q u e n c e (2.3.0.1) c o m m u t e s w i t ha r b i tr a r y c h a n g e o f b a s e S ' -- * S .

Proo f . B y ( 2.3 .2 ), t h e c o n j u g a t e s p e c t r a l s e q u e n c e i s d e g e n e r a t e a t E 2 ,a n d b y ( 2 . 3 . 1 . 2 ) t h e f o r m a t i o n o f i t s E 2 t e rm c o m m u t e s w i th a r b it ra r yc h a n g e o f S ' ~ S . T h e r e s u l t f o l l o w s b y (2 .2 .1 .1 1).( 2 .3 .3 ) W e a re n o w in a p o s i t i o n to e x p la i n t h e t e r m i n o l o g y " c o n j u g a t e "s p e c tr a l s e q u e n c e . W i t h t h e a s s u m p t i o n s o f p r o p o s i t i o n (2 .3 .2 ), s u p p o s ef u r t h e r t h a t S i s t h e s p e c t r u m o f a f ie ld K o f c h a r a c t e r i s t i c p , a n d t h a t t h ed i v is o r D is v o id . T h e H o d g e ~ D e R h a m s p e ct ra l s e q u e n ce(2.3.3.1) E~'b = H b ( X , ~ X / K ) =*" H a + b ( X , ~ "~ '/ K)b e i n g d e g e n e r a t e a t E t , w e h a v e(2 .3 .3 .2) g ~ (H n(X , f2~/r ) ~ H " - ~ X , f~a'/~)-T h e d e g e n e r a c y a t E 2 o f t h e c o n j u g a t e s p e c tr a l s e q u e n c e(2.3.3.3) co,E~2'b= F ~ H ~ ( X , f2~ ,/~ ) = ~ H~+~(X, f2~/r)w h i c h w e p r e f e r t o r e w r i t e a s(2.3.3.4) r = H ~ ( X ~p~,O~c~/K)=~ Ha+b(X, f2~//~),g iv e s a n i s o m o r p h i s m(2.3 .3.5) g ~ o n n ~ (X , f2~c/K ~-- H a (X ~p~, ~}?~,~/K)9Put t ing toge ther (2 .3 .3 .5 ) and (2 .3 .3 .3 ) ( fo r X tp~)w e f in d a n i s o m o r p h i s m(2.3 .3 .6 ) g ~ o H " (X , f 2 ~ / r ) " ~ g ~ - ~ H ~ ( X ~v~, 2],~,/r ) .


26/118

26 N.M. Katz:

In o rd e r t o exp la in t he t r ansc end en ta l a na lo gu e o f (2 .3 .3 .6 ), l et Y bea a p r o p e r a n d s m o o t h C - s c h e m e , a n d d e n o t e b y Y "" t h e " u n d e r l y i n g "complex mani fo ld . By GAGA ( [39 , 36] ) and Po inca r6 ' s l emma ( [14] ) ,w e h a v e i s o m o r p h i s m s(2.3.3.7) H"(Y , ,O~ , / c ) ~ , H"(Y~",f2~/~), ~ H "(Y~ ",C), ~ H " ( Y ' ~ " , Z ) |b y m e a n s o f w h i c h a n y a u t o m o r p h i s m o f t h e f ie ld C o p e r a te s o n H "(Y ,O~,/c(by t ranspor t ing by (2 .2 .3 .6) i t s ac t ion on H " ( Y a", Z ) | t h r o u g h t h es e c o n d f a ct o r) . I n p a r t i c u la r , t h e a u t o m o r p h i s m " c o m p l e x c o n j u g a t i o n ",d e n o t e d(2.3 .3.8 ) F~bs: C ---} C ,o p e r a t e s o n H " ( Y , I2~,/c), furn ishin g a ca no nic al (albei t t ra ns ce nd en tal)i s o m o r p h i s m(2.3.3.9) Fa* H " (Y, f2~/c)~ H " (Y, f2~./c .Th e com plex con juga t e o f t he H od ge f il tr a ti on , no t e d ,onF , is by de f in i ti onthe image under (2 .3 .3 .8) of the f i l t ra t ion F * s ( U ) o f F * s H " ( Y , O ~ , / c ).A cco rd ing to H od ge the ory ( [42, 5 ]), we have , fo r i= 1 . . . . n , a d i r ec ts u m d e c o m p o s i ti o n(2.3.3 .10) H"(Y, f2~,/c)=FiH"(Y, 9 , + 1 - , , 9r/c) G Fdo, H (Y, O r/c)o r , w h a t is t h e s a m e , a b i g r a d u a t i o n ( H o d g e d e c o m p o s i t io n )

n(2.3.3.10 bis) H "(Y , t2~,/c)= @ ( F i n Fc ~ ') (H" ( Y, f2~,/c)).~=oT h i s b i g r a d u a t i o n g iv e s (t r a n s ce n d e n t a l ) i s o m o r p h i s m s(2 .3 .3 .11) g~oo, (H"(Y, 12~/c) , ~ , F " -" n V g o , , N gr~_~(H,(y ,swhich we r ega rd as t he t r an scen den ta l an a log ue o f (2 .3 .3 .6 ).(2 .3 .4 ) W e m u s t h o w e v e r h a s t e n t o p o i n t o u t t h a t t h e a n a l o g u e o v e ra f ie ld K o f cha rac t e r i s ti c p o f the H od ge dec om po s i t i on (2 .3 .3 .10) p ro -v ided by F an d Fr ove r C is gene ra l l y f a l s e . Indeed , t he ex t en t to w hichthe f i l tra t ions F an d Fr fail to be t ransv ersa l is an in teres t ing ar i th m et icinvar i an t . I n o rde r t o exp la in t h i s po in t more fu l l y , i t i s conven ien t t of ir st r eca l l t he Hass -W i t t "op er a to r s " an d the i r r e l a t i on t o ' t he con juga t espect ra l sequence .(2.3 .4 .1) W e re tur n to the shel ter ing hy po thes es of pro po s i t ion (2 .3 .2).F i x in g a n i n t e g e r n > 0 , t h e d e g e n e r a t i o n o f t h e c o n j u g a t e s p e c tr a l se -quence a t E2 gives us an inc lus ion

~ - 1(2.3.4.1.1) F*~ R " f . (d~x) ~ ~~o,E~'0~.._,R "f . (O~/s(log D))


27/118


28/118

28 N.M . Katz :t o g e t h e r w i t h t h e r e s p e c t i v e d e g e n e r a t i o n s ( 2 . 3 . 2 . 2 - 3 ) i m p l y , f o r e v e r y0 < i < n , t h e f o rm u l a( 2 .3 .4 .1 .11 ) d im K H " ( X , f2~./~ log D) ) = d im K F i + t + d im r Fr i .F o r t h e h o m o m o r p h i s m s (2 .3 .4 .1 .8 ), s o u r c e a n d t a rg e t h a v e t h e s a m ed i m e n s i o n , j u s t a s f o r (2 .3 .4 .1 .9 ), a n d b o t h h o m o m o r p h i s m s h a v e t h es a m e k e rn e l , n a m e l y F 1 n F ~ , . Q . E . D .(2 .3 .4 .2 ) W e n o w d is c u ss t h e s o m e t i m e s d e f in e d " h i g h e r " H a s s e - W i t to p e r a t i o n s ( w h i c h i n c l u d e t h e u s u a l o n e a s a s p e c ia l ca s e) , s ti ll s u p p o s i n gt h e h y p o t h e s e s o f (2 .3 .2 ). A s b e f o r e , w e fix a n i n t e g e r n > 0 . F o r e a c hi n te g e r i, w e d e n o t e b y h (i) t h e c o m p o s i t e m a p p i n g

n- -i ?1 9 FI 9~ o n ( R f , (F2~c/s l o g D ) ) ) ~ R f , ( f 2 x / s( l o g D ) )(2.3.4.2 .1) . [

R " f . (f2~,./s log D ) ) / F i+ 1(2.3.4.2.2) P r o p o s i t i o n . H y p o t h e s e s a s i n (2.3.2) , a n d n > O f i x e d a sa b o v e , s u p p o s e t h a t f o r a n i n t e g e r i , t h e m a p p i n g h ( i) (2.2.4.2.1) i s an i so -m o r p h i s m . T h e n t h e r e is a u n i q u e m a p p i n g o f lo c a l l y f r e e C s - m o d u l e s , t h ei + 1- s t H a s s e - W i t t o p e r a t io n(2 .3 .4 .2 .3 ) H - W ( i + I ) : F , ~ s R " - i - t f , ( f 2 i x - ~ s l ( l o g D ) ) - ~ R " - i - ~ f , ( f 2 ~ T s ~ ( l o g D ) )w h i c h re n d e r s c o m m u t a t i v e t h e f o l l o w i n g d i a g r a m :

H- W ( i+ I )

Fa~s R " - ' - ~ , ( f~ ix -~ l o g D ) )) ~ ~ cohEn2 i- 1,i+1 E i + 1 , n - i - 1 ~ R n - i - I f , ( ~ r ~ - S 1 l o g V ) )

