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DUKE MATHEMATICAL JOURNALVol. 110, No. 2, c©2001

LOGARITHMIC DIFFERENTIAL FORMS ON p-ADICSYMMETRIC SPACES

ADRIAN IOVITA AND MICHAEL SPIESS

AbstractWe give an explicit description in terms of logarithmic differential forms of the iso-morphism of P. Schneider and U. Stuhler relating de Rham cohomology of p-adicsymmetric spaces to boundary distributions. As an application we prove a Hodge-type decomposition for the de Rham cohomology of varieties over p-adic fields whichadmit a uniformization by a p-adic symmetric space.

1. IntroductionLet K be a finite extension ofQp, and letX = X (d+1) be Drinfeld’s p-adic sym-metric space of dimensiond overK . It is a rigid analytic space, and it is defined as thecomplement of allK -rational hyperplanes inPd

/K . Unlike real symmetric spaces,X

is not simply connected. Its cohomology groupsH•(X ) for any cohomology theorysatisfying certain axioms (examples are de Rham or`-adic cohomology) have beencomputed by Schneider and Stuhler. For 1≤ k ≤ d, Hk(X ) is infinite-dimensionaland carries a natural PGLd+1(K )-action. In [SS] it is shown thatHk(X ) is canoni-cally and PGLd+1(K )-equivariantly isomorphic to a certain space of locally constantfunctions on some flag manifold.

In the following we consider the de Rham cohomology groups only. For ourpurposes the first computation ofHk(X ) in [SS] in terms of the cohomology ofa certain profinite simplicial set is important. LetH be the set of allK -rationalhyperplanes inPd

/K , and letY(k)· be the simplicial set given by

Y(k)r =

{(H0, . . . , Hr ) ∈H r+1

∣∣∣ dim( r∑

i=0

K`Hi

)≤ k

},

with the obvious face and degeneracy maps (where`Hi denotes a linear form definingHi ). It is shown in [SS] that there is a natural isomorphism betweenHk(X ) and

DUKE MATHEMATICAL JOURNALVol. 110, No. 2, c©2001Received 15 March 2000. Revision received 14 November 2000.2000Mathematics Subject Classification. Primary 11F85; Secondary 14F40.Authors’ work partially supported by Engineering and Physical Sciences Research Council grant number GR/M

95615.Iovita’s work partially supported by National Science Foundation grant number DMS-0070464.

253

254 ADRIAN IOVITA AND MICHAEL SPIESS

the dual of the reduced cohomologyHk−1(|Y(k)· |, K ) of the geometric realization of

Y(k)· . This can be interpreted (see Lemma3.2) as an isomorphism between the space

of distributions onH k+1 modulo a certain subspace of distributions (which we calldegenerated) andHk(X ):

8 : D(H k+1, K )/ D(H k+1, K )deg−→ Hk(X ) (1)

(see Section 4 for the definition of D(H k+1, K )deg). The proof in [SS] is based onthe axiomatic properties of the cohomology theory and is therefore not explicit at all.Our first main result provides an explicit description of8. For a distributionµ onH k+1 we have

8([µ]) =

∫H k+1

[dlog(H0, . . . , Hk)

]dµ, (2)

where dlog(H0, . . . , Hk) := dlog(`H1/`H0) ∧ · · · ∧ dlog(`Hn/`Hn−1). The bracketsdenote the class of the distribution, respectively, of the differential form.

In particular, we see that the differential form∫H k+1

dlog(H0, . . . , Hk) dµ (3)

is a canonical representative of the cohomology class8([µ]) whenever the integralis well defined. We see in Section 4 that this is the case ifµ is a measure, that is,a bounded distribution. The existence of such a canonical form representing a givende Rham cohomology class has been known previously only for the top-dimensionalcohomology groupHd(X ) (see [Te] for the cased = 1 and [ST1] for arbitraryd). Let �k

log,b(X ) be the subspace of the space of holomorphic differentialk-forms

�k(X ) onX of type (3) (whereµ is a measure). Then our result can be stated in theform that�k

log,b(X )→ Hk(X ), ω 7→ [ω] is “almost” an isomorphism. We want tomention that logarithmic forms onX were also studied recently by E. de Shalit. In[dS] he proves that they generate a dense subspace inHk(X ).

In the last section we give an application to the de Rham cohomology ofp-adically uniformized varieties, that is, of varieties of the formX0 := 0\X where0⊆ PGLd+1(K ) is a cocompact discrete subgroup. Their cohomolgy groups havebeen computed in [SS] using a covering spectral sequence

Er,s2 = H r (0, Hs(X )

)=⇒ H r+s(X0). (4)

It turns out that the only interesting cohomology group is the middle one,Hd(X0).Denote byHd(X0) = F0

0 ⊇ F10 ⊇ · · · ⊇ Fd+1

0 = 0 the filtration induced by (4). Weshow that it is opposite to the Hodge filtrationF i

H on Hd(X0). (This was conjecturedin [SS].) As a consequence one gets a Hodge-type decomposition forHd(X0):

Hd(X0) =

d⊕i=0

F i0 ∩ Fd−i

H .

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 255

We also deal with the corresponding question for the cohomology of certain local sys-tems onX0 induced by finite-dimensionalK [0]-modules which contain a0-stablelattice.

For convenience we briefly describe the contents of the other sections. In Section2 we prove a compatibility property between the cup product and an edge morphismof a certain spectral sequence associated to a system of closed subschemes of a variety.In Section 3 this is applied to prove the analogue of (2) for the de Rham cohomologyof the complement of finitely many hyperplanes in the projectived-space. Finally,in Section 4 we show that the integrals above are well defined and prove formula(2) by reducing it to the corresponding result for the complement of finitely manyhyperplanes.

2. Cup product and a spectral sequenceLet K be a field of characteristic zero. For a pair(X, Y) consisting of an algebraicK -schemeX and a closed subschemeY ⊆ X, we write H•(X), H•Y(X) as shorthandfor the de Rham cohomology ofX, respectively, the de Rham cohomology ofXwith support inY. (However, the results of this section are valid for any reasonablecohomology theory, e.g.,-adic or singular cohomology.) A general reference for deRham cohomology of algebraic varieties is [Ha1].

Let X be as above, and letY = (Y0, . . . , Yn) be an (n + 1)-tuple of closedsubschemes ofX. There exists a spectral sequence

E−r,s1 =

{ ⊕0≤i0<···<ir≤n Hs

Yi0∩···∩Yir(X) if 0 ≤ r ≤ n, 0≤ s,

0 otherwise,

}=⇒ Hs−r

∪Y (X), (5)

where∪Y := Y0∪· · ·∪Yn. It is constructed as follows: Let�·X → I · be an injectiveresolution. Denote byC·+(Y , I ·) the double complex

C−r+ (Y , I s) :=

{ ⊕0≤i0<···<ir≤n 0Yi0∩···∩Yir

(X, I s) if 0 ≤ r ≤ n,0 otherwise.

For a fixeds the complex

C·+(Y , I s) −→ 0∪Y (X, I s) −→ 0

is exact; by induction this can be reduced to the case wheren = 1, that is,Y := Y0∪

Y1 where it follows from [Ha2, Chap. III, Exer. 2.3] using the fact that the cohomologygroups ofX with support in a closed subset and coefficients in an injective sheafvanish. Hence the augmentation map

C·+(Y , I ·) −→ 0∪Y (X, I ·)

256 ADRIAN IOVITA AND MICHAEL SPIESS

induces a quasi-isomorphism TotC·+(Y , I ·) → 0∪Y (X, I ·), and therefore the co-homology of TotC·+(Y , I ·) is equal toH•

∪Y (X). The spectral sequence (5) is thesecond spectral sequence of the double complexC·+(Y , I ·).

