Sheaves of bounded <span class="mathjax-formula">$p$</span>-adic logarithmic differential forms

4^esérie, t. 40, 2007, p. 351 à 386.

SHEAVES OF BOUNDED p -ADIC LOGARITHMIC DIFFERENTIAL FORMS

B

E

GROSSE-KLÖNNE

ABSTRACT. – LetKbe a local field,Xthe Drinfel’d symmetric space of dimensiondoverKandX the natural formalOK-scheme underlyingX; thusG= GLd+1(K)acts onXandX. Given aK-rational G-representationM we construct aG-equivariant subsheafM⁰_OK˙ ofOK-lattices in the constant sheafM onX. We study the cohomology of sheaves of logarithmic differential forms onX(orX) with coefficients inM⁰_OK˙. In the second part we give general criteria for two conjectures of P. Schneider onp-adic Hodge decompositions of the cohomology of p-adic local systems on projective varieties uniformized by X.

Applying the results of the first part we prove the conjectures in certain cases.

©2007 Elsevier Masson SAS

RÉSUMÉ. – Soient K un corps local, X l’espace symétrique de Drinfel’d de dimension d sur K et XleOK-schéma formel canonique sous-jacent à X; le groupe G= GLd+1(K) agit sur X etX.

Soit M une représentation K-rationnelle. Dans le faisceau constantM sur X, on construit un sous- faisceauG-équivariantM⁰_O˙

KdesOK-réseaux. On s’intéresse à la cohomologie des faisceaux des formes différentielles logarithmiques à coefficients dansM⁰_OK˙. Dans la deuxième partie, on donne des critères généraux pour deux conjectures de P. Schneider sur des décompositions de Hodge p-adiques de la cohomologie des systèmes locauxp-adiques sur des variétés projectives uniformisées parX. En appliquant les résultats de la première partie, on démontre ces conjectures dans certains cas.

©2007 Elsevier Masson SAS

0. Introduction

Letpbe prime number andd∈N, letK/Qpbe a finite extension. In connection with the search for a Langlands type correspondence between suitable p-adically continuous representations of the group GLd+1(K)onp-adic vector spaces on the one hand, and suitable p-adic Galois representations on the other hand, thep-adic cohomology (de Rham, crystalline, coherent,p-adic étale) of Drinfel’d’s symmetric space X overK and its projective quotientsX_Γ= Γ\X, with coefficients in rational representationsM ofGL_d+1(K), has recently found increasing interest.

We mention the first spectacular results due to Breuil [2] who uses the cohomology ofX and X_Γ with coefficients in M = Sym^k(Q²_p) (somek∈N) to establish a partial correspondence in case d= 1, K=Qp, and the work of Schneider and Teitelbaum [17] where (for any d andK) the GLd+1(K)-representation on the space Ω^•_X(X)of top differential forms onX is determined. Substantial as these works are, they call for generalizations. On the one hand one hopes to generalize the constructions from [2] to cases whered >1. Since a decisive ingredient in [2] is the work with p-adic integral structures in equivariant sheaf complexes onX, the investigation of such integral structures should be a starting point. On the other hand one hopes

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

to generalize the analysis of [17] to more general equivariant vector bundles onX(instead of the line bundleΩ^d_X), e.g. to the vector bundlesΩⁱ_X, or evenM⊗Ωⁱ_X, for anyi; this would be done best by finding and analysing equivariant subsheaves inM⊗Ωⁱ_X, e.g. those of exact, closed or logarithmic differential forms. All this motivates the main objective of the first part of this paper, the study of equivariant integral structures in the vector bundlesM⊗Ωⁱ_Xand in their subsheaves of logarithmic differential forms. The central question concerning the de Rham cohomology with coefficients inM of a projective quotientX_ΓofXis that for the position of its Hodge filtration (e.g. due top-adic Hodge theory its knowledge in cased= 1is another crucial point in [2]); the second part of this paper is devoted to this question.

We discuss the content in more detail. Let now more generallyKbe a non-Archimedean local field with ring of integers OK, uniformizer π∈ OK and residue fieldk. LetM be a rational representation ofG= GLd+1(K), i.e. a finite-dimensionalK-vector spaceM together with a morphism ofK-group varietiesGLd+1→GL(M). It is well known that for any compact open subgroupH ofGthere exists anH-stable freeOK-module lattice inM; we fix one such choice M⁰ for H= GLd+1(OK). We choose a totally ramified extension K˙ of K of degreed+ 1 and twist the action ofGonM⊗KK˙ unramified by a suitableK-valued character of˙ G. We show that the choice ofM⁰determines for any other maximal open compact subgroupH⊂G a distinguished H-stable OK˙-lattice in M ⊗K K˙ and the collection of these lattices can be assembled into aG-equivariant coefficient system on the Bruhat–Tits buildingBT ofPGL_d+1. In fact this is only a reinterpretation of our Proposition 3.1. We do not mentionBT at all, we rather work with the G-equivariant semistable formal OK-scheme X underlying Drinfel’d’s symmetric spaceXoverKof dimensiond+ 1, as constructed in [14]. It is well known that the intersections of the irreducible components ofX⊗kare in natural bijection with the simplices of BT. Thus what we do is to construct fromM⁰a constructibleG-equivariant subsheafM⁰_OK˙ of the constant sheaf with valueM⊗KK˙ onXsuch thatM⁰_OK˙(U)for quasicompact openU⊂X is anOK˙-lattice inM⊗KK.˙

We then consider the coherent OX⊗_OK OK˙-module sheaf M⁰_O˙

K⊗_OK OX and compute explicitly its reduction (M⁰_OK˙ ⊗OK OX)⊗OK˙ k. See Theorem 3.3 for our result. Similarly, letΩ^•_Xbe the logarithmic de Rham complex ofXand letLog^s(M⁰_OK˙)be theπ-adic completion of the subsheaf ofM⁰_OK˙ ⊗OKΩ^s_X consisting of logarithmic differentials-forms; we compute explicitlyLog^s(M⁰_OK˙)⊗_OK˙ k. See Theorem 4.4 for our result.