(2.3.4.2.4) F~'o~i- l(R"f , ( f2~c/s( log D ) ) ) h ( i + l ) , R , f , ( f 2 ~ c / s ( l o g D ) ) / F , + 2

F ~ : ' ( R " f , (f2 ~c/s(l3g D)) ) h(0 ) f * ( i x / s ( l ~ D ) ) / FR ~ f 2 " i + 1

!0


29/118


30/118

3 0 N . M . K a t z :

is a s e c t i o n of t he i nc lus ion F~= i r ~ -1 , wh ose ke rne l is no ne o th e rthen F i + l n F ~ o ~ ~ -1 . Thi s shows tha t t he canon ica l mapp ing F~+lc~F~o~ - 1 ~ F ~o~ - a / F f o ~ i i s an i som orp hism , w hich pro ves (2 .3 .4 .4 .2).(2 .3 .5) In th i s sec t ion we wish to exp la in the d i f ferent ia l eq uat io nssa ti sf ied by ce r t a in o f t he h igher H asse -W i t t ma t r i ces i n t rod uce d in(2 .3 .4) . These d i f ferent ia l equat ions were f i r s t not iced by Igusa [22] inthe case o f e l li p ti c cu rves , t hen l a t e r exp la ined qu i te gen era l l y by M an in[28] .(2 .3 .5 .1 ) W e p l ace our se lves un de r t he hyp o theses o f (2 .3 .2 ), an d as sum efur the r t ha t t h e d iv i so r D i s vo id , and tha t t he geom et r i c f ib res o f X / S a rec o n n e c t e d , o f d i m e n s i o n N . U n d e r t h e s e h y p o t h e se s , w e h a v e(2.3.5.1.1) R2Nf , (~Q~c/s ) - -%RNf , (O~/s ) is l oca l ly f r ee o f r ank 1 and evencano n ica l ly i som orph ic t o t he s t ruc tu ra l she a f Cs, v ia t he t r a c e m o r p h i s m(cf . [15] an d [42] )(2.3.5.1.2) tr : RaNf , (Y2]/S) ~ d)w h i c h c a r r i e s t h e G a u s s - M a n i n c o n n e c t i o n o n R Z N f , ( f 2 ] / s ) i n to t hes t andard connec t ion on d ) ( t he one g iven by ex t e r io r d i f f e r en t i a t i ond: (9 ' -~ 12~/r).T h e c u p - p r o d u c t p a i r in g s

tr(2.3.5.1.3) R n f , ([2*x/s)@ R 2 N - ' f , (fd ~c /s)~ R2N f (Q~ c/s ) " ' (g sa re pe r f ec t dua l i t i e s o f coheren t l oca l ly f r ee S modules , fo r which thef i l tra t ions F an d F~o~ are bo th se l f -dual in th e sense tha t( 2 . 3 . 5 . 1 . 4 ) ( F ' R n f , ( O ~ / s ) ) = F N + 1 - n R a N - n , ( f2 ~ /s )(2.3.5.1.5) (F~'onR ' f , (O~c/s)) = F~o, ~-n R a N -n f , (12~;/s).

T h e a s s o c i at e d g r a d e d p a ir in g si n * N - - i 2 N - n * ~ N 2 N 9r R 12 | R f2 r R [2F ( f * ( X / S ) ) g F ( f , ( x / s ) g F ( f , ( s)))

n - i i N + i - n N - i

a n d

(2.3.5.1.7) /lF ~ * n - - i i * N + i - n N - - i,bs(R f , ( ~ x / s ) ) | f , ( f 2 x / s ))

, grfoo. (R 2S f, (f2~:/s))/ l~ * N N. F . b sR f , ( f 2 x / s ) " " __~ (9


31/118


32/118

32 N.M. Katz:

i s d e f i n e d ( b e c a u s e , i n d i m e n s i o n n , a l l t h e " l o w e r " o n e s H - W ( / ) , i < a ,h a v e s o u r c e a n d t a r g e t th e z e r o m o d u l e , h e n c e a r e is o m o r p h i s m s ) . I nf a c t , i t i s n o n e o t h e r t h a n t h e c o m p o s i t e ( a n a l o g o u s t o ( 2 . 3 . 4 . 1 . 3 ) )

(2.3.6.2)1s,n--a,a ~. ~l ~n f [r ~ a, n- -ac o n a . ~ 2 Jt~, j , t ' ~ X / S ) ~ E 1

F ~ ( R " - " f , ( f~x /s )) ......... n-w(.) .+. .. .. .. .. .. .. .. .. .. .. .. . :: R " - " f , ( ~ x / s ) .( 2.3 .6 .3 ) P r o p o s i t i o n ( I g u s a , M a n i n ) . A s s u m p t i o n s a s i n (2.3.6), s u p p o s e Sa f fi n e a n d s o s m a l l th a t a l l t h e H o d g e c o h o m o l o g y s h e a v e s a r e f r e e (9 -m o d u l e s . L e t c o1 . . . . m e b e a b a s e o f R ~ - a , ( ~ a x / s ) ' a n d d e n o t e b y ( a i j ) t h em a t r i x i n M e ( F ( S , (_gs))o f H - W ( a ) w i t h r e s p e c t t o t h i s b a s e :( 2 . 3 . 6 . 4 ) H - W ( a ) ( F ~ s ( c o i) ) = Z a j i o ) j .

i

(2.3.6.5) C o n s i d e r t h e d u a l b a s i s o 9 " , . . . , o 9 " o f R N + a - n f ~ ( ~ ' ~ N ~ s a ) . B yd u a l i t y a n d t h e d e f i n i t i o n o f t h e i n t e g e r a , i t i s a l s o t h e l e a s t i n t e g e r w i t hR N + " - " f , ( 1 2 ~ 7 s~) n o n - z e r o , s o t h a t t h e d e g e n e r a t i o n o f t h e H o d g e = ~ D eR h a m s p e c t r a l s e q u e n c e g i v e s a n i n c l u s i o n(2.3.6.6) R " + ~ " f , ~ ( a ; , / s ) .(2.3.6.7) L e t ~ 1 . . . . . ~ e b e T - l in e a r d i ff e r e n ti a l o p e r a t o r s o n S w h i c h a r ei n t h e a l g e b r a g e n e r a t e d b y D e r ( S / T ) , w h i c h o p e r a t e o n t h e D e R h a mc o h o m o l o g y s h e a v e s R i f , ( O x / s) v i a t h e G a u s s - M a n i n c o n n e c t i o n 17. S u p p o s eR 2 N - n : t ~ * ~ s a t i s f y t h eh a t m * . . . . , co * , c o n s i d e r e d a s s e c t i o n s o f J , ~ x /s 1 ,d i f f e r e n t i a l e q u a t i o n(2 .3 .6 .8 ) Z V ( ~ j ) ( ~ o * ) = 0 i n R z u - " f , ( ~ 2 ] / s ) .

J

T h e n e a c h c o l u m n o f t h e m a t r i x ( a ij ) s a t is f ie s t h e d i ff e r e n ti a l e q u a t i o n(2 .3 .6 .9 ) Z ~ j ( a j i ) = 0 , f o r i = 1 . . . . . ( .J

P r o o f T h e p r o o f is b a s e d o n t h e f a ct t h a t t h e c o m p o s i t e p - l i n e a rm a p p i n g(2 .3 .6 .9bis ) R " - " f , ( f P x / s ) F'~b. , F , s ( R , - , f , (fP x/s)) n'W(")~ R " - ~ f , (fP x/s)

R , r q2o ~v o f h o r i z o n t a l s e c t i o n sa y b e f a c t o re d t h r o u g h t h e s u b sh e a f J , ~ x / s )( e l . ( 2 . 3 . 1 . 3 ) ) , a s e x p r e s s e d i n t h e c o m m u t a t i v e d i a g r a m


33/118

A l g e b r a i c S o l u t i o n s o f D i f fe r en t ia l E q u a t i o n s 33H-W(a)

n - a , a ~ n 9 n - a a

\ T I // / / / p r\ F ~ J , ( f? ~ ,s ) ) a " f , ( f 2 x ~)v/ / 7

F~bs // / a//n- -a a /R f,(f2x/s)

L et ( , ) d e n o t e t h e c u p - p r o d u c t p a i r i n g ( 2 . 3 . 5 . 1 . 3 ) o f D e R h a m c o h o -m o l o g y . W e h a v e , f o r e a c h i,(2 .3 .6 .10) 0 = (~ (co,) , E V (~j)(co* ))Ja n d , b e c a u s e ~(coi) is horizontal, w e h a v e(2 .3 .6 .11) 0= }- " ~ j ( ( ~ (co l) , co* . ]J /"J

B e c a u s e e a c h c o * h a s H o d g e f i l t r a t i o n > N - a , th e c u p - p r o d u c t(~(coi), co *) d e p e n d s o n l y o n t h e c l a s s o f ~ (c oi) m o d u l o F " + 1 R " c q 2 "* ~ X / S I ,by (2 .3 .5 .1 .4 ), wh ich i s to say on p r . ~ ( co l)= H -W (a ) (F~*s coi))-F u r t h e r m o r ed e n o t i n g b y ( , ) t h e c u p - p r o d u c t p a i r in g (2 .3 .6 .1 .6 ) o f H o d g e c o h o m o l o g y ,w e h a v e (~ (co i), co * )= (p r . ~ ( co i) , co* ) = (H -W (a )(F ,* s ( co i )) , co* )(2.3.6 .12) = ( ~ ak ic ok,c o* )=a j i. Q . E . D .