Let eY = esY : Hs

∪Y (X) −→ Hs+n∩Y (X) be the edge morphism of the above

spectral sequence (∩Y := Y0 ∩ · · · ∩ Yn). The aim of this section is to prove acompatibility property between the cup product and the morphismses

Y .Note that in the caseY = (Y0, Y1) the mapes

Y is just the boundary map of theMayer-Vietoris sequence

· · · −→ HsY0∩Y1

(X) −→ HsY0

(X)⊕ HsY1

(X) −→ HsY0∪Y1

(X)

−→ Hs+1Y0∩Y1

(X) −→ · · ·

and is given by the composition

HsY0∪Y1

(X) −→ HsY0−Y1

(X − Y1)

∼= HsY0−(Y1∩Y0)

(X − (Y1 ∩ Y0)

)−→ Hs+1

Y0∩Y1(X), (6)

where the second arrow is the boundary map in the long exact cohomology sequenceof the triple(X, Y0, Y0 ∩ Y1).

LEMMA 2.1LetY = (Y0, . . . , Yn) be an(n+ 1)-tuple of closed subschemes of X, and setY ′ :=

(Y0, . . . , Yn−2, Yn−1 ∪ Yn), Y ′′ := (Y0 ∩ · · · ∩ Yn−2 ∩ Yn−1, Y0 ∩ · · · ∩ Yn−2 ∩ Yn).Then esY = es+n−1

Y ′′ ◦ esY ′ .

ProofThere is a canonical map of double complexesf : C·+(Y , I ·) → C·+(Y ′, I ·) givenon the summand0Yi0∩···∩Yir

(X, I s), 0≤ i0 < · · · < i r ≤ n, of C−r+ (Y , I s) by

0Yi0∩···∩Yir(X, I s)

id−→ 0Yi0∩···∩Yir

(X, I s) ⊆ C−r+ (Y ′, I s)

if i r ≤ n− 2, by

0Yi0∩···∩Yir(X, I s)

incl−→ 0Yi0∩···∩Yir−1∩(Yn−1∪Yn)(X, I s) ⊆ C−r

+ (Y ′, I s)

if i r−1 ≤ n− 2 andi r ≥ n− 1, and by zero ifi r−1 = n− 1 andi r = n.Denote the horizontal and vertical differentials inC·+(Y , I ·) by d−r,s

h andd−r,sv ,

respectively. Explicitly, the mapesY : Hs

∪Y (X) −→ Hs+n∩Y (X) can be described as

follows.For x ∈ Hs

∪Y (X) there are elementsx0 ∈ C0+(Y , I s), x1 ∈ C−1

+ (Y , I s+1),

. . . , xn ∈ C−n+ (Y , I s+n) such thatd−r,r+s

h (xr ) = d−(r−1),s+r−1v (xr−1) for r =

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 257

1, . . . , n and such that the image ofx0 under the augmentation map⊕n

i=00Yi (X, I s) → 0∪Y (X, I s) represents the cohomology classx. Thenes

Y (x) = [xn],the class ofxn in Hs+n

∩Y (X).The analogous description ofes

Y ′ givesesY ′(x) = [ f (xn−1)]. Now by definition

of the boundary map of the Mayer-Vietoris sequence forY ′′ we have

es+n−1Y ′′

(esY ′(x)

)= es+n−1

Y ′′([ f (xn−1)]

)= [xn] = es

Y (x).

Let∪ : H i

Y(X)× H jZ(X) −→ H i+ j

Y∩Z(X)

denote the cup product for de Rham cohomology. It is induced by the multiplicativestructure�·⊗�·→ �· of the de Rham complex. We need that it be compatible withrespect to long exact relative cohomology sequences (compare [Iv, Chap. II.9]): IfY′, Y, Z⊆ X are closed subschemes withY′⊆Y and ifη ∈ H j

Z(X), then the diagram

. . . H iY′(X) −−−−−→ H i

Y(X) −−−−−→ H iY−Y′(X − Y′) −−−−−→ H i+1

Y′ (X) . . .y∪η y∪η y∪η|X−Y′

y∪η. . . H i+ j

Y′∩Z(X) −−−−−→ H i+ jY∩Z(X) −−−−−→ H i+ j

(Y−Y′)∩Z(X − Y′) −−−−−→ H i+ j+1Y′∩Z (X) . . .

(7)

commutes.

LEMMA 2.2Let Y = (Y0, . . . , Yn) be an(n + 1)-tuple of closed subschemes of X, let Z⊆ X beanother closed subscheme, and letη ∈ H j

Z(X). Then the following diagram com-mutes:

H i∪Y (X) −−−−→

eiY

H i+n∩Y (X)y∪η y∪η

H i+ j∪(Y ∩Z)

(X) −−−−→eiY ∩Z

H i+ j+n∩(Y ∩Z)

(X)

where we have setY ∩ Z := (Y0 ∩ Z, . . . , Yn ∩ Z).

ProofAccording to Lemma2.1 it is enough to consider the casen = 1. Then the assertionfollows from the commutativity of diagram (2) and from the description (6) of theboundary map in the Mayer-Vietoris sequence.

The following result is crucial for the description of the isomorphism (2) in terms oflogarithmic forms.

258 ADRIAN IOVITA AND MICHAEL SPIESS

PROPOSITION2.3Let Y = (Y0, . . . , Yn) be an(n + 1)-tuple of closed subschemes of X, and let xi ∈

Hmi (X − Yi ) for i = 0, 1, . . . , n. Then

eY(δ(x0 ∪ x1 ∪ · · · ∪ xn)

)= δ(x0) ∪ δ(x1) ∪ · · · ∪ δ(xn).

Hereδ denotes the respective boundary maps H•(X − ∪Y )→ H•+1∪Y (X), H•(X −

Yi )→ H•+1Yi

(X).

ProofWe start with the casen = 1. The description (6) of the boundary map in the Mayer-Vietoris sequence shows that we can expresseY ◦δ : H•

(X−(Y0∪Y1)

)→ H•+2

Y0∩Y1(X)

as the composition

H•(X − (Y0 ∪ Y1)

) δ1−→ H•+1

Y0−Y1(X − Y1)

∼= H•+1Y0−(Y0∩Y1)

(X − (Y0 ∩ Y1)

) δ2−→ H•+2

Y0∩Y1(X).

Hereδ1 (resp.,δ2) is the boundary map in the relative cohomology sequence of thepair (X − Y1, Y0 − Y1) (resp., of the triple(X, Y0, Y0 ∩ Y1)). Therefore, using thecommutativity of the third square in (2), we obtain

eY ◦ δ(x0 ∪ x1) = δ2 ◦ δ1(x0 ∪ x1) = δ2(δ1(x0|X−(Y0∪Y1)) ∪ x1

)= δ2

(δ(x0)|X−Y1 ∪ x1

)= δ(x0) ∪ δ(x1).

