Compositio Mathematica 105: 55–64, 1997.

55
c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Pfaffian equations and the Cartier operator

MINHYONG KIM
Department of Mathematics, Columbia University, New York, NY, 10027
Current address: Dept. of Mathematics, University of Arizona, Tucson, AZ85750.

Received 25 April 1995, accepted in final form 30 August 1995

Key words: Pfaffian equations, Cartier operator, de Rham–Witt complex.

1. Notation
k = algebraically closed field of characteristic p > 0
X = smooth, irreducible, projective variety over k
K = function field of X

iX = sheaf of differential forms of degree i on X

iK = i-differential forms on the function field of X , also considered as a constant sheaf on X

C [n] = a complex C shifted in such a way that C [n]i = C i+n
F n C = a complex C truncated (brutally) so that F n C i = 0 for i < n and F n C i = C i for i > n

X = the algebraic de Rham complex of X

A Pfaffian equation on X is an invertible subsheaf L ,!

1X of the sheaf
of differential 1-forms on X . It can thus be thought of as a global section of

1X
L 1 or, by choosing an isomorphism L 1 ' OX (E )  K , as a meromorphic
differential form ! on X . We say that the Pfaffian equation has a first integral if
there is a non-empty open subset U  X and a smooth map f : U !P1 such that
LU ' f 
1P1 as subsheaves of
1X ; that is, if L ‘comes from’ a rational map to a
curve. In this case, f , viewed as a rational function, is called a first integral. Note
that as rational differential forms, df ^ ! = 0.
A (reduced and irreducible) subvariety of codimension one i: D ,! X is said
to be a hypersurface solution for ! , if i ! = 0, as a section of
D
L 1 . This is
the same as requiring the composed map

L!
1X !i
1D ;
to be zero. Note that if ! has a first integral f , then the closure of the irreducible
components of f = hp are solutions, for any rational function h. The reader is
referred to [4] for a discussion of relations to classical differential equations.
If D is represented as a Cartier divisor by a collection (fi ; Ui ), then D is a
solution for ! if and only if (dfi =fi ) ^ ! 2 (Ui ;
2X
L 1 ) for each i.

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56 MINHYONG KIM

The paper [4] studies Pfaffian equations on compact complex manifolds sat-
isfying certain conditions on its Hodge-to-de Rham spectral sequence. The main
result there is that for Pfaffian equations on such manifolds, there are infinitely
many irreducible hypersurface solutions only in the trivial case when ! admits a
first integral. We wish to generalize this to include the case of varieties in posi-
tive characteristic. For compact smooth varieties in characteristic zero, Jouanolou’s
condition is automatically satisfied. However, an hypothesis is necessary over fields
of positive characteristic:

THEOREM 1. Suppose all global one-forms on X are closed and ! does not have
a first integral. Then there are only finitely many irreducible hypersurface solutions
for ! .

The proof of this theorem is largely modeled on the complex case studied
by Jouanolou. However the existence of non-constant d-closed functions, namely
p-powers, keeps the translation from being entirely straightforward. It was thus
quite surprising to the author that a pleasant resolution arises from the systematic
use of the Cartier operator. That is, this endows the eventual proof with a nature
particular to characteristic p. The relation between the solution varieties of a closed
differential form and of its Cartier descendants seems to deserve careful study.
This theorem has been used by Vojta (in characteristic zero) [6] to obtain
bounds for heights on algebraic points for curves over function fields. It also
can be used to bound families of curves on surfaces satisfying certain numerical
conditions, as a consequence of Bogomolov’s inequality [1]. Although the class
of surfaces satisfying Bogomolov’s inequality in positive characteristic is still
unknown, this paper illustrates that intimately related results can be obtained,
provided certain ‘ordinarity hypotheses’ are made. One can conjecture, then, that
the counterexamples to Bogomolov’s inequality in positive characteristic arise from
a failure of ordinarity.

2. Preliminaries
We will need a few facts about the de Rham cohomology HDRi
(X ) := Hi (X;

X )
of X (the boldface denotes hypercohomology) as well as the crystalline cohomo-
logy Hcri
(X=W ) of X with coefficients in the Witt vectors W of k. The latter
can be realized as the hypercohomology of the de Rham–Witt complex W

X [3].
There is a map of complexes

W

X !