k

(2 .3 .7 ) A N u m e r i c a l E x a m p l e - C e r t a i n H y p e r s u r fa c e s o f G e o m e t r icG e n u s O n e(2 .3 .7 .0 ) R e c a l l th a t f o r a n y b a s e s c h e m e S , a n d a n y h y p e r s u r f a c e X i np ~ + 1 w h i c h is s m o o t h o v e r S , t h e H o d g e c o h o m o l o g y s h e av e s. R " f. (f~x/s)( f : X ~ S t h e st r u c t u r a l m o r p h i s m ) a r e l o c a l l y fr ee t~ s - m o d u l e s o f f i n it er a n k ( w h o s e f o r m a t io n c o n s e q u e n t l y c o m m u t e s w i th a ll c h a n g e o f b a s e)a n d t h e H o d g e = ~ D e R h a m s p e c tr a l s e q u e n c e d e g e n e r a t e s a t E 1 ( cf . [ 5 a ] ).I n t er m s o f a s ys te m o f h o m o g e n e o u s c o o r d in a te s X 1 , . . . , X , + 2 o np ~ + l , w e c a n w r it e a n e q u a t i o n f o r X (i.e ., a n i s o m o r p h i s m b e t w e e n9 p ( - d ) , d b e in g t h e degree o f X , a n d t h e i d e al I ( X ) def in ing X in P~ 1),

3 lnventionesm ath., Vo l. 18


34/118

3 4 N . M . K a t z :

a t le a s t l o c a ll y o n S . L o c a l i z in g o n S , w e m a y a n d w i ll a s s u m e t h a t X isd e f i n ed b y a h o m o g e n e o u s f o r m o f d e g r e e d ,

H = H ( X 1 . . . . . X , + 2 ) e F ( S , C ) [ X 1 . . . . . X , + 2 ] .T h e c o r r e s p o n d i n g s h o r t e x a c t s e q u e n c e o n P d ~ , ,p ~ + 1(2.3.7.1) 0---~ (_O r(-d) ~ , d~ --* Cx ~ 0g iv e s a n i s o m o r p h i s m o f c o h o m o l o g y s h e a v e s o n S ( n: P ; 1 ~ S d e n o t i n gt h e p r o je c t i o n ) v i a c o b o u n d a r y :( 2.3 .7 .2 ) R " f , ( e x ) ~ , R " +1 n , ( O r ( - d ) ).U s i n g t h e s ta n d a r d c o v e r i n g o f p r o j e c t iv e s p a ce , t h e ~ s - m o d u l eR " + I n , ( O r ( - d ) ) is e as il y c o m p u t e d : i t ' s t he f r ee C s -m o d u l e(2 .3 .7 .3 ) " f o r m s " o f d e g r e e - d i n / t h e C gs-sp an o f t h o s e m o n o m i a l s--1 --1 W Wl Wn+2 "(~s[X1 . . . . . Xn+2 X 1 . . . . . Xn+ 2]/ X = X 1 . . . Xn+ 2 f o r w h i c h/ ~ W / = - d b u t W /=>O f o r s o m e iw h i c h a d m i t s a s b a s e t h e m o n o m i a l s(2 .3 .7 .4) m ( W ) = X w = X W ' . , v w ,+ 29 ~ ,+ z , ~ W i = - d , W i < 0 f o r a l l i .W e d e n o t e b y th e s a m e s y m b o l s r e ( W ) t h e c o r r e s p o n d i n g b a s i s o fR " f , ( ~ x ) v i a t h e i n v e r s e o f t h e i s o m o r p h i s m (2 .3 .7 .2 ).

T h e " r e s i d u e " e x a c t s e q u e n c e i n h i g h e s t d e g r e e0 ~ ~ 1 ~ ~"~+S1( l o g X ) --~ f2]/s --* 0

(2.3.7.5) II+1 -1~ / s |

g iv e s u s an i s o m o r p h i s m , t h e " P o i n c a r 6 r e s i d u e "(2.3.7.6) n , (Y~,~s I ( X ) - I ) _=_, f , (~x / s ) .I n t e r m s o f t h e l o c al c o o r d i n a t e s x , = X i / X . + 2, t h e g l o b a l s e c t i o n o f

+ IO ~ /s ( n + 2 ) g i v e n b ydxl dxn +1(2 .3.7 .7) X 1 . . . X . + 2 - - ^ " " ^ - -Xl Xn+l

d e f i n e s a n i s o m o r p h i s m(2.3.7.8) O r ~ ; f~I/~1 n + 2 )


35/118


36/118

36 N .M. Ka tz :

Then the H ass-W ~tt ma trix is given by(2 .3.7.1 6) H - W ( F * s ( m ( - W ) ) ) = Z A p w _ v m ( - V ).

(2 .3 .7 .18 ) Coro l l a ry .S p e c ( F p [ 2 ] [ 1 / 1 - 2 d ]) .P s - 1 o f equation

(2 .3 .7 .1 7 ) S p e c i a l C a s e . H y p o t h e s e s a s i n ( 2.3 .7 .1 4 ), s u p p o s e t h a t X h a sd e g r e e d = n + 2 . T h e n R " f , ((g x) is f r e e o f r a n k o n e o n S , w i t h b a s em ( - 1 , - 1 . . . . , - 1 ) a n d t h e H a s s e - W i t t m a t r ix o f X / S i n d i m e n s i o n n( o r t h e H a s s e i n v a r i a n t , a s w e s h a l l c a l l i t i n t h is c a s e ) w i t h r e s p e c t t o t h eb a s i s m ( - 1 , - 1 . . . . - 1 ) is g i v e n b y t h e c o e f f i c i e n t o f ( X 2 . .. X , + 2 ) p - 2i n H p - 2 .

W e n o w a p p l y t h is t o a p a r t i c u l a r l y b e a u t if u l fa m i ly o f h y p e r s u r f a c e s .Let d > 2 be relatively pr im e to p, and put S =Consider the smo oth (over S ) hypersurface X in

d(2.3.7.19) ~ , X d - d 2 X 1 .. . X d.i=1

It's Ha sse invariant is given by the truncated hypergeometrie series~ , ( 1 / d )a (2 / d )a . . .( d -1 / d ) , 2_ad(2 .3 .7.20) (d2 )P - 2 a ! a ! . . . a !

a~O

where for ~4=0 , we put ( ~ )o = 1 , ( ~ ) , = ~ ( ~ + l ) . . . ( ~ + n - 1 ) / f n > l . Thisma y be expressed in terms o f the " full" hypergeom etrie series in Z p [ [ 2 - 2 ]] [ 2 ]by means of Dw ork's congruence (c f. [ 8 a ] , p p . 3 6 - 3 7 )( 2 .3 . 7 .2 1 ) ( d 2 ) p - 2 y ' (1 /d )a (2 /d ) , . . . (d - 1/d)a 2_ad=_G(2)/G(2P)modulo p)

a = o a ! a ! . . . a !where G ( 2 ) = ( d 2) - 1 F [ 1/d, 2/d . . . . d - 1/d.given by \ 1 , . . . , 1 , 2 - d ) is the elemen t o f Z p [ [ 2 - 1 ] ](2 .3 .7 .2 2) G ( 2 ) = 2 - 1 a ~ o ( - l ) a 'd -1 ) ( - 1 / d ) ( - 2 / d ) . . . ( - ( d - a l ) / d ) 2 _ a d "

P r o o f B y d i r e c t c a l c u l a t i o n , o n e f i n d s t h a t ( 2 . 3 . 7 . 2 0 ) is t h e c o e f f ic i e n to f ( l- -I x , ) v - 1 i n ~ X ( - d 2 I-I X , . T o a p p l y t h e c o n g ru e n c e s o f D w o r k ,

i iw e n e e d o n l y o b s e r v e t h a t i n th e s u m (2 .3 .7 .2 0), w e c o u l d h a v e l e t a r u nf r o m 0 a l l th e w a y t o p - 1 ( w h i c h is t h e usual f i r s t t r u n c a t i o n p o i n t f o rh y p e r g e o m e t r i c - t y p e s er ie s) , b e c a u s e , f o r a ~ p - l b u t a d > p , w e h a v e

(ad)! -0 m o d u l o p . Q . E .D .2.3.7.23) (1/d)a(2/d)a...(d-1/d)a= dad.a!