For the general casen ≥ 2 we proceed by induction. SetY ′ := (Y0, . . . , Yn−2,

Yn−1 ∪ Yn), Y ′′ := (Y0 ∩ · · · ∩ Yn−2 ∪ Yn−1, Y0 ∩ · · · ∩ Yn−2 ∩ Yn). Then we have

eY(δ(x0 ∪ · · · ∪ xn)

)= eY ′′ ◦ eY ′

(δ(x0 ∪ · · · ∪ (xn−1 ∪ xn))

)(by Lemma 1)

= eY ′′(δ(x0) ∪ · · · ∪ δ(xn−2) ∪ δ(xn−1 ∪ xn)

)(by the induction hypothesis)

= δ(x0) ∪ · · · ∪ δ(xn−2) ∪ eY ′′(δ(xn−1 ∪ xn)

)(by Lemma 2)

= δ(x0) ∪ · · · ∪ δ(xn−2) ∪ δ(xn−1) ∪ δ(xn)

(the case of two subschemes).

There is a variant of the above spectral sequence which was introduced in [SS, Sec.2]. Let F = {0, . . . , n}, and denote byC·(Y , I ·) the double complex

C−r (Y , I s) :=

{ ⊕(i0,...,ir )∈Fr+1 0Yi0∩···∩Yir

(X, I s) if r ≥ 0,

0 otherwise.

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 259

Again the augmentation map

C·(Y , I ·) −→ 0∪Y (X, I ·)

induces a quasi-isomorphism TotC·(Y , I ·) → 0∪Y (X, I ·) (see [SS, Sec. 2]). Thesecond spectral sequence of the double complex thus yields a spectral sequence

E−r,s1 =

{ ⊕(i0,...,ir )∈Fr+1 Hs

Yi0∩···∩Yir(X) if r, s ≥ 0,

0 otherwise

}=⇒ Hs−r

∪Y (X). (8)

The canonical injective map

C·+(Y , I ·) −→ C·(Y , I ·)

induces a map from the spectral sequence (5) to (8). It is already an isomorphism onthe E2-terms since for a fixeds the map

C·+(Y , I s) −→ C·(Y , I s)

has a homotopy inverse that is compatible with the differentialsI s→ I s+1 (see the

proof of [SS, Sec. 2, Prop. 6]).

3. Logarithmic differential forms on complements of finitely many hyperplanesLet K be a field, letd be an integer≥ 1, and letP be a finite set ofK -rational hyper-planes inPd. Let XP be the complementPd

/K −⋃

H∈P H . By `H ∈ (K d+1)∗ weusually denote a functional defining a given hyperplaneH . For a profinite setS, letE·(S) be the simplicial setEr (S) = Sr+1 with faces and degeneracy maps given,respectively, by partial projections and diagonals. The partial projectionSr+1

→

Sr , (s0, . . . , sr ) 7→ (s0, . . . , si , . . . , sr ) is denoted bypri . For k ≥ 1, let Y(P,k)

· bethe simplicial subset ofE·(P) given by

Y(P,k)r =

{(H0, . . . , Hr ) ∈Pr+1

∣∣∣ dim( r∑

i=0

K`Hi

)≤ k

}.

Consider the spectral sequence (8) for the set of closed subschemesP of Pd,

E−r,s1 =

{ ⊕(H0,...,Hr )∈Pr+1 Hs

H0∩···∩Hr(Pd) if r, s ≥ 0,

0 otherwise

}=⇒ Hs−r

∪P (Pd). (9)

As in [SS, Sec. 3] theE2-terms can be described in terms of the cohomology of thesimplicial setY(P,k)

· . Since Schneider and Stuhler work with rigid analytic rather thanalgebraic varieties, we briefly recall their arguments and modify them accordingly.

260 ADRIAN IOVITA AND MICHAEL SPIESS

LEMMA 3.1For an (n+ 1)-tuple(H0, . . . , Hn) of K -rational hyperplanes inPd we have

HsH0∩···∩Hn

(Pd) =

{K if s is even withdim

( ∑ni=0 K`Hi

)≤ s/2≤ d,

0 otherwise.

ProofThis can be proved as in [SS, Sec. 3, Lem 1]. We only have to remark that ifm :=dim(

∑ni=0 K`Hi ) and`H0, . . . , `Hm are linearly independent—we may assume this

after renumbering theH0, . . . , Hn if necessary—then

Pd− (H0 ∩ · · · ∩ Hn) −→ Pm, z= [z0 : · · · : zd] 7→

[`H0(z) : · · · : `Hm(z)

]is a locally trivial fibration with fiber∼= Ad−m.

It follows that

E−r,s1 =

{ ⊕(H0,...,Hr )∈Y(P,s/2)

rK if s is even and 2≤ s ≤ 2d,

0 otherwise.

Thed1-differentials are induced by the face maps of the simplicial setY(P,s/2)· , and

so we obtain

E−r,s2 =

{Hr (Y

(P,s/2)· , K ) if s is even and 2≤ s ≤ 2d,

0 otherwise.

As in [SS, Sec. 3, remarks following Lem. 1], one shows that

Hr (Y(P,k)· , A) = 0 unlessr = 0 or= k− 1,

for any abelian groupA. By [SS, Sec. 3, Prop. 5] one sees thatHr (Y(P,k)· , K ) can be

identified with the dual Hom(H r (Y(P,k)

· , Z), K)

of the cohomology. Lets be even,and let 2≤ s ≤ 2d. The composition

E0,s2 −→ Es

= Hs∪P(Pd) −→ Hs(Pd)

is an isomorphism fors > 2 and surjective fors = 2 (see [SS, Sec. 3, Lem. 7]).It follows that the edge morphismE−(k−1),2k

2 → Ek+1/F0 for k ≥ 1 induces anisomorphism

Hom(Hk−1(Y(P,k)

· , Z), K)−→ Hk(XP). (10)

(F i denotes the filtration onHs∪P(Pd) induced by the spectral sequence.) The source

of (10) can be described in terms of functions or—equivalently—of distributions on

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 261

Pk+1; we prefer to work with the latter formulation since this point of view is in linewith the corresponding result forp-adic symmetric spaces (see Theorem4.5below).

Let us briefly recall the notion of distributions and measures on a profinite setS.Let A be an abelian group. AnA-valued distributionis a map

µ : {U ⊆ S | U compact and open} −→ A

which is additive for finite disjoint unions of compact open sets:

µ( ·⋃

i∈I

Ui

)=

∑i∈I

µ(Ui ).

We denote by D(S, A) the abelian group ofA-valued distributions onS and byC∞(S, A) the group of locally constant functionsS→ A. Any f ∈ C∞(S, A) canbe integrated against distributions, and the map

D(S, A) −→ Hom(C∞(S, Z), A

), µ 7→

(f 7→

∫S

f (s) dµ)

(11)

is an isomorphism.If K is a p-adic field with absolute value| · | and valuation ringO, then a

boundedK -valued distributionµ is called ameasure(i.e., there exists aC > 0 suchthat |µ(U )| ≤ C for all compact openU ⊆ S). We denote by M(S, K ) the spaceof K -valued measures onS. Under map (11) for A = K , M(S, K ) is mapped ontoHom

(C∞(S, Z), O

)⊗O K . Of course, ifS is finite, anyK -valued distribution is a

measure and D(S, K ) can be identified with the space of functionsS→ K .

LEMMA 3.2Let k≥ 1, let S be a profinite set, and letι : Y· ↪→ E·(S) be a closed simplicial subsetof E·(S) with Yr = Sr+1 for all r ≤ k−1. Then there exists a canonical isomorphism

Hom(Hk−1(|Y·|, Z), K

)∼= D(Sk+1, K )/

(ι∗(D(Y(P,k)

k , K ))+ ∂k∗ (D(Sk+2, K ))

).