X ;
which induces a map from the slope spectral sequence

H q (W
pX ) ) Hcrp+q (X );

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PFAFFIAN EQUATIONS AND THE CARTIER OPERATOR 57

to the Hodge-to-de Rham spectral sequence

H q (
pX ) ) HDR
p+q
(X ):
Recall the following facts about the slope spectral sequence ([3] Corollary II.3.3
and Proposition II.3.11)

E1p;0 = E1p;0 = H 0 (W
pX );
()
E10;1 = E10;1 = H 1(W OX ):
Given an invertible sheaf L on X , we can associate to it a first Chern class c1 (L)
in the crystalline cohomology of X by using the map of complexes

OX [ 1]!W

X ;
given by the logarithmic derivative f 7! df=
~ f~ (f~ = (f; 0; 0; : : :) 2 W OX ). That
is, this map of complexes induces the Chern class map

c1: H 1 (OX )!Hcr2 (X ):

The map of complexes factors through

OX [ 1]!F 1 W

X  W

X ;
and () implies, in fact, that

H2 (F 1 W

X )  Hcr
2
(X );
so that the Chern class can be seen as lying in the first group.
There is also a quotient map F 1 W

X [1]!W
1X , sending c1 (L) to the Chern–
Hodge class ch(L) 2 E1 1;1  H 1 (W
1 ). Its image ch0 (L) inside H 1 (
1 ) is the
X X
usual Chern–Hodge class, which may be interpreted as the class of the
1X -torsor
given by the connections on L. In particular, L admits a connection iff ch0 (L) = 0.
Recall that if L is associated to the Cartier divisor (fi ; Ui ) with transition functions
gij , then a connection is equivalent to the data of regular 1-forms Ai on Ui satisfying
Ai Aj = (dfi=fi ) (dfj =fj ) = dgij =gij . We shall refer to such a collection
also as connection forms for the Cartier divisor, as well as for the invertible sheaf
it defines. The relation between ch0 (L) and connections on L follows from a
straightforward computation using Cech cocyles for the de Rham complex, and the
same computation for W

X yields the fact that if ch(L) = 0, then we can find
local sections A~i of W
1X , which satisfy A~i A~j = dg~ij =g~ij ; and which therefore
reduce to connection forms for L. Since W

X forms a differential graded algebra,
we get d(dg~ij =g~ij ) = 0, so we see that dA~i = dA~j . Therefore, the collection (dA~i )

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58 MINHYONG KIM

~ of W
2X , which reduces to the usual curvature form R
defines a global section R
of the connection (Ai ). As a special case of ()

E12;0 = E12;0 = H 0 (W
2X )  Hcr2 (X );
and in the situation just described when ch(L) = 0, another Cech computation
show us that c1 (L) = R ~ with respect to this inclusion. In particular, if c1 (L) =
0() ch(L) = 0), then we can find a collection A~i so that R ~ = 0. This implies the
following

LEMMA 1. Suppose c1 (L) = 0 in crystalline cohomology. Then L admits a con-

nection with vanishing curvature.

Our preceding remarks amount to the fact that it admits something stronger, namely,
an ‘integrable de Rham–Witt connection.’
This lemma applies, for example, in the case where L is algebraically equivalent
to zero. Note that the de Rham–Witt complex is used to circumvent the possible
non-degeneration of the Hodge-to-de Rham spectral sequence.
Since any two connections differ by a global one-form we see that under the
hypothesis that all global one-forms are closed, any connection for a line bundle
with vanishing first Chern class will have zero curvature.
Recall that the Cartier operator C ([5], 7.2) fits into an exact sequence

iX 1 !
d
Z
iX !
C

iX !0;
where Z
iX refers to the sheaf of closed differential i-forms. Together with this
exact sequence (which includes a theorem of Cartier), it is characterized by

C (f p 1
df ) = df; C (f p) = fC (); C (1) = 1;
and

C ( ^  ) = C () ^ C ( );
for any function f and closed forms  and  .
Taking the stalk at the generic point gives an exact sequence of rational differ-
ential forms

iK 1 !
d
Z
iK !
C

K !0:
Now when all global forms on X are closed, if (Ai ) defines a connection form
for a Cartier divisor D = (fi ) with vanishing first Chern class, then as noted above,
the Ai are closed so we may apply C to them. The fact that C (dfi =fi ) = dfi =fi
implies that C (Ai ) also defines a connection form for D , and hence, is again closed.
Thus all iterations C k (Ai ) are defined, and are all closed.

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PFAFFIAN EQUATIONS AND THE CARTIER OPERATOR 59

We will use the map of exact sequences

0 - O X
- K - K  =O X
-0
;
? ? ?
0 -
1 X
-
1K
-
1 =
1
K X
-0
where the vertical arrows are induced by the logarithmic derivatives.

LEMMA 2. Logarithmic differentiation induces an injection

K ,!

K1 :
1

(K  )(p) O X

Proof. Let f be a rational function such that df=f is regular. We will show that
f is locally a unit times a p-power.
It suffices to show this in the local ring R of a point. But since R is a unique
factorization domain, we may write f = u1n


knk where u is a unit and the
1

i ’s are distinct primes. So df=f = du=u + i ni (di =i ). Since df=f and du=u
are regular, we see by localizing at each prime in turn that all the ni ’s must be
multiples of p. 2
Thus, we get an injection
  !
K ,!
1K
:
(K  )(p) O
1X
Denote by Div the group of Weil divisors on X . It is a free abelian group
generated by the irreducible Weil divisors and we have Div/p Div ' Div
Fp =
the Fp vector space generated by the irreducible Weil divisors.