37/118

Algebraic Solutions of Differential Equations 37(2.3.8) 7he example, continued. A c c o r d i n g t o (2 .3 .6 ), t h e H a s s e i n v a r i a n to f t h e h y p e r su r f a c e (2.3 .7 .1 9 ) is t o s a t is fy e v e ry d i f f e r e n t i a l e q u a t io nsa t is f i e d b y th e d i f f e r e n t i a l o ~=c o (1 . . . . . 1 ) ( i n t h e n o t a t i o n (2.3 .7 .1 0 ))c o n s id e re d a s a s e c t i o n o f R a - 2 f . ( f2 ]./s)" By D w o rk ' s c o n g ru e n c e (2.3 .7 .2 1 ),t h i s i s e q u iv a l e n t t o t h e fo rma l s e r i e s G(2 ) (2 . 3 . 7 . 2 1 ) s a t i s fy in g e v e rys u c h d i f f e r e n t i a l e q u a t i o n . B e c a u s e t h e f o r m a l s e r i e s G ( 2 ) i s u n i v e r s a l ,i .e ., i n d e p e n d e n t o f p , a n d t h e f o r m a t i o n o f t h e H o d g e a n d D e R h a mc o h o m o l o g i e s o f a s m o o t h h y p e r s u r fa c e c o m m u t e s w i th a r b it r ar y c h a n g eo f b a s e , it f o ll o w s t h a t w e h a v e :(2 .3 .8 .1) Corol lary . Consider the hypersurface smooth over the spectrumS o f Z [2 ] [ 1 / d ( 1 - 2d )] g iven by the equa tion (2.3.7.19). For every dif-ferential equation d i(2.3.8.2) 2 a i ( 2 ) 1 7 ( d - f ) ( r i n R d - 2 f , ( ~ / s )satisfied by c0=09(1, . . . , 1) in f , (f2~rTs2), a i (2 ) ~Z [~. ] [1 /d ( l -A a ) ] , we have(2.3 .8 .3 ) ~ a i ( 2 ) ~ (G (2 ) ) =0 in Z [ [ 2 - 1 ] ] ,

w h e r e G ( 2 ) E Z [ 1 ] [ [ 2 - 1 ] ] i s t h e s e r i e s(2 .3 .8 .4 ) a ( R ) = ( d 2 ) - I ~ ( 1 / d ) a ' " ( d - 1 / d ) aa ! . . . a !(2.3.8.5)(2.3.8.6)

1~ a d .R e m a r k . I n f a c t, " t h e " d i f f e re n t i a l e q u a t i o n s a t is f ie d b y co i s

( Z ( - ' d , " - 'a s m a y b e d e d u c e d f r o m 1-23a ] a n d a n i m m e d i a t e c a l cu l a ti o n s h o w s t h a ti n d e e d [ d ~ - 1 ( 2 d ~ - 1(2.3.8.7) ~-d2-! (G(2)) = -d2-! (2G (2)).(2 .3 .9 ) W e re f er t o f o r t h c o m i n g w o r k s o f B . M a z u r f o r t h e c o n g r u e n c er e l a ti o n s b e t w e e n t h e h i g h e r H a s s e - W i t t m a t r i c e s a n d t h e z e t a f u n c t io n ,w h i c h g e n e ra li ze t h e " o r d i n a r y " c o n g r u e n c e f o r m u l a [2 51 .2 .4 . T h e Q u e s t i o n o f Q u a s i eo h e r e n c e o f th e C o n j u g a t e S p e c t r a l S e q u e n c e(2 .4 .0 ) W e r e t u r n n o w t o t h e g e o m e t r i c s i t u a t i o n o f 1 .0 , a n d , t o fixi d e a s, w e s u p p o s e t h a t S is a ff in e . T h e c o n j u g a t e s p e c t r a l s e q u e n c e(2,4.0.1) E~2'b=Raf,(Jfb(f2[/s(logD))) =~ R a + b f , ( ~ / s ( 1 O g D ))


38/118


39/118

Algebraic Solutions of Differential Equations 39(2 .4 .1) In the specia l case wh en S is a sch em a of cha rac ter i s t i c p , theconjugate spect ra l sequence (2 .4 .0 .1) is t he second spec t r a l s equence o fhy pe rc oh om ol og y for the (,0xcp)-linear com plex F , ( O } / s ( l o g D ) ) o n X ~p),an d the func to r s R"J~ ~ . Because f~P) : X ~p)-- , S i s q u a s i - c o m p a c t a n dsepara t ed , t he va r ious F , ( f ~ x / s ( l o g D ) ) are quas i -coherent (gx-modules ,a n d F , ( d ) i s Cxc~,- linear, the sec on d spe ctral seq ue nc e (2.4.0.8) of theC e c h b i c o m p l e x d o e s m a p i s o m o r p h i c a l l y t o t h e c o n j u g a t e s p e c t r a lsequence (2.4.0.9) , which is a spe ctral seq ue nc e o f qu asi- co he ren t (_9 -m o d u l e s .The s imple in t e rp re t a t i on o f t he con juga t e spec t r a l s equen ce i ncharac t e r i s t i c p a s t he sec ond spec t r a l s equen ce o f t he C ech b i com plex(2.4.0,3) m ak es possib le effect ive calcu lat ions , as w e shal l see. O n th ec o n t r a r y , t h e c o n j u g a t e s p e c t r a l s e q u e n c e o v e r C r e m a i n s i n s h a d o w .

3 . T h e M a i n T e c h n i c a l R e s u l t o n t h e p - C u r v at u r eo f t h e G a u s s - M a n i n C o n n e c t i o n3 .0 . W e r e tu rn t o t he geom et r i c s i t ua t ion 1 .0 , an d as sum e as be fo re t ha tS, and hence T , i s a scheme of charac ter i s t i c p . Recal l tha t for any (9 -m od ule w i th i n t egrab l e T -conn ec t ion , (M, V), i ts p -curva tu re f f is t hep- l inea r ho m om or ph i sm of Cs -modules ( c f. [ 24] , 5 .2 )(3.0.1) ~: D e r ( S / T ) - - , E n d e s ( M )d e f i n e d b y(3.0.2) ~k (D ) = (V (D )) p - V (DP) .E q u i v a le n t ly , w e m a y v ie w ~ a s d e fi n in g b y t r a n s p o s i ti o n a h o m o m o r -ph ism of (~s-modules, a l so no ted ~,,(3.0.3) r M - ~ F ~ ( ~ 2 ~ / T ) | M(w he re F~bS d e n o t e s t h e a b s o lu t e F r o b e n i u s e n d o m o r p h i s m o f S).

Th e s igni f icance of p-c urv atur e (due to Ca r t ie r ; c f . [24] , 5 .1) is thefo l lowing : (M, V) has p -curv a tu re ze ro i f and on ly i f M is spann ed over(9 by the subshea f M ~ o f hor i zo n ta l s ec t ions ; m ore p rec ise ly , den o t ingb y S w ) t he f i b re p rod uc t o f g : S - -~ T an d the a bso lu t e Fro ben ius F~b~:T--~ T , i f an d on ly i f t he cano n ica l m app ing(3.0.4) (M e) | (~s - -* Mi s an i somorph i sm.3 .1 . T a k i n g f or M t h e D e R h a m c o h o m o l o g y R n f , ( f 2 x / s ( l o g D ) ) , a n dfor V the G au ss -M an in con nec t ion , we r eca l l once a ga in (c f. [24] , 3 .5)tha t t h e en t i re con juga t e spec tr a l s equence( 3 . 1 . 1 ) E"2"b=R"f , (~ t~ ~ R a+ bf , (f2~c/s(logD))


40/118

4 0 N . M . K a t z :

is e n d o w e d w i t h th e a c t i o n o f t h e G a u s s - M a n i n c o n n e c t i o n , a n d t h a ton t he E2 t e rms , t he p - cu rv a tu re is zero . Thi s impl ie s t ha t t he p - cu rva tu rea l so vanishes o n th e Eo~ te rms , and hen ce tha t(3 .1 .2) ~ ( F ~ o , R " f , ( g J ~ , / s ( l o g D ) ) ) c F * ~ ( f J ~ / r ) | + 1 R " f , (gJ~r (log D)).Pas s ing t o t he a s soc i a t ed g raded , t he re i s an i nduced mapp ing , aga ind e n o t e d r(3.1.3) ~J: gr~oo, R "f , (O~/s(lo g O))) --* F~* (~2s~/7-) gr~ooo R "f, (~J~?s(logD))).Our ma in t echn ica l r e su l t 3 .2 iden t i f i e s t h i s mapp ing , unde r su i t ab l ehyp o these s wi th a twi s t ed fo rm o f the K od a i r a -S pen ce r map p ing (1 .3 .2 .1 ).3 , 2 . T h e o r e m . U n d e r t h e h y p o t h e s e s o f (2.3.2), t h e d i a g r a m b e l o w i sc o m m u t a t i v e

g~oo,(Ra+O f,(CJ~./s(logD))) ~ , F *~ (fJ~ /r) | )

c o n E ~ b 1 7 . [ O 1 ] t ' ~ g ' a + l , b - 1a a b s ~ a ' ~ S / T ] ~ c o n ~

e o n g ~ ,b b ' * ~O 1 " ~ t ~ t T ' a + l , b - 1. t a b s I, a a S / T ] ~ e o n ~ L ' 2

~k a b --F~b (R f , (Q x / s ( log D))) ( 1)b+'F~*b~(p~Fa~ (~ /T ) | F*~ (R ~+ ~ f , (~ b / j ( log D) ))bs s ~ F a * b s

R o f , D ) ) ( - ' ' . . . . 9 | g o + ' / , ( lo g D ) )i n w h i c h p i s t h e c u p - p r o d u c t w i t h t h e K o d a i r a - S p e n c e r c l a s s (1.1.3),v i e w e d a s a g l o b a l s e c t i o n o v e r S o f 1 1s / r | R f , ( D e r o (X / S )) .