Here|Y·| denotes the geometric realization of Y·.

ProofConsider the commutative diagram

0 −−−−→ Z −−−−→ C∞(S1, Z) −−−−→ · · ·∂r−1

−−−−→ C∞(Sr+1, Z) −−−−→ · · ·y y y0 −−−−→ Z −−−−→ C∞(Y0, Z) −−−−→ · · ·

∂r−1

−−−−→ C∞(Yr , Z) −−−−→ · · ·(12)

262 ADRIAN IOVITA AND MICHAEL SPIESS

where the rows are complexes. The differentials∂r are given by

∂r ( f ) =

r+1∑i=0

(−1)i f ◦ pr+1i ,

and the vertical maps are induced by the inclusionι. They are isomorphisms forr ≤k − 1 and surjective for allr (see [SS, Sec. 3, remark at the bottom of p. 67]). By[SS, Sec. 3, remark on p. 66] the cohomology groups of the rows are isomorphicto the reduced cohomology groups of|E·(S)| and|Y·|, respectively. Since|E·(S)| iscontractible, the upper row is exact. A diagram chase shows that the differential

∂k−1: C∞(Yk−1, Z) = C∞(Sk, Z) −→ C∞(Sk+1, Z)

induces an isomorphism

Hk−1(Y·, Z) ∼= Ker(C∞(Sk+1, Z)

(ι∗,∂k)−→ C∞(Yk, Z)⊕ C∞(Sk+2, Z)

).

Dualising it gives the assertion.

Thus the isomorphism (10) can be interpreted as a map

8 = 8P : D(Pk+1, K )/ D(Pk+1, K )deg−→ Hk(XP), (13)

where we have set D(Pk+1, K )deg := Im(ι∗) + Im(∂k∗ ). If P ′⊆P, then we get a

natural map from the spectral sequence (9) for P ′ to the one forP. Consequently,the diagram

D((P ′)k+1, K

)/ D

((P ′)k+1, K

)deg

8−−−−→ Hk(XP

′)y yD(Pk+1, K )/ D(Pk+1, K )deg

8−−−−→ Hk(XP)

(14)

commutes. We now give an explicit description of8 in terms of logarithmic differen-tial forms. Let(H0, . . . , Hn) be an (n+1)-tuple of hyperplanes inPd. For 0≤ i, j ≤ nthe ratio`H j /`Hi defines a rational function onPd which is regular onXP and is welldefined up to a unit inK . Hence the differential form

dlog

(`H j

`Hi

)= d

(`H j

`Hi

)/`H j

`Hi

is independent of the choice of`Hi , `H j .

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 263

Definition 3.3The logarithmicn-form of (H0, . . . , Hn) is defined as

dlog(H0, . . . , Hn) := dlog

(`H1

`H0

)∧ · · · ∧ dlog

(`Hn

`H0

).

It has the following properties:

dlog(Hσ(0), . . . , Hσ(n)) = sign(σ )dlog(H0, . . . , Hn) (15)

for every permutationσ of {0, . . . , n}, and

n+1∑i=0

(−1)i dlog(H0, . . . , Hi , . . . , Hn+1) = 0. (16)

In fact, by using (15), one shows that the sum is equal to d(dlog(H0, . . . , Hn+1)

)and

hence equal to zero since all logarithmic forms are closed. Moreover,

dlog(H0, . . . , Hn) = 0 if `H0, . . . , `Hn are linearly dependent. (17)

THEOREM 3.4For k ≥ 1 the isomorphism8P is given by the formula

8([µ]

)=

∫Pk+1

[dlog(H0, . . . , Hk)

]dµ.

Here the brackets[ ] denote, respectively, the class of the distribution and of the form.Consequently,8P factors through�k(XP)d=0 = 0(XP , �k)d=0 and any elementin Hk(XP) has a canonical representative in�k(XP)d=0.

ProofClearly, it is enough to prove the assertion for a point measureµ = ε(H0,...,Hk) ofa (k + 1)-tuple(H0, . . . , Hk) in P. Moreover, since8 is functorial with respect toinclusionsP ′⊆P (diagram (14) above) and because of (17), we may assume thatP = {H0, . . . , Hk} and`H0, . . . , `Hk are linearly independent (hencek ≤ d).

As explained at the end of Section 2, the isomorphism8P can also be evaluatedusing the spectral sequence (5) for the set of closed subschemesP of Pd,

E−r,s1 =

{ ⊕0≤i0<···<ir≤k Hs

Hi0∩···∩Hir(Pd) if r, s ≥ 0,

0 otherwise

}=⇒ Hs−r

∪P (Pd), (18)

264 ADRIAN IOVITA AND MICHAEL SPIESS

where by (3.1) we have

E−r,s1 =

{ ⊕0≤i0<···<ir≤k K if s is even and max

(2, 2(r + 1)

)≤ s ≤ 2d,

0 otherwise.

In particular,E−k,2k1 = 0 and

E−(k−1),2k1 =

k⊕i=0

H2kH0∩···∩Hi∩···∩Hk

(Pd) ∼=

k⊕i=0

K =k⊕

i=0

K ei ,

wheree0, . . . , ek denote the standard basis ofK k+1. The basis elementei correspondsto the image of 1 under the canonical isomorphism

K∼=−→ H2k(Pd)

∼=←− H2k

H0∩···∩Hi∩···∩Hk(Pd). (19)

Recall that the first map is given by 17→ ξk whereξ is the image of the canonicalline bundle onPd under the cycle map Pic(Pd)→ H2(Pd).

It is easy to keep track of the image ofµ = ε(H0,...,Hk) under the sequence ofmaps

D(Pk+1, K ) −→ Hom(Hk−1(Y(P,k)

· , Z), K)∼= E−(k−1),2k

2

∼= E−(k−1),2k2 ⊆ E−(k−1),2k

1 =

k⊕i=0

K ei . (20)

The element on the right-hand side corresponding toε(H0,...,Hk) is µ :=∑k

i=0(−1)i ei .

Let Ad:= Pd

− H0, and puthi := Hi ∩ Ad for i = 1, . . . , k. We need thefollowing:

The composition of (19) with the restriction map

K −→ H2kH1∩···∩Hk

(Pd) −→ H2kh1∩···∩hk

(Ad) (21)

is given by1 7→ δ([dlog(H0, H1)]

)∪ · · · ∪ δ

([dlog(H0, Hk)]

).

Indeed, it follows immediately from the definition of the cycle map Pic(Pd)

→ H2(Pd) for de Rham cohomology that under the isomorphism

H2(Pd)∼=←− H2

Hi(Pd)

∼=−→ H2

hi(Ad),

ξ is mapped ontoδ([dlog(H0, Hi )]

)and consequently 17→ ξk

7→ δ([dlog(H0,

H1)])∪ · · · ∪ δ

([dlog(H0, Hk)]

)under (21).

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 265

The spectral sequence (5) for thek-tuple of hyperplanesA = (h1, . . . , hk) in Ad

has a very simple shape. For an (r + 1)-tuple 1≤ i0 < · · · < i r ≤ k we have

Hshi0∩···∩hir

(Ad) =

{K if s= 2(r + 1),

0 otherwise.