As an application, assume now thatM|SLd+1(K)is the trivial representationK, the standard representationM =K^d+1or its dual(K^d+1)^∗. We show (Proposition 4.5)

H^j

X,Log^s

M⁰_OK˙∼=H^j

X,M⁰_OK˙ ⊗_OKΩ^s_X (1)

for anyjand anys. Using the above computations the proof of (1) is reduced to the statement that for any irreducible componentY of X⊗k—such aY is the successive blowing up ofP^d_k in allk-linear subspaces—with logarithmic de Rham complexΩ^•_Y we haveH^j(Y,Ω^s_Y) = 0 if j= 0, andH⁰(Y,Ω^s_Y)consists of global logarithmic differentials-forms onY.

In the second part of this paper (Sections 5 and 6) we develop general criteria for conjectures of Schneider raised in [16]. LetΓ⊂SLd+1(K)be a cocompact discrete (torsion-free) subgroup;

thus the quotientXΓ= Γ\XofXis a projectiveK-scheme [14]. LetM be aK[Γ]-module with dimK(M)<∞. Using theΓ-action (induced from theΓ-action onM) on the constant local system onXgenerated byM we get a local systemM^ΓonX_Γ. The Hodge spectral sequence

E^r,s₁ =H^s

XΓ,M^Γ⊗KΩ^r_XΓ

⇒H^r+s

XΓ,M^Γ⊗KΩ^•_XΓ (2)

gives rise to the Hodge filtration H^d=H^d

XΓ,M^Γ⊗KΩ^•_XΓ

=F_H⁰ ⊃F_H¹ ⊃ · · · ⊃F_H^d+1= 0.

If char(K) = 0 Schneider [16] conjectures that a splitting of F_H^• is given by the covering filtration F_Γ^• of H^d arising from the expression of H^d through the Γ-group cohomology of M⊗KH_dR^∗ (X). Concretely, he expectsH^d=F_Hⁱ⁺¹⊕F_Γ^d−ifor0id−1.

If M underlies a K-rational representation of G (and char(K) = 0), Schneider defines another sheaf complex, quasiisomorphic with M^Γ ⊗K Ω^•_XΓ, hence again a corresponding Hodge filtrationF_red^• onH^d. He then conjecturesF_red^• =F_H^•, thus in particular he conjectures H^d=F_redⁱ⁺¹⊕F_Γ^d−i. The particular interest in this last decomposition is that combined with yet another conjecture from [16]—the degeneration of the ‘reduced’ Hodge spectral sequence—it would allow the computation ofΓ-group cohomology spacesH^∗(Γ, D)for certain ‘holomorphic discrete series representations’DofG.

For the trivial representationM =Kthe conjectures were proven first by Iovita and Spiess [10], later proofs were given by Alon and de Shalit ([1], using harmonic analysis) and the author ([6], using p-adic Hodge theory). The main tool in the approach of Iovita and Spiess is a certain subcomplex of Ω^•_X(X)consisting of bounded logarithmic differential forms onX. For more generalM this complex does not seem to generalize well, essentially because there is no integral structure in the complex M ⊗K Ω^•_X(X) of globalforms. This led us to consider a K-vector space subsheaf complex L^•(M) of M ⊗KΩ^•_X on X which should replace the global logarithmic differential forms. We show that the filtrationF_Γ^•can be redefined in terms of L^•(M)and obtain criteria for the above splitting conjectures and the degeneration of (2) which avoidΓ-group cohomology of global objects. A certain variant ofL^•(M), theK-vector space sheaf complex L^•_D(M), leads to a similar criterion for the splitting H^d=F_redⁱ⁺¹⊕F_Γ^d−i and the degeneration of the ‘reduced’ Hodge spectral sequence. The general hope is that, working as indicated with integral (or bounded) structures insideL^•(M)orL^•_D(M), we can reduce to problems in characteristic pand work locallyon the reduction of the natural formal scheme underlying X. This approach worked out in [7] in dimension d= 1 where we used integral structures insideL^•_D(M)to proveH¹=F_red¹ ⊕F_Γ¹ and the degeneration conjecture. Here, as suggested above, we use integral structures insideL^•(M)provided by the first part of this paper to prove (for arbitrary dimensiond):

THEOREM (see Corollary 5.1, Theorem 6.4 and the remarks given there). –Suppose that M|SLd+1(K)=K,M|SLd+1(K)=K^d+1orM|SLd+1(K)= (K^d+1)^∗.

(a) For arbitrary char(K)the Hodge spectral sequence(2) degenerates inE1. The Hodge filtrationF_H^• has a canonical splitting defined through logarithmic differential forms.

(b) Ifchar(K) = 0we haveF_H^• =F_red^• and the splitting in(a)is given by the filtrationF_Γ^•: H^d

XΓ,M^Γ⊗KΩ^•_XΓ

=F_Hⁱ⁺¹⊕F_Γ^d−i (0id−1).

It seems that even for M =K the degeneration in (a) in case char(K)>0 was unknown before.

Notations. – We fix d∈N and enumerate the rows and columns of GLd+1-elements by 0, . . . , d. We denote by U the subgroup of GLd+1 consisting of unipotent