Before p roceed ing t o t he p roof , w e wi ll r eca ll som e bas i c f ac t s abo u tt h e m o d u l a r r e p r e s e n t a t i o n t h e o r y o f f in i te g ro u p s o f o r d e r p r i m e t o p ,a n d t h e n r e s t a t e 3.2 " w i t h a g r o u p o f o p e r a t o r s ".(3.2 .1) Le t G be a f in i te gro up of or de r pr im e to p, an d k a f ie ld ofcha rac t e r i s t ic p . Le t V be a f i n i t e -d imens ion a l k - space on w hich G ac t sa s a g r o u p o f k - a u to m o r p h i s m s , t h r o u g h a h o m o m o r p h i s m Z : G ~ G L ( V ) .D e n o t i n g b y F , bs: k ~ k t h e a b s o l u t e F r o b e n i u s e n d o m o r p h i s m o f k , t h erep rese n t a t i on Z v) o f G on F . ' ~ , s ( V ) = V | (where k i s a modu le ove rk


41/118


42/118

42 N.M. Katz:

B e c a u s e G is f in it e, a n y r e p r e s e n t a t i o n o f G o n a f i n it e d i m e n s i o n a lK - s p a c e c o m e s b y e x t e n s io n o f s c a la r s f r o m a r e p r e s e n t a t i o n o f G in af r ee (9 - m o d u l e o f f in i te r a n k ( t a k e t h e l a t ti c e g e n e r a t e d b y t h e G - t r a n s l a t e so f a n y l a tt ic e ) , a n d h e n c e i t s c har ac t e r t a k e s v a l u e s i n ( 9 . I t f o l l o w s t h a tt h e f u n c t i o n s e i t a k e v a l u e s i n ( 9 ; i . e . , l i e i n (gEG]. B e c a u s e t h e Z~ a r ea b s o l u t e l y i r re d u c ib l e , w e h a v e(3 .2 .2 .4) deg (xi ) l ~ G , h en ce p4Vdeg(gi) .H e n c e t h e v a l u e s o f th e e~ a t t h e i d e n t i t y e l e m e n t 1G G a r e u n i t s i n (9,b e c a u s e d e g ( z i ) ( d e g ( g i) ) 2( 3 .2 .2 . 5 ) e i ( 1 ) = 9 r a c e ( g ~ ( 1 ) )=~ : G : ~ GT h u s i f f G (9 [ G ] is c e n t ra l , f = ~ , a i ei a n d

(a i e i ) (1) ( f e i )(1) G(9 .(3 .2 .2 .6) a i e i (1) - e i (1)T h u s t h e e~ g i v e a n ( 9 - b a se o f t h e c e n t e r o f (9 [ G ] .

N o w l e t f ~ G k [ G ] b e t h e i n d e c o m p o s a b l e c e nt ra l i d e m p o t e n t c o r -r e s p o n d i n g t o a n i r r e d u c i b l e r e p r e s e n t a t i o n X- W e c a n c e r t a i n l y l if t f~ t oa c e n t r a l f u n c t i o n f w i t h v a l u e s i n (9, w h i c h w e m a y w r i te(3.2.2.7) f = ~ a i ei , a i G ( 9 .B e c a u s e f l " f: = f l , w e h a v e(3.2.2.8) ~ a { ei = f 2 = _ f = ~ a i e , m o d ( p [ G ] )a n d m u l t i p l y i n g b o t h s i d es b y e~, w e g e t(3 .2 .2 .9) a~ ei --- ai ei m o d p [ G ] .E v a l u a t i n g a t t h e i d e n t i t y e l e m e n t l e G , w e h a v e ( b y (3 .2 .2 .4 ))(3 .2 .2 .10) a ~ = a i m o d p ,s o t h a t e a c h c o e f f i c ie n t a i is c o n g r u e n t t o e i t h e r 0 o r 1 m o d p . T h u s w em a y lif t f~ t o a c e n t r a l i d e m p o t e n t(3.2.2.11) ~ , e i e i , e i = 0 o r 1i n ( 9 [ G ] . B e c a u s e t h e e~ a r e a bas i s o f t h e c e n t e r o f (9 [ G ] , t h e i n d e c o m p o -s a b i l i t y o f f~ a s c e n t r a l i d e m p o t e n t i n k I -G ] i m p l i e s t h a t e~ d i ff e r s f r o mz e r o f o r o n l y o n e v a l u e o f i, s a y i = 1 . T h i s s h o w s t h a t e i i s t h e un i quec e n t r a l i d e m p o t e n t i n (9 [ G ] l if ti ng f ~ . T h i s s h o w s t h a t Z is t h e " r e d u c t i o nm o d p " o f a u n i q u e i r r e d u c i b l e r e p r e s e n t a t i o n X l o f G in a f in i te -d i m e n s i o n a l K - s p a c e . Q . E . D .


43/118

Algebraic Solutions of Differential Equations 43(3 .2 .3 ) Coro l l a ry . H y p o t h e s e s a n d n o t a t i o n s a s i n (3.2.2), t h e i n d e c o m -p o s a b l e c e n t r a l i d e m p o t e n t s in K [ G ] , which a l l l i e in (P [ G ] , h a v e r e d u c t i o n smodulo t~ in k [ G ] w h i c h r e m a i n i n d e c o m p o s a b l e .

P r o o f . T h e n u m b e r o f e i is e q u a l t o t h e n u m b e r o f f~ , b o t h b e in g t h en u m b e r o f c o n j u g a c y cl as se s in G . H e n c e e v e r y e i li ft s a n f t . Q . E . D .

I n t e r m s o f r e p r e s e n t a t i o n s , t h i s g iv e s :(3.2.3 b i s ) C or o l l a ry . H y p o t h e s e s a s i n (3 .2 .2b i s ) , t h e r e d u c t i o n m o d p o fa n y K - i r r e d u c i b l e r e p r e s e n t a t i o n o f G in a f r e e C - m o d u l e o f f i n i t e r a n k i si rr e d u c ib l e ( a n d e v e r y i r r e d u c i b le r e p r e s e n t a t i o n o f G in a f i n i t e - d i m e n -s i o n a l k - s p a c e a r i s e s t h i s w a y ) .(3 .2 .3 .1 ) Coro l l a ry . H y p o t h e s e s a s i n (3 .2 .2his ) , e v e r y i r r e d u c i b l e r e p r e -s e n t a t i o n o f G in a f i n i te - d i m e n s i o n a l k - s p a c e i s a b s o l u t e l y i rr e d u c ib l e.

P r o o f . T h i s f o l l o w s f r o m ( 3 . 3 . 0 b i s ) , a p p l i e d t o a r b i t r a r y f i n i t e e x t e n -s io n s K ' o f K , a n d a r b i t r a r y e x t e n s i o n s o f t h e v a l u a t i o n o f K t o K ' , b yw h i c h w e c a n r e a l iz e a r b i t r a r y f in i te e x t e n s i o n s k ' o f k a s r e s i d u e f ie ld .(3 .2 .3 .2 ) Coro l l a ry . H y p o t h e s e s i n (3.2.2 h is) , t h e i n d e c o m p o s a b l e c e n t r a li d e m p o t e n t ( n o t e d P ( X ) ) a s s o c i a t e d t o a n i r r e d u c i b l e r e p r e s e n t a t i o n ;( i n af i n i te - d i m e n s i o n a l k - s p a c e is g i v e n b y t h e f o r m u l a(3.2.3.3) P (Z ) = deg@G(Z_) ~ tr a c e ( z ( g - 1)). g .(3 .2 .4 ) W e r e t u r n t o t h e h y p o t h e s e s o f (2 .3 .2 ), a n d s u p p o s e g i v e n i na d d i t i o n :(3 .2 .4 .0 ) a fi n it e g r o u p G o f o r d e r p r i m e t o p , w h i c h a c t s a s a g r o u p o fS - a u t o m o r p h i s m s o f X a n d p r e se r v es th e d i v is o r D ( t h o u g h n o t n e c es -s a r il y t h e i n d i v i d u a l D i );(3 .2 .4 .1 ) a s u b f i e l d k o f F ( T , (,O r) a n d a n a b s o l u t e l y i r r e d u c i b l e r e p r e -s e n t a t i o n X o f G i n a f i n i t e - d i m e n s i o n a l k - s p a c e .(3 .2.4 .2 ) B y f u n c t o r ia l i ty , t h e g r o u p G a c t s o n t h e D e R h a m c o h o m o l o g ys h e a v e s R " f , ( g 2 1 /s ( lo g D ) ) a s a g r o u p o f (~ s -l in e ar h o r i z o n t a l ( fo r th eG a u s s - M a n i n c o n n e c t i o n ) , a n d r e sp e c ts b o t h t h e H o d g e a n d t h e c o n -j u g a t e f il tr a ti o n s . ( In f ac t, G ac t s o n b o t h t h e H o d g e =~ D e R h a m a n d t h ec o n j u g a t e s p e c t r a l s e q u e n c es .)(3 .2 .4 .3 ) T h e f u n c t o r i a l i t y o f t h e d i a g r a m (3 .2 .0 ) s h o w s i m m e d i a t e l yt h a t i t is a d i a g r a m o f G - m o r p h i s m s ( t h o u g h t h e l o w e r v e r t ic a l a r r o w sF ~ s , b e in g p - li n ea r , a r e n o t k [ G ] - m o r p h i s m s ) .