Therefore the spectral sequence degenerates and we get an isomorphism⊕1≤i1<···<ir≤k

H2rhi1∩···∩hir

(Ad) −→ H r+1∪A (Ad)

for everyr ≥ 0. Consider the diagram

Hk(XP)δ

−−−−→ Hk+1∪P (Pd) −−−−→ E−(k−1),2k

2yid

y yHk(XP)

δ−−−−→ Hk+1

∪A (Ad) −−−−→ H2kh1∩···∩hk

(Ad)

(22)

The vertical maps are induced by the inclusionAd ↪→ Pd. The right horizontal mapsare induced by the spectral sequence (5) for (Pd, P) and(Ad, A ), respectively. Thecomposition of the two upper horizontal maps is an isomorphism. Its inverse is8P

if we identify E−(k−1),2k2 with

E−(k−1),2k2 = Hom

(Hk−1(Y(P,k)

· , Z), K)∼= D(Pk+1, K )/ D(Pk+1, K )deg.

The right vertical map is induced by the map

k⊕i=0

H2kH0∩···∩Hi∩···∩Hk

(Pd)proj−→ H2k

H1∩···∩Hk(Pd) −→ H2k

h1∩···∩hk(Ad),

and therefore it sendsµ to the image of 1 under (21), that is, toδ([dlog(H0, H1)]

)∪

· · · ∪ δ([dlog(H0, Hk)]

).

The right lower horizontal map in (22) is just the inverse ofek+1A , and the left

lower horizontal map is bijective. Using the commutativity of (22) and Proposition2.3, we obtain

8([µ]

)= δ−1((ek+1

A )−1(δ([dlog(H0, H1)]) ∪ · · · ∪ δ([dlog(H0, Hk)])))

=[dlog(H0, H1) ∪ · · · ∪ dlog(H0, Hk)

]=

[dlog(H0, . . . , Hk)

]=

∫Pk+1

[dlog(H0, . . . , Hk)

]dµ.

Remark.It is easy to see that the image of

D(Pk+1, K ) −→ �k(XP), µ 7→

∫Pk+1

dlog(H0, . . . , Hk) dµ

266 ADRIAN IOVITA AND MICHAEL SPIESS

is the space generated by the logarithmick-forms on XP , which we denote by�k

log(XP). By (16) and (17) it factors through D(Pk+1, K )/ D(Pk+1, K )deg, andTheorem3.4 implies that the induced map

D(Pk+1, K )/ D(Pk+1, K )deg−→ �klog(XP)

is an isomorphism and when composed with�klog(XP) → Hk(XP), ω 7→ [ω] is

8P . Therefore every cohomology class has a unique representative in�klog(XP).

This is an old theorem of E. Brieskorn [Br].

4. The main theoremFrom now on, letK be ap-adic field, that is, a finite extension ofQp. Let | · | denoteits valuation, and letO denote its valuation ring. We fix a prime elementπ ∈ O. LetCp be the completion of a fixed algebraic closure ofK .

We start this section with some remarks about the integration theory on profinitesets and about Frechet spaces which are needed later. LetF be a topologicalK -vector space.F is calledcompleteif it is sequentially complete, that is, every Cauchysequence inF has a limit. A complete topologicalK -vector spaceF is calledFrechetspaceif there exists a sequence of seminorms{ρn}n which defines the topology onF , that is, a basis of open neighbourhoods of zero is given by the sets{x ∈ F |

ρn(x) ≤ ε for all n ≤ N } for ε > 0 andN a positive integer. IfF is a Frechet spaceandU a closed subspace, then the quotient spaceF/U is a Frechet space too. Also,the inverse limitF = lim

←−Fi of a countable inverse system of Frechet spaces with

the inverse limit topology is a Frechet space.Let S be a profinite set. The spaces of distributions D(S, K ) and measures

M(S, K ) have a natural structure as topological vector spaces. In fact, if we writeS = lim

←−Si for an inverse system of finite sets{Si , Sj → Si }, then we have

D(S, K ) = lim←−

D(Si , K ). Every D(Si , K ) is finite-dimensional and thus a Banach

space. We provide D(S, K ) with the inverse limit topology. IfS can be written as aninverse limit of a countable inverse system of finite sets—which we assume from nowon—then D(S, K ) is even a Frechet space. The point measuresεs, s ∈ S generate adense subspace in D(S, K ). The topology on M(S, K ) is defined as follows. For eachintegern we provide D(S, π−nO) with its topology as a subspace of D(S, K ) andM(S, K ) = lim

−→D(S, π−nO) with the direct limit topology. Note that this topology is

in general finer than the topology induced by the inclusion M(S, K ) ⊆ D(S, K ). Nev-ertheless, the point measuresεs, s ∈ S still generate a dense subspace of M(S, K ).We need the following elementary lemma from the theory ofp-adic integration whoseproof is left as an exercise to the reader.

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 267

LEMMA 4.1(a) Let F be a Frechet space, and let f: S→ F be a continuous map. Then

there exists a unique continuous K-linear map∫f : M(S, K ) −→ F , µ 7→

∫S

f (s) dµ

such that∫

S f (s) dεt = f (t) for all point measuresεt , t ∈ S.(b) Assume thatF := lim

←−Fi for a countable inverse system of Frechet spaces

{Fi , F j → Fi }, and assume that the composition Sf−→ F

proj−→ Fi is

locally constant for every i . Then f can be integrated against distributions aswell; that is, (a) remains true if we replaceM(S, K ) byD(S, K ).

Let V denote the category of smooth separated rigid analytic varieties overK . For apair (X,U ) consisting of a varietyX in V and an admissible open subvarietyU ⊆ X,we denote byH•(X) (respectively,H•(X,U )) the de Rham cohomology ofX (re-spectively, of the pair(X,U )). Any smooth separatedK -varietyX gives rise to a rigidanalytic varietyXan, and there is a canonical morphism

H•(X) −→ H•(Xan) (23)

from algebraic to rigid analytic de Rham cohomology which is an isomorphism ifX isprojective by the rigid analytic GAGA (geometrie algebrique et geometrie analytique)principle (see [Ki1]). More generally, ifY is a closed subscheme ofX and U itscomplement, then we have a canonical morphism

H•Y(X) −→ H•(Xan,Uan). (24)

Let H = Pd((K d+1)∗

)be the set of allK -rational hyperplanes inPd. Let X =

X (d+1)= Pd

/K −⋃

H∈H H denote Drinfeld’sp-adic symmetric space. We take over

some notation and recall some facts from [SS, Sec. 1]. Forz ∈ Pd(K ), a coordinaterepresentationz = [z0, . . . , zd] is always assumed to be unimodular, that is,|zi | ≤ 1for 0 ≤ i ≤ d and |zi | = 1 for somei . Similarly, for any hyperplaneH ∈ H

we consider now only those functionals`H defining it which are unimodular, that is,`H ∈ (Od+1)∗ and`H 6≡ 0 mod π . Let

redn : Pd(K )→ Pd(O/πn), H 7→ line through(`H mod πn)

be the reduction map modπn. Forε > 0, let H(ε) = {z ∈ Pd(Cp) | |`H (z)| ≤ ε} bethe epsilon neighbourhood ofH ∈H . Recall that

H ≡ H ′mod πn⇐⇒ redn(H) = redn(H ′)⇐⇒ H

(|πn|)= H ′

(|πn|).

268 ADRIAN IOVITA AND MICHAEL SPIESS

For n ≥ 1, let Xn be the admissible open subvarietyPd/K −

⋃H∈H H(|πn

|) of

Pd/K . The increasing sequence of open subvarieties. . . ⊆Xn⊆Xn+1⊆ . . . X is an

admissible covering ofX . For realρ > 0 we further setX ρ = {z ∈ Pd(Cp) |

|`H (z)| ≥ |π |ρ for all H ∈ H }. As in [SS, Sec. 1], one shows thatX ρ is an openaffinoid subvariety ofPd.