LEMMA 3. Div/p Div ,! (K  =(K  )(p) O  ).

Proof. We need only prove the corresponding statement for Cartier divisors,
that is, that
 K    K   K 
O p ,! O (K  )(p) O
;
via the canonical map.
Let D be a Cartier divisor which is (for a fine enough covering) locally rep-
resented by p-powers: D = [(fip ; Ui )]. Since (fi =fj )p = fip =fjp is a unit on

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60 MINHYONG KIM

the overlaps, so is fi=fj . Thus, (fi ; Ui) defines a Cartier divisor D0 such that
pD0 = D. 2
Composing the two lemmas above, we get an injection
!

1K
Div
p Div ,!
1X
:

PROPOSITION 1. The inclusion above induces an injection

  !

1K
: Div
Z k =
k ,! :
Div
p Div Fp
1X
Proof. Let D1 ; : : : ; Dr be Weil divisors. We will show by induction on r that
a non-trivial k -linear relation between the (Di ) implies a non-trivial Fp -linear
relation between the Di . This will show that if D1 ; : : : Dr are linearly independent
mod p, then, for any non-trivial set of coefficients a1 ; : : : ; ar in k , (i ai Di ) =
i ai (Di ) 6= 0. The case r = 1 is the injection above. Assume now that
X
r
ai (Di ) = 0;
i=1

where none of the (Di ) are zero. (Otherwise, we are done, by induction.) Since
(D1 ) 6= 0, D1 has an irreducible component E with multiplicity prime to p. Let
U be an affine open set which intersects E and on which each Di has a defining
equation fi . Our assumption implies that ri=1 ai (dfi =fi ) is regular on U . Now,
choose a curve C in U which intersects E tranversally at a simple point x0 2 E –
(other components of D1 ) and is not contained in the support of any Di . Such a
C exists by Bertini’s Theorem. So each fi restricts to a rational function fi0 on C
and if we put mi = Resx0 (dfi0 =fi0 ), then r1 ai mi = 0. Also, m1 6= 0 in Fp , by our
choice of C , E , and x0 . Thus, we get

X
r X
r
ai (m1 Di mi D1) = m1 ai (Di ) = 0:
i=2 1

Hence, by the induction hypothesis there exist ni 2 Fp not all zero, such that
X
r
ni (m1Di miD1) = 0: (mod p):
2

But then, since some ni m1 6= 0, this gives a non-trivial Fp-linear relation among
the Di . 2
Assume, henceforward, that all global one-forms on X are closed.

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PFAFFIAN EQUATIONS AND THE CARTIER OPERATOR 61

Now denote by Div0 the subgroup of Div whose associated invertible sheaves
have vanishing Chern classes in crystalline cohomology and let D 0 := Im(Div0

k!Div
k): The inclusion constructed above
!

1K
Div
k ,! ;

1X

when composed with the boundary map  in the exact sequence

!

1K
0! (
X )! (
K ) !
1 1
! H 1 (
1X );

1X

gives nothing but the Chern–Hodge class ch0

1k . Thus, it is zero on D 0 . We use
this fact to lift the map above to an inclusion

,! (

1 )
: D0 K
:
(
1X )

Given a divisor D 2 Div0 , a rational differential form whose class is (D ) can

be explicitly described as follows: Choose a Cartier representative (fi ; Ui ) for D .
We may then find a collection of 1-forms (Ai ) defining a connection for D . Then

Ai Aj = dffi dfj
fj ) Ai
dfi
fi = Aj
dfj
fj ;
i

on the overlaps of the Ui ’s. Thus, these local forms glue to give a rational one form,
. In this case, since c1 (D) = 0,  is closed, as discussed above, so we may apply
the Cartier operator to it. But locally, C () is just dfi =fi C (Ai ) which is again
closed, since the C (Ai ) define a connection for the same divisor.

LEMMA 5. If D 2 Div0 , then, given any rational representative  for (D ), we

may apply the Cartier operator repeatedly to get closed one-forms C k ().

This lemma follows from the preceding discussion and the fact that any two
representatives in (
K ) for (D ) will differ by a global one-form.
If  and  are rational 1-forms and a; b 2 k , then C (a + b ) = a1=p C () +
b C ( ), so the property mentioned in the lemma is closed under k-linear combi-
1=p

nation, in an obvious sense.

COROLLARY 1. For any x 2 D0, and any representative  for (x), all the
C k () are defined and closed.