T h i s s a id , w e m a y n o w " r e s t a t e " 3 .2.


44/118

44 N.M . Katz:( 3 . 3 ) ( = 3 . 2 b i s ) T h e o r e m . Under the hypotheses o f (3.2.4), the fo l lowingsubdiagram of (3 .2 .0 ) is commutative.

( p ) * 1 a , bp(z~v,)(~o,,E~b)_ o , (I | ) )

P ( Z p))F ~ (R ~ f, ( fdbx/s)) ~- ) b + ' Fa*b'( )~' (1 | P(x~ P')) F~,~ f2~/r) ) F*~ (R ~+ ' f , (s ~ log D)))b s b s a b ~

P( x) R ~ f , ( ~ x / s ( l o g D)) ~ _ l ) b + , p , ( l | 1 7 4P r o o f ( A s s u m i n g 3 .2 ). T h u u p p e r s q u a r e i s d e d u c e d f r o m t h e u p p e rs q u a r e o f (3 .2 .0 ), w h i c h i s a d i a g r a m o f k [ G ] - m o d u l e s , b y a p p l y i n g t h ep r o j e c t o r P (xC ~ T h e l o w e r s q u a r e is d e d u c e d f r o m t h e lo w e r s q u a r e o f(3 .2 .0 ) b y a p p l y i n g P ( Z ) t o t h e l o w e r h o r i z o n t a l l in e , a n d n o t i n g t h a t( de g (Z) g tF * s o ( P ( z ) )= F * s \ ~ ~ , t r a c e ( z (g - ~ ) ) 9 / oF~s

(3.3.1) = ( d e g (Z) '~P\ # G I Z( tracez(g-1))Vg~ Q . E . D .(3 .3 .2 ) Coro l la ry . Hypotheses as in (3 .2 .4) , suppose that the absoluteFrobenius endom orphism o f C is injective (w hic h is the case i f T is reduced,for exam ple ) . I f fo r a f i x ed integer n , the p -curva ture o f the Gauss -M aninconn ection on the "p a rt o f R" f,(s D)) w hich transform s by Z~P. ' ',i .e . , on the submodule P(ztP~)(R"f, ( (2] /s ( log D))) is zero, then the inducedH odg e f i l t ra t ion on P(z)(R "f , ( f2"x /s( log D ))) i s hor izontal , i .e ., s table b ythe G auss -M anin connec tion .

3 .4 . T h e P r o o f o f 3 . 2 : R e d u c t io n S t e p s(3 .4 .0 ) T h e q u e s t i o n b e i n g lo c a l o n S , w e m a y a n d w i ll a s s u m e S a ffin e.A s e x p l a i n e d i n 2 .4 , t h e D e R h a m c o h o m o l o g y sh e a v e s o n S a re t h e t o ta lh o m o l o g y o f t h e C e c h b i c o m p l e x a s s o c ia t e d t o a n y a ff in e o p e n c o v e r i n go f X , a n d t h e H o d g e a n d c o n j u g a t e s p ec t r al s e q u e n c e s a r e t h e " f i r s t"a n d " s e c o n d " s p e ct ra l s e q u en c e s o f t h e C e c h b i co m p l e x . T h e p r o o f w ew i ll g i v e i s n o t a t a l l i n t ri n s i c , b u t r a t h e r d e p e n d s o n e x p l i c it c a l c u l a t io n so n t h e l e v e l o f t h e C e c h b i c o m p l e x i ts elf. I t w o u l d b e o f c o n s i d e r a b l ei n t e r e s t t o g i v e a n i n t r i n s i c p r o o f .( 3 .4 .l ) W e c h o o s e a f in i te c o v e r i n g o f X b y a ff in e o p e n s e t s V / ( a s i n( I.0 .3 .1 )) , e a c h 6 t a l e o v e r A ~ v i a " lo c a l c o o r d i n a t e s " xi(i) , . . . , x ,( i) , s u c ht h a t t h o s e b r a n c h e s o f D m e e t i n g V~ a r e d e f i n e d b y t h e e q u a t i o n sx l ( i ) = 0 . . . . . x ~ ( i) = 0 , ~ d e p e n d i n g o n Vi.


45/118

Algebraic Solutions of Differential Equations 45W e n e x t d e f i n e a ( n o t n e c e s s a r i ly i n t e g r a b l e ) T - c o n n e c t i o n 17 o n t h es i m p l e c o m p l e x d e d u c e d b y " t o t a l i z a t i o n " f r o m t h e C e c h b i c o m p l e x ,

w h i c h y i e l d s t h e G a u s s - M a n i n c o n n e c t i o n u p o n p a s s a g e t o h o m o l o g y(cf. [24 , 35]).F o r a n y D e D e r ( S / T ) , w e d e n o t e

(3 .4 .1 .1 ) D ( i ) = t h e u n i q u e e l e m e n t o f D e r o ( V j T ) w h i c h e x t e n d s D a n dw h i c h a n n i h i la t e s t h e c h o s e n l o c a l c o o r d i n a t e s x ~ ( i ) . . . . , x , ( i ) , a n d b y(3.4 .1.2) Lie(D (/)) : f2~, s( l og D ) ~ f2~,/s( log D)t h e " L i e d e r i v a t i v e w i t h r e s p e c t t o D i " .

F o r e a c h p a i r o f i n te g e r s i < j , w e d e n o t e b y9 ~ o- -1(3.4.1,3) I ( D ( i ) - D(j)) : f 2 v , ~ v j / s ( l o g D ) f J v , n v j / s ( l o g D )

t h e o p e r a t o r " i n t e ri o r p r o d u c t w i t h t h e S - d e r iv a t i o n D ( i ) - D (j) " (cf. (1.2.2)).T h e c o n n e c t i o n 17 o n C ' ( { V i } , g 2 ] / s ( l o g D ) ) i s g iv e n a s fo l l o ws . F o r af ix e d in t e g e r b , a s e c t i o n(3 .4 . l .4) o)~ Z C"({V~}, ~ b/s ( lo g D))

a

a n d a s i m p l e x i o < " ' < i q ,( V ( D ) ( o ) ) ) ( i o , . . . , iq )= L i e ( D ( i o ) ) ( o o ( i o . . . . . iq ) )(3.4.1.5) + ( - 1)b I ( D ( i o ) - D ( ix ) )( o g ( i l . . . . . i q ) ) .

T h u s 1 7 ( D ) i s t h e su m o f tw o t e rm s . T h e f ir s t, o f b id e g re e (0 , 0 ), i s t h ec u p - p r o d u c t w i t h t h e 0 - c h a i n { L ie (D ( /))} . T h e s e c o n d , o f b i d e g r e e(1 , - 1 ) , is t h e c u p - p r o d u c t w i t h t h e 1 - c o c y c le I ( D ( i ) - D ( j ) ) . T h i s e x p l i c itc o n s t r u c t i o n m a k e s c l e a r c o m p u t a t i o n a l l y t h e t r u t h o f (1 .4 .1 .6 ), (1 .4 .1 .7 ),and (2 .3 .0 .1) .In o rd e r t o p ro v e 3 . 2 , i t s u f f i c e s b y l i n e a r i t y t o e s t a b l i sh t h e c o m-m u t a t i v i t y o f t h e o u t e r m o s t s q u a r e o f ( 3.2 .0 ), a n d a f te r " c o n t r a c t i o n "w i t h a n y D ~ D e r ( S / T ) . T h u s w e m u s t p r o v e c o m m u t a t i v e t h e d ia g r a m

a b 9R f , ( J Y ( f 2 ~ / s ( l o g D ) ) ) r ~ R , + l f , ( g f b _ l ( f 2 ~ / s ( l o g D ) ) )(3.4.1.6) l~e-l~ l~e-I ~

R " f , ( a b / s ( l o g O)) (-1)b+10(D),R O + l f ,( f~ x S s ~ ( l o g D ) ) .L e t u s e x p l i c a t e t h e a r r o w s i n t h i s d i a g r a m . T h e v e r t i c a l o n e s a r ed e d u c e d f r o m t h e m o r p h i s m s o f s h ea v e s

(3.4.1.7) f ~ b x / s lo g D ) " * , f ~ x , , , / s ( log D ~ " ) ) -- ~ 2 1 , o ~ b ( a } / s ( log D )).