The spacesX , Xn are Stein spaces. ForX this is proved in [SS, Sec. 1, Prop.1.4], where it is shown that the sequence{X n} is an admissible covering with thedefining property of a Stein space (see [Ki2, Def. 2.3]). Similarly, one can show that{X n−(1/m)}m∈N is such an admissible covering forXn. The spaces ofK -valued holo-morphick-forms�k(X ), �k(Xn) carry naturally the structure of Frechet spaces, andwe have�k(X ) ∼= lim

←−�k(Xn) (as Frechet spaces).

Indeed, letX ∈ V be an arbitrary Stein space, and letM be a freeOX-moduleof finite rank, that is,M = V ⊗K OX for a finite-dimensionalK -vector spaceV .Fix a norm‖ · ‖ on V . For every open affinoid subvarietyU ∈ X we consider thesupremums norm

‖ f ‖U = sup{‖ f (x)‖ | x ∈ U

}on M (U ). Then the set of seminormsρU ( f ) := ‖ f |U‖U , whereU runs through allaffinoid open subvarieties ofX, defines a topology on the set of global sectionsM (X)

of M giving it the structure of a Frechet space. It is easy to see that the topology isindependent of the choice of the trivialisationM = V ⊗K OX .

LEMMA 4.2(a) The differentialsd : �k(X ) → �k+1(X ), d : �k(Xn) → �k+1(Xn) are

continuous, and their images are closed in�k+1(X ) and�k+1(Xn), respec-tively.

(b) By (a) we can equip Hk(X ) and Hk(Xn) with the quotient topology of�k(X )d=0 and �k(Xn)d=0, respectively. They are Frechet spaces, and thecanonical map

Hk(X ) −→ lim←−

Hk(Xn)

is an isomorphism of topological vector spaces.

Proof(a) This follows from [G-K, Lem. 7.6].(b) The mapsHk(X ) → Hk(Xn) induced by the restrictionsω 7→ ω|Xn are

clearly continuous, and thus

Hk(X ) −→ lim←−

Hk(Xn)

is continuous as well. According to [SS, Sec. 3, remarks before Lem. 2] it isan isomorphism. By Banach’s theorem its inverse is continuous too.

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 269

For k ≥ 1, letY(k)· be the profinite simplicial subset ofE·(H ) given by

Y(k)r =

{(H0, . . . , Hr ) ∈H r+1

∣∣∣ dim( r∑

i=0

K`Hi

)≤ k

}.

In [SS, Sec. 3] Schneider and Stuhler have defined a PGLd+1(K )-equivariant isomor-phism

Hom(Hk−1(|Y(k)

· |, Z), K)−→ Hk

d R(X ). (25)

According to (3.2) we can replace the vector space on the left by D(H k+1, K )/

D(H k+1, K )deg; that is, we have an isomorphism

8 : D(H k+1, K )/ D(H k+1, K )deg−→ Hk(X ) (26)

where, as usual,ι : Y(k)· → E·(H ) denotes the inclusion and D(H k+1, K )deg :=

Im(ι∗)+ Im(∂k∗ ).

LEMMA 4.3(a) The map8 is continuous (and therefore a homeomorphism).(b) LetP be a finite subset ofH . Then the diagram

D(Pk+1, K )/ D(Pk+1, K )deg8P−−−−→ Hk(XP)y y

D(H k+1, K )/ D(H k+1, K )deg8

−−−−→ Hk(X )

(27)

commutes. Here the first vertical arrow is induced by the inclusionPk+1→

H k+1, whereas the second one is given by the composition of the compari-son map Hk(XP) → Hk(XP

an) with the canonical map Hk(XPan) →

Hk(X ).

Proof(a) By Lemma4.2 it is enough to prove that the composition

D(H k+1, K )proj−→ D(H k+1, K )/ D(H k+1, K )deg

8−→ Hk(X ) −→ Hk(Xn)

(28)is continuous. As in [SS, Sec. 3], letHn = Pd

((Od+1/πnOd+1)∗

), and letY(n,k)

· bethe simplicial subscheme ofE·(Hn) given by

Y(n,k)r =

{(H0, . . . , Hr ) ∈H r+1

n

∣∣∣ rank( r∑

i=0

(O/πnO)`Hi

)≤ k

}.

270 ADRIAN IOVITA AND MICHAEL SPIESS

Here`Hi ∈ (Od+1/πnOd+1)∗ now denotes a representative ofHi ∈ Hn, and therank of a finitely generatedO/πnO-module is defined as the minimal number ofgenerators. A careful analysis of the proof of [SS, Th. 1] reveals that we have homo-morphisms

Hom(Hk−1(|Y(n,k)

· |, Z), K)−→ Hk

d R(Xn) (29)

and that (25) is actually the inverse limit of them. Again the left-hand side of (29)can be identified with the corresponding quotient of D(H k+1

n , K ), and we obtain acommutative diagram

D(H k+1, K )8◦proj−−−−→ Hk(X )y y

D(H k+1n , K ) −−−−→ Hk(Xn)

(30)

Since the vector spaces D(H k+1n , K ) andHk(Xn) are finite-dimensional, the lower

horizontal map is continuous. The first vertical arrow is continuous for trivial reasons.Therefore (28) is continuous as well.

(b) This follows immediately from the functorial properties of the comparisonhomomorphism (23) and of the spectral sequence [SS, Sec. 2, (*)].

For a (k+ 1)-tuple(H0, . . . , Hk) of K -rational hyperplanes inPd/K we now consider

the logarithmick-form dlog(H0, . . . , Hk) as a rigid analytic differential form onX .The following lemma allows us to define the integrals

∫H k+1 dlog(H0, . . . , Hk) dµ

(resp.,∫H k+1[dlog(H0, . . . , Hk)] dµ) for a measure (resp., a distribution)µ on

H k+1.

LEMMA 4.4(a) The map

H k+1−→ �k(X ), (H0, . . . , Hk) 7→ dlog(H0, . . . , Hk)

is continuous.(b) For every n, the map

H k+1−→ Hk(Xn), (H0, . . . , Hk) 7→

[dlog(H0, . . . , Hk)

]is locally constant.

ProofBecause of

dlog(H0, . . . , Hk) = dlog(H0, H1) ∧ · · · ∧ dlog(H0, Hk),

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 271

it suffices to consider the casek = 1. Let Z0 be the hyperplane{z0 = 0}. Sincedlog(H0, H1) = dlog(Z0, H1)− dlog(Z0, H0), it is enough to show that for everynwe have the following:(a′) the mapH −→ �1(X n), H 7→ dlog(Z0, H) is continuous;(b′) the mapH −→ H1(Xn), H 7→ [dlog(Z0, H)] is locally constant.

For ` ∈ (Od+1)∗ − π(Od+1)∗, let f`(z) := `(z)/z0. Then (Od+1)∗ −

π(Od+1)∗ → O(X n), ` 7→ f` is continuous and|πn| ≤ ‖ f`‖ ≤ |π−n

|, where‖ · ‖ denotes the supremums norm onO(X n). As

(Od+1)∗ − π(Od+1)∗ −→ �1(X n), `(z) =d∑

i=0

ai zi 7→ d( f`) =d∑

i=1

ai d( zi

z0

)is obviously continuous too, we conclude that` 7→ d f`/ f` is continuous as well. Thisproves(a′).