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62 MINHYONG KIM

3. Proof of theorem
Denote by N the group of divisors generated by the irreducible solutions of ! . Thus,
N is a free abelian subgroup of Div and the number of irreducible solutions is equal
to the rank of N or, equivalently, the dimension of N
F. Let N0  N \ Div0 be
the subgroup of N consisting of divisors algebraically equivalent to zero.
Note now that N being generated by irreducible divisors, is a saturated subgroup
of Div, allowing

N
Fp  Div
Fp :
Also, the exact sequence

N0
Fp !N
Fp ! NN
Fp!0;
0

together with the finite generation of N=N0  NS (X ) (the Neron–Severi group

of X ) tells us that it suffices to prove that

M := [Im(N0
Fp !Div
Fp )]
k  D0;
is finite-dimensional. We constructed above an injection, still denoted by the same
letter

,! (

1 )
:M K
:
(
1 ) X

Composing with wedge product by ! , we get a map

h : M ! !(
^
(
L1 ) ) :
2 1

The image lies in the given subspace because elements of M are linear combinations
of solutions for ! .
Let x be in the kernel of h and let  be a rational form representing (x). Then
there exists a global 1-form such that  ^ ! = ^ ! , or ( ) ^ ! = 0. That
is, x 2 Ker h iff (x) has a representative which is proportional to ! as a rational
1-form.
We now distinguish two distinct cases:
(1) There exists an x 2 M and a representative  for (x) such that  ^ ! = 0
and some iteration C k () has a first integral.
In this case, note that  itself does not have a first integral by assumption, since
it is proportional to ! . Write C k () = f dg for rational functions f; g . Then

C k 1() = f pgp 1
dg + dxk 1 ;

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PFAFFIAN EQUATIONS AND THE CARTIER OPERATOR 63

for some rational function xk 1

C k 2() = f p gp(p 1)gp
2 2 1
dg + xpk 1
x
1d k 1 + dxk 2;
and, continuing down the powers
k
 = f pk gpk dg + xpk dxk +


+ xp1 1 dx1 + dx0 :
1 1
1
1 1

Now, suppose i : V ,! X is a subvariety such that i ! = 0, and hence i  = 0.

Then, by the naturality of the Cartier operator, C n (i ) = i C n () = 0 for all n.
So

i (f dg) = 0 ) i dxk 1 = 0 )


) i dx0 = 0:

That, is all the differential form summands in the formula above for  must vanish.
However, since  does not have a first integral, at least two of the summands are
generically linearly independent. Therefore, there is a Zariski closed subset Z  X
such that all V as above not lying in Z has codimension at least two. This proves
the theorem in this case.
(2) For any x 2 M , if  is a representative for (x) such that  ^ ! = 0, then
none of the C k () have first integrals.
For this case, suppose x and y are contained in the kernel of h, and let 
and  (respectively) be rational 1-forms representing (x) and (y ) such that
 ^ ! = 0;  ^ ! = 0.
Then  = f for some rational function f . But since  and  are both closed,
this gives us df ^ = 0. Thus df = 0 since  does not have a first integral, and
hence, f = f1p , for some function f1 . So  = f1p  ) C () = f1 C ( ). But C ( )
and C () are also closed, and also do not have first integrals. Thus f1 is also a
p-power. Continuing in this way, we see that f must be a infinite p-power. So f is
a constant and ;  are k -linearly dependent. By injectivity of , this implies that
x and y are linearly dependent.
That is, in this second case, dim(Ker h) 6 1. So we are done again, because

(
2X
L 1 )
! ^ (
1) ;
is finite-dimensional. 2
Acknowledgement
This paper is dedicated to my New York friends, especially Bonjung Goo, Sungjin
Suh, Gweeyup Son, Hyeonjoo Lee, Dosang Joe, and Jaehak Hahm, in gratitude for
their continuing inspiration.

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64 MINHYONG KIM

References
1. Deschamps, M.: Courbes de genre géométrique borné sur une surface de type général (d’aprés
F. A. Bogomolov), in Lecture Notes in Mathematics 710, Springer-Verlag, Berlin, Heidelberg,
New York, 1978.
p
2. Deligne, P. and Illusie, L.: Relèvements modulo 2 et décomposition du complex de de Rham,
Invent. Math. 89 (1987) 247–270.
3. Illusie, L.: Complex de de Rham–Witt et cohomology crystalline, Ann. Scient. Ec. Norm. Sup.
12 (1979) 501–661.
4. Jouanolou, J. P.: Hypersurface solutions d’une equation de Pfaff analytique, Math. Ann. 232
(1978) 239–245.
5. Katz, N.: Nilpotent connections and the monodromy theorem; application of a result of Turrittin,
Publ. Math. I.H.E.S. 39 (1970) 355–232.
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