46/118

46 N.M . Katz:I n o r d e r t o r e n d e r t h e v e r t ic a l a r r o w s s ti l l m o r e c o n c r e t e , w e i n t r o d u c e ,f o r e a c h o p e n s e t V i, a p - l i n e a r e n d o m o r p h i s m O 5 o f ~ v ,/ s ( lo g D ), w h o s ei m a g e l ie s i n th e s u b s h e a f o f c l o s e d f o r m s , a n d w h i c h " l i f ts " t h e m a p p i n g3 . 4 . 1 . 7 . I n d e e d , t h e f o r m u l a s ( 2 . 1 . 2 . 1 ) g i v e s u c h a l i f t i n g : w e r e q u i r e t h a t

o 5 ( 1 ) = 1( d x ~ ( i ) ~ = d x v ( i )O 5 \ x v ( i ) ] x ~ ( i ) f o r i = 1 . . . .

o s ( d x ~ ( i ) )= ( x ~ ( i ) ) p - ~ d x ~ ( i) f o r i = ~ + 1 . . . . n(3.4.1.8)o s ( c o ^ ~ ) = o s ( c o ) ^ O 5 ( ~ )o s ( c o + ~ ) = o s ( c o ) + O 5 ( ~ )o s ( h c o) = h p o s( co )

W e t h e n h a v e o u r d e s i r e d l if ti ng(3.4.1.9)

f o r h~( gv ~ .

f ~ v , / s ( l o g D ) ~=' , c lo se d fo rm s ~ v , / s ( l o g D )tcanonicalprojection

~ b ' ~ o ~ 1 D ~( v , / s ( o g ) )(3 .4 .1 .1 0 ) T h e l o w e r h o r i z o n t a l a r r o w i n ( 3.4 .1 .6 ) is , a s p r e v i o u s l y(cf . (1.1.3 ) ) no ted , ( - 1 )b + 1 t i m e s t h e c u p - p r o d u c t w i t h 1 - c o c y c l e

{ I ( D ( i ) - O ( j ) ) } .T h e u p p e r h o r i z o n t a l a r r o w i n (3 .4 .1 .6 ) is sl i g h t ly le ss s t r a i g h t f o r w a r d

t o e x p l i c a t e . L e t ~ b e a s e c t i o n o f R a f . ( j~ f fb i 2~ / s ( lo g D ))). W e m a y r e p r e s e n tb y a n a - c o c h a i n ~ o f c l o s e d f o r m s(3.4.1.11) z ~ c a ( { v ~ } , P ~ x / s ( l o g D ) ) , d z = Ow h o s e i m a g e in c a ( { Vii}, ~ b (~ 2 ] /s ( lo g D)) )is a c o c y c l e r e p r e s e n t in g ~ . B y t h e d e g e n e r a t i o n o f t h e c o n j u g a t e s p e c tr a ls e q u e n c e a t E z , w e m a y c h o o s e z t o b e t h e c o m p o n e n t o f b i d e g r e e (a, b )in a t o t a l a + b c o c y c l e a w h i c h l ie s i n(3.4.1.12) F ~ , = 2 c a + i ( { V i } ' f~x~-s'(logD)).i=>OT h e s e c t i o n ~ , ( D ) ( ~) o f R a + i f , ( oc g b- 1 f2 ~ /s ( l o g D )) ) i s t h e n r e p r e s e n t e db y t h e c o m p o n e n t o f b i d e g r e e (a + 1, b - 1) in t h e t o t a l c o c y c l e(3.4.1.13 ) ( V D )p - V D ' ) ) ( a ) .


47/118

Al gebr a i c So l u t i ons o f Di f f e ren t ia l Eq ua t i ons 47

R e c a l l t h a t(3.4.1.14) ( V ( D ) p - V ( D P ) ) ( F r 1 7 61 7 6 ' ' )( b e c a u s e V ( D ) ( F ~ , ( C ~ 1 7 6 1 7 6 ) , a n d V ( D ) o n gffvr ~176s a n i n t e -g r a b l e c o n n e c t i o n o f p - c u r v a t u r e z e r o , g i v e n o n Vio c ~ .- . n V i, b y D - -+L ie (D( i0) )) . F ~ o, ( C ) , I t f o l l o w s f r o m ( 3.4 .1 .1 3)i n c e a is c o n g r u e n t t o z m o d u l o ,+ 1 , , 9t h a t(3.4.1.15) ( V ( D ) p - V ( D p ) ) ( a )- - - (V ( D ) p - V ( D ' ) )( z ) m o d V ~ o+ 2.T h u s t h e s e c t io n O ( D ) (a ) is re p r e s e n t e d b y t h e c o m p o n e n t o f b i d e g r e e( a + 1, b - 1 ) i n ( V ( D F - V ( D P ) ) ( r ) .( 3 . 4 . 2 ) L e m m a . L e t z e C " ( { V i } , Y ~ x / s (l o g D )) , a n d i o < i a < . . . < i , + 1 . F o re a c h i n t e g e r n > 1 , t h e e l e m e n t(3.4.2.0) V ( D ) " ( z ) e ~ C"+' ({V~}, ~x Ts ' ( log D))i> Oh a s i ts c o m p o n e n t s o f b i d e g r e e ( a, b) a n d ( a + 1, b - 1) g i v e n b y(3.4.2.1) V ( D ) " ( z )( i o . . . . , i , ) = L i e ( D ( i o ) ) " ( Z ( i o , . . . , i~ )),

V ( D ) ' ( z ) ( i o , . . . , i ,)(3.4.2.2) - - - ( - 1 ) b ~ L ie (D ~ o)k t ( D ( i o ) - D ( iO ) L i e ( D ~ f ( z ( i I . . . . . i , ) ).k + d ~ n - - 1

P r o o f T h e p r o o f , b y i n d u c t i o n , is i m m e d i a t e ; t h e c a s e n = 1 is t h ed e f i n i t i o n . Q . E . D .

P u t t i n g n = I ', w e f i n d( V ( D ) p - V ( D P ) ) (z ) ( io , . . . , i~ )

(3.4.2.3) = [( L ie (D ( i0)))P - L ie (D ( io )P ) ] ( z ( i o . . . . . i ,) ) - 0 ,s i n c e ( D ( i o ) ) p = - D P ( i o )

( V ( D ) p - V ( D P ) ) (z )( io . . . . . ia + O(3 .4 .2 .4 ) = ( - 1 ) b ~ L i e ( D ~ o ) k I ( D ( i o ) - D ( i O ) L i e ( D , y ( z ( q . . . . , i ,+ ~ ))

k + d = p + l- ( - - 1 ) b I ( D ( i o ) p - D ( i O p ) ( z ( il . . . . . i . + 1 ) ) .

T h u s ( 3 .4 .2 .4 ) g i v e s a f o r m u l a f o r a c o c y c l e r e p r e s e n t i n g ~ h(D )(a ) i nR " + l f . ( 3 r D ))), w h e n w e t a k e f o r z a p a r t i c u l a r r e p r e s e n t i n gc o c h a i n f o r ~.(3 .4 .3 ) C o m b in in g a l l o u r ex p l i ca t io ns (3 .4 .1 .9 ), (3 .4 .1 .10), a n d (3 .4 .2 .4 ),w e s e e th a t t h e c o m m u t a t i v i t y o f (3 .4 .1 .6 ), a n d h e n c e t h e t r u t h o f 3 .0 , isi m p l i e d b y t h e f o l l o w i n g a s s e r t io n :


48/118


49/118


50/118

50 N.M. Katz:

( 3 . 5 . 1 . 3 )( 3 . 5 . 1 . 4 )( 3 . 5 . 1 . 5 )( 3 . 5 . 1 . 6 )

P r o o fs i m p l y c a l c u l a tei f i e ~ ,

L i e (Q ) (a ( /) ) = 0 ,I ( P - Q ) ( z ( i ) ) = g ( i ) ,I ( P - Q ) ( a ( i ) ) = f ( i ) ,I ( P p - Qp) (a ( i) ) = h ( i) .