For (b′), note that‖1 − ( f`0/ f`1)‖ < 1 if `0, `1 ∈ (Od+1)∗ − π(Od+1)∗ aresufficiently close to each other. Sincez 7→ log(z) is analytic on the open unit discwith center 1, we have[d f`1

f`1

]−

[d f`0

f`0

]=

[d( f`1/ f`0)

f`1/ f`0

]=

[d(log( f`1/ f`0))

]= 0

in H1(Xn).

We are now in the position to prove our main result.

THEOREM 4.5Let k be a positive integer. Then the isomorphism

8 : D(H k+1, K )/ D(H k+1, K )deg−→ Hk(X )

is given explicitly by the formula

8([µ]) =

∫H k+1

[dlog(H0, . . . , Hk)

]dµ.

If µ is a measure onH k+1, then the integral∫H k+1

dlog(H0, . . . , Hk) dµ

is well defined. Therefore, when restricted toM(H k+1, K )/ M(H k+1, K )deg, themap8 factors as

M(H k+1, K )/ M(H k+1, K )deg−→ �k(X )d=0 −→ Hk(X ),

[µ] 7→

∫H k+1

dlog(H0, . . . , Hk) dµ 7→

[∫H k+1

dlog(H0, . . . , Hk) dµ

],

272 ADRIAN IOVITA AND MICHAEL SPIESS

whereM(H k+1, K )deg := M(H k+1, K ) ∩ D(H k+1, K )deg = ι∗(

M(Y(k)k , K )

)+

∂k∗

(M(H k+2, K )

).

ProofBy Lemmas4.1, 4.2, and4.4, the homomorphisms

D(H k+1) −→ Hk(X ), µ 7→

∫H k+1

[dlog(H0, . . . , Hk)

]dµ

M(H k+1) −→ �k(X ), µ 7→

∫H k+1

dlog(H0, . . . , Hk) dµ (31)

are well defined and continuous, and by (16) and (17) they factor throughD(H k+1, K )/ D(H k+1, K )deg and M(H k+1, K )/ M(H k+1, K )deg, respectively.Since8 and (31) are both continuous, it is enough to see that they agree on a pointmeasureε(H0,...,Hk) for (H0, . . . , Hk) ∈ H k+1

− Y(k)k . By using Lemma4.3 above,

this can be deduced from the corresponding statement Theorem3.4for complementsof finitely many hyperplanes.

Definition 4.6We denote the image of the PGLd+1(K )-equivariant homomorphism

M(H k+1, K ) −→ �k(X )d=0, µ 7→

∫H k+1

dlog(H0, . . . , Hk) dµ

by �klog,b(X ) and call it the space of bounded logarithmic differentialk-forms onX .

Theorem4.5 implies that�klog,b(X ) −→ Hk(X ), ω 7→ [ω] is injective and that its

image corresponds to Hom(Hk−1(Y(P,k)

· , Z), O)⊗O K under the inverse of (25).

5. Hodge decompostion for de Rham cohomology ofp-adically uniformized va-rieties

Let0⊆ PGLd+1(K ) be a discrete cocompact subgroup acting without fixed points onX . ThenX0 := 0\X is a proper smooth rigid analytic variety overK (in fact, it iseven algebraic; compare [Mu]) andX → X0 is anetale covering. LetM be a finite-dimensionalK [0]-module (i.e., finite-dimensional as aK -vector space).M gives riseto a local systemM on theetale site ofX0 (see [Sch]), and we can consider the deRham cohomology groupsH•(X0, M ) of X0 with values inM . They are definedas theetale hypercohomology groupsH•et(X0, �· ⊗K M ) and can be identified withthe hypercohomology groupsH•(0, �·(X )⊗K M) sinceX is a Stein space. If0 istorsion-free or ifM = K , then the de Rham cohomology groups can be computed inthe usual rigid analytic cohomology. In this case they are defined even algebraicallyby the GAGA theorems of R. Kiehl [Ki1].

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 273

According to [SS, Sec. 5, Prop. 2] and [Sch, Sec. 1] we have a covering spectralsequence

Er,s2 = H r (0, Hs(X )⊗K M

)=⇒ H r+s(X0, M ). (32)

In [Sch, Sec. 1] it is shown that it degenerates. This result is then used to computethe dimensions of the de Rham cohomology groups. It turns out that the only in-teresting cohomology is the middle cohomology groupHd(X0, M ). We denote byHd(X0, M ) = F0

0 ⊇ F10 ⊇ · · · ⊇ Fd+1

0 = 0 the filtration induced by (32). Then wehave ford ≥ 2, i = 0, 1, . . . , d + 1 (see [Sch, Th. 2]),

dim(F i0) =

{(d + 1− i )µ(0, M) if d is odd or 2i > d,(d + 1− i )µ(0, M)+ dimK (M0) if d is even and 2i ≤ d.

(33)

Hereµ(0, M) denotes the dimension ofHd(0, M). From (33) we deduce

dim(F i0)+ dim(Fd+1−i

0 ) = dim(Hd(X0, M )

). (34)

There is a second natural filtration onHd(X0, M ), the Hodge filtration. It isinduced by the spectral sequence

Er,s1 = Hs

et(X0, �r⊗K M ) =⇒ H r+s(X0, M ). (35)

We denote it byHd(X0, M ) = F0H ⊇ F1

H ⊇ · · · ⊇ Fd+1H = 0. It is conjectured (com-

pare [SS, remark after Th. 5] and [Sch, Sec. 2]) that the two filtrations are opposite,that is, that

Hd(X0, M ) = F iH ⊕ Fd+1−i

0 (36)

holds for everyi , 0 ≤ i ≤ d + 1. In this section we prove this ifM is integral, thatis, if M contains a0-stable lattice. Ford ≤ 2 this is already known (see [Sch, Sec. 2,Cor. 6]). Therefore we can and do assume from now on thatd is at least 2.

To begin with, we state two simple results from homological algebra whoseproofs are straightforward and are left to the reader.

LEMMA 5.1Let R be a commutative ring with1, let V, M be R[0]-modules, and assume that Vis free as an R-module. Then

H•(0, HomR(V, M)

)∼= Ext•R[0](V, M).

LEMMA 5.2Let R, V, M be as above, and assume moreover that V has a resolution F· → Vwhere for every n≥ 0, Fn is a finitely generated free R[0]-module. Let S be a flatR-algebra. Then the canonical map

Ext•R[0](V, M)⊗R S−→ Ext•S[0](V ⊗R S, M ⊗R S)

274 ADRIAN IOVITA AND MICHAEL SPIESS

is an isomorphism.

The following proposition, which is the key ingredient in the proof of (36) for in-tegral M , is based on the existence of certain nice resolutions of the0-modulesHk−1(|Y(k)

· |, Z) which is proved in [SS, Sec. 6].

PROPOSITION5.3Let M be an integral K[0]-module. Then the canonical map�k

log,b ⊗K M →

Hk(X )⊗K M induces an isomorphism on cohomology groups

H•(0, �klog,b ⊗K M) −→ H•

(0, Hk(X )⊗K M

).

Consequently, the canonical map of hypercohomology groups

H•(0, �·log,b(X )⊗K M

)−→ H•

(0, �·(X )⊗K M

).

is an isomorphism too.

ProofAs remarked at the end of the last section, the map�k

log,b(X )⊗K M → Hk(X )⊗K

M corresponds under isomorphism (25) to the inclusion

Hom(V, O)⊗O M −→ Hom(V, K )⊗K M, (37)

whereV := Hk−1(|Y(k)· |, Z). For any ringR we putV(R) := V ⊗Z R. In [SS, Sec.

6] it is proved thatV admits a resolution by finitely generated freeZ[0]-modules. LetL be a0-invariant lattice inM . Since Hom(V, O)⊗O M = HomO

(V(O), L

)⊗O K

and Hom(V, K )⊗K M = HomK(V(K ), L⊗ K

), we have to show that the inclusion

HomO(V(O), L

)⊗O K −→ HomK

(V(K ), L ⊗ K

)induces isomorphism on cohomology. This now follows by consecutive applicationof Lemmas5.1and5.2above:

H•(0, HomO(V(O), L)⊗O K

)= Ext•K [0]

(K , HomO(V(O), L)⊗O K

)∼= Ext•O[0]

(O, HomO(V(O), L)

)⊗O K

∼= Ext•O[0](V(O), L

)⊗O K

∼= Ext•K [0](V(K ), L ⊗O K

)∼= H•

(0, HomK (V(K ), L ⊗O K

), (38)

and we get the first assertion. Note that to deduce (38) from Lemma5.2we have to usethe above-mentioned result about the existence of a resolution by finitely generatedfreeZ[0]-modules again, this time for the trivial0-moduleZ (the casek = 0).

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 275

The second assertion follows from the first one by considering the map of thehypercohomology spectral sequences associated with the inclusion�·log,b(X ) ⊗K

M −→ �·(X )⊗K M of complexes of0-modules.

THEOREM 5.4Let M be an integral finite-dimensional K[0]-module. Then for i= 0, 1, . . . , d + 1we have

Hd(X0, M ) = F iH ⊕ Fd+1−i

0 .

Proof

For a complexC· = (C0 d−→ C1 d

−→ · · · ) we introduce the following notation forvarious truncated complexes:

t≤i C·:= (C0

−→ · · · −→ Ci−1−→ Ker(d) −→ 0−→ · · · ),

C·≤i := (C0

−→ · · · −→ Ci−1−→ Ci

−→ 0−→ · · · ),

t≥i C·:= (0−→ · · · −→ 0−→ Coker(d) −→ Ci+1

−→ · · · ),

C·≥i := (0−→ · · · −→ 0−→ Ci

−→ Ci+1−→ · · · ).

We have

F iH = Im

(Hd(0, �·

≥i (X )⊗K M) −→ Hd(0, �·(X )⊗K M))

and

Fd+1−i0 = Im

(Hd(0, t≤i−1�

·(X )⊗K M) −→ Hd(0, �·(X )⊗K M)).

The inclusion map�·log,b(X )⊗K M → �·(X )⊗K M factors through�·≥i (X )⊗K

M+t≤i−1�·(X )⊗K M since all differentials in�·log,b(X )⊗K M are zero. Therefore

we obtain

F iH + Fd+1−i

0 ⊇ Im(Hd(0, �·log,b(X )⊗K M) −→ Hd(0, �·(X )⊗K M)

),

and by Proposition5.3we get

Hd(X0, M ) = F iH + Fd+1−i

0 . (39)

Together with (34) it follows that

dim(F iH ) ≥ dim

(Hd(X0, M )

)− dim(Fd+1−i

0 ) = dim(F i0), (40)

and it remains to prove equality to see that the sum in (39) is direct.

276 ADRIAN IOVITA AND MICHAEL SPIESS

For that, consider the commutative diagram

Hdet(X0, �·

≥i ⊗K M )×Hdet(X0, �·

≤d−i ⊗K M ∗)∪

−−−−→ Ky x yid

Hd(X0, M ) × Hd(X0, M ∗)∪

−−−−→ K

Both pairings are nondegenerate by Serre duality. (Strictly speaking, we can applySerre duality only if the sheaves involved are algebraic; this can be achieved by pass-ing from0 to a normal torsion-free subgroup0′ of finite index. By then taking0/0′-invariants, one sees that the pairings are nondegenerate in the general case as well.)

The diagram shows thatF iH

(Hd(X0, M )

)⊥= Fd+1−i

H

(Hd(X0, M ∗)

)and

therefore

dim(F i

H (Hd(X0, M )))= dim

(Hd(X0, M ∗)

)− dim

(Fd+1−i

H (Hd(X0, M ∗)))

≤ dim(Hd(X0, M ∗)

)− dim

(Hd(X0, M ∗)

)+ dim

(F i

0(Hd(X0, M ∗)))

= dim(F i

0(Hd(X0, M ))).

For the last two (in-)equalities, note thatM∗ = Hom(M, K ) is also integral—so wecan apply (40)—and that computation (33) holds forF i

0

(Hd(X0, M ∗)

)as well since

µ(0, M) = µ(0, M∗). If M0= 0, this is [Sch, Sec. 1, Prop. 2 ]. The general case

can be deduced from it because we have a0-stable decompositionM = K r⊕M ′ for

somer ≥ 0 with M ′0 = 0 as it is explained in the proof of [Sch, Sec. 1, Cor. 2].

As remarked in [Sch], we obtain a Hodge-type decomposition forHd(X0, M ) fromTheorem5.4.

COROLLARY 5.5We have

Hd(X0, M ) =

d⊕i=0

F i0 ∩ Fd−i

H

with

dim(F i0 ∩ Fd−i

H ) =

{µ(0, M) if 2i 6= d,µ(0, M)+ dimK (M0) if 2i = d.

In fact, the proof of Theorem5.4shows that

F i0 ∩ Fd−i

H∼= H i (0, �d−i

log,b(X )⊗K M).

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES 277

Remark.By using a more refined version ofp-adic integration than Lemma4.1, itis likely that one can prove conjecture (36) for other types of0-representations aswell. In the one-dimensional case J. Teitelbaum has proved (36) for certain algebraicrepresentations, that is, representations coming from a representation of PGLd+1(K )

by using ap-adic integration theory for certain distributions (see [Te]). In their recentpaper [ST2], Schneider and Teitelbaum have developed an integration theory thatenables them to integrate locally analytic functions against a very general class ofdistributions. We hope that this theory will be useful to prove (36) for algebraic0-representations and plan to return to this question in a later work.

Acknowledgment.We want to thank P. Schneider for useful comments.

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[Sch] P. SCHNEIDER, The cohomology of local systems on p-adically uniformized varieties,Math. Ann.293(1992), 623–650.MR 93k:14032 272, 273, 276

[SS] P. SCHNEIDERandU. STUHLER, The cohomology of p-adic symmetric spaces, Invent.Math.105(1991), 47–122.MR 92k:11057 253, 254, 258, 259, 260, 262, 267,268, 269, 270, 273, 274

[ST1] P. SCHNEIDERandJ. TEITELBAUM, An integral transform for p-adic symmetricspaces, Duke Math. J.86 (1997), 391–433.MR 98c:11048 254

[ST2] , Locally analytic distributions and p-adic representation theory, withapplications toGL2, preprint,arXiv:math.NT/9912073277

278 ADRIAN IOVITA AND MICHAEL SPIESS

[Te] J. TEITELBAUM, Values of p-adic L-functions and a p-adic Poisson kernel, Invent.Math.101(1990), 395–410.MR 91m:11034 254, 277

IovitaDepartment of Mathematics, University of Washington, Box 355754, Seattle, Washington98105, USA;[email protected]

SpiessSchool of Mathematical Sciences, University of Nottingham, University Park, NG7 2RDNottingham, United Kingdom;[email protected]

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