( 3 .5 . 1 .1 ) i s b y d e f i n i t i o n ( 3 .5 . 0 .1 ) o f ~ A s f o r (3 .5 .0 .2 ) , w eL i e ( P ) ( a ( i ) ) = L i e ( e ) ( d x , I = L i e ( e ) ( d x i ) P ( x , ) d x~

\ x i ] x i x i x i( 3 . 5 . 1 . 7 ) d P ( x i ) P ( x i ) d x i

X i X i X i

= d \ - ~ - i ! = t i T ( i) "( 3 . 5 . 1 . 8 ) i f ir w e m u l t i p l y t h e a b o v e c a l c u l a t i o n (3 .5 .1 .7 ) b y x~ .( 3 . 5 . 1 . 3 ) h o l d s b e c a u s e Q ( x i ) = 0 b y d e f i n i t i o n ( 3 .5 .0 ) o f Q . A s f o r ( 3 .5 .1 .4 ),w e c a l c u l a t e( 3 . 5 . 1 . 9 ) i f i e~ , l i P - Q ) ( z (i )) = I ( P - Q ) ( d x i ] - ( P - Q ) (x i) - P ( x , ) = g ( i ) ,

\ x i ! x i x i( 3 . 5 . 1 . 1 o ) i f i r l(P-Q)(z(i))=I(P-Q)(dxO=(P-Q)(xi)=P(x,)=g(i)a n d ( 3 . 5 . 1 . 5 ) f o l l o w s s i m i l a r l y :( 3 . 5 . 1 . 1 l ) i f i ~ , I ( P - Q ) ( , ~ ( i ) ) - - I ( P - Q ) ( , i x , ] = e l x , ) = f ( i ) ,\ x l ! x i( 3 . 5 . 1 . 12 ) i f i r I ( P - Q ) ( a ( i ) ) = I ( f - Q ) ( x ~ - l d x i ) = x e , - 1 p ( x i ) = f ( i ) .T h e p r o o f o f (3 . 5 .1 . 6 ) i s i d e n t i c a l t o t h a t o f ( 3 .5 .1 . 5 ), d e p e n d i n g o n l y o n t h ef ac t t h a t Q p ( x i )= 0 . Q . E . D .( 3.5 .2 ) W e n o w r e t u r n t o t h e v e r i f i c a t io n o f t h e c o n g r u e n c e (3 .5 .0 .3 ) f o ra p r o d u c t o f t h e z(i). T o b e g i n , w e ' ll c h e c k t h e c a s e z = r ( 1 ) . I n t h is c a se( b = 1 ), t h e c o n g r u e n c e ( 3 .5 .0 .3 ) b e c o m e s a n a s s e r t i o n o f e q u a l i t y :

L i e ( P ) k I ( P - Q ) Lie (Q )e ( ~ (z ( 1 ) ) ) - I ( f p - Q P ) ~ ( z (1))( 3 . 5 . 2 . 0 ) k+:=p-a= - ~ ( I ( P - Q ) ( ~ ( 1 ) ) ) .

S u b s t i t u t i n g v i a (3 .5 .1 ), ( 3 .5 .2 .0 ) b e c o m e s( 3 . 5 . 2 . 1 ) Z L i e ( p ) R I ( p - Q ) L i e ( Q ) e ( a ( 1 ) ) - I ( P P - Q P ) ( a ( 1 ) ) = - ~ ( g ( 1 ) )

k + g = p - 1


51/118


52/118

52 N.M . Katz:A f i n a l s u b s t i t u t i o n g i v e s

L ie ( P )P - 1 ( f ( 1 ) a (2) - f ( 2 ) a (1)) - (h (1) a (2) - h (2) a (1))(3.5.3.3) = g ( 1 ) P a ( 2 ) + g ( 2 ) P a ( 1 ) m o d u l o d F(V ,(_ % ).A v a i l i n g o u r s e l v e s o f (3 .5 .2 .3 ), ( 3 .5 .3 .3 ) b e c o m e s

L i e (P ) ~ - t ( f ( l ) a ( 2 ) - f ( 2 ) a ( t ) ) = W - t ( f ( t ) } a ( 2 ) - p ~ - i ( f( 2 ) ) a ( l )(3.5.3.4) m o d u l o dF (V , (9~).N o w t h e r i g h t h a n d m e m b e r o f( 3.5 .3 .4 ) is t h e " f ir s t t e r m " in t h e e x p a n s i o no f t h e l e ft h a n d m e m b e r b y L e i b n i z ' s r u l e , s o th a t , e x p a n d i n g , ( 3.5 .3 .4 )b e c o m e s(3.5.3.5) Z ( P k l ) [ p k ( f ( 1 ) ) L i e ( P ) e ( a ( 2 ) )k+~=p-l ,8~:O

_ p k ( f ( 2 ) ) L i e (p)e (or (1)) ] ~ d r ( E rB y ( 3.5 .1 .2 ), w e t r i a y s u b s t i t u t e

L i e ( F )e ( a (2 )) = L i e ( p ) e - 1 d ( 2 ) = d P t - ~ ( f ( 2 ) )(3 .5 .3 .6) L ie (p)e (a (1)) = L ie (p)e_ ~ d f ( 1 ) = d p e _ l ( f ( 1 ) ) .D o i n g so , a n d r e m e m b e r i n g t h a t(3.5.3.7)w e m u s t s h o w

( P k 1) - - - ( - 1)k m o d p f o r O g k < _ p - 1

~ , ( - t) k [ p ~ f ( 1 ) d P e - : f ( 2 )( 3 . 5 . 3 . 8 ) k + e = p - l , e * 0 - - P k ( f ( 2 ) ) d e e - ' f ( 1 ) ] e dr(V , (gv )R e - i n d e x i n g t h e s u m m a t i o n b y k a n d r e = Y - 1 , (3 .5 .3 .8) b e c o m e s (r e-m e m b e r i n g t h a t ( - 1)k +l - ( - ! ) m m o d p )

( - 1 )k n k f ( 1 ) d n " f ( 2 )(3 .5 .3 .9) k+m=p-Z+ ~ ( - 1 ) " 1 P k ( f( 2 ) ) d e ' f ( l ) e d F ( V , ( g v ) .

k+m=p--2T h i s i s t h e c a s e ; i n f a c t, t h e l e ft m e m b e r o f (3 .5 .3 .9 ) i s(3 .5 .3 .1 0) d ( E ( _ 1 ) k p k ( f ( 1 ) ) P ' ( f ( 2 ) ) ) .

k+m=p -2T h i s p r o v e s (3 .5 .0 .3 ) i n c a s e b = 2, a n d g i v e s a h i n t o f t h e c o m b i n a t o r i a lr e a r r a n g e m e n t s n e c e s s a r y in t h e g e n e r a l c a se .(3 .5 .4 ) W e n o w t u r n t o th e g e n e r a l c a se . W e a d o p t t h e c o n v e n t i o n t h a ta p r o d u c t i n d e x e d b y a s u b s e t o f Z is t o b e t a k e n i n i n c r e a s i n g o r d e r , a n d


53/118

A l g e b r a i c S o l u t i o n s o f D i f f e r en t i a l E q u a t i o n s 5 3

t h a t , u n l e s s o t h e r w i s e s p e c i f i e d , a l l i n d e x i n g v a r i a b l e s r u n o v e r t h e s e t{1, . . . ,h i .

W e m u s t v e r i f y t h e c o n g r u e n c e (3 .5 .0 .3 ) i n t h e c a s e(3 .5 .4.0) ~ =-c (1) /x ~ (2) A . . . /x ~ (b) = I ] ~ (v).

vU s i n g t h e s u b s t i t u t i o n s ( 3 . 5 . 1 ) a n d t h e i d e n t i t y ( 3 . 5 . 2 . 3 ) j u s t a s i n t h e c a s eb = 2 , w e s e e t h a t t h e c o n g r u e n c e i n q u e s t i o n is e q u i v a l e n t t o t h e c o n -g r u e n c e

L i e ( P ) e - 1 ( 2 ( - 1 ) ' + ~ f ( i ) I - I t r (v ) )i v * i(3.5.4.1) - - - ~ ( - 1) + l P P - l ( f ( i ) ) H t7 (v ) m o d u l o d F ( V , ~?~/s ( log O ~)).

i v * iE x p a n d i n g t h e l ef t m e m b e r o f (3 .5 .4 .1 ) b e , L e i b n i z ' s r u l e , a n d r e -m e m b e r i n g ( 3 . 5 . 3 . 7 ) , ( 3 . 5 . 4 . 1 ) b e c o m e s

p - - 1( - 1 ) + ' ( - 1 ) k P P - ~ - k ( f ( i ) )(3.5.4.2) ' k= l

9L i e ( p ) k ( [ I a ( v )) ~ d F ( V , f ~ / s ( l o g D r) ) .v * iO u r n e x t t a s k i s t o e x p a n d e a c h o f t h e te r m s

(3.5.4.3) L ie (p)k ( 1-I a (v))v~:i

u s i n g L e i b n i z ' s r u le . T o f a c i l it a te t h i s, w e i n t r o d u c e t h e n o t a t i o n s( 3.5 .4 .4 ) t~ = ( t'l . . . . . ( b ) , a b - t u p l e o f n o n - n e g a t i v e i n t e g e r s(3 .5 .4 .5) I l l = ~ ( ,(3 .5 .4 .6 ) ( [~ 1) = ( I fD ' ( ~ ) '

i(3.5.4.7) S ( f ) = { i l f ~ . O }(3 .5.4 .8) f o r a n y n o n e m p t y s u b s e t A c { 1 . . . . , b } - { i } w e d e n o t e b ysgn (A, i ) t he s i g n o f t h e p e r m u t a t i o n

A , {1 . . . . . b } - { i } - A ~{ l , . . . , b } - {i} f

( W e m a k e t h e c o n v e n t i o n t h a t s u b s e t s o f Z a r e t o b e e n u m e r a t e d i ni n c r e a s i n g o r d e r . )


54/118

5 4 N . M . K a t z :

R e t u r n i n g t o t h e e x p a n s i o n o f (3 .5 .4 .3 ), w e f i n dL ie (P )R( l - I a (v ) ) = ~ (1~ } ) l - I L i e ( P ) e ~ ( ~ r ( v ) )

v * i ~ '; [ l I = k , i C s ( O x "

Alg Solutions of Diff.eq

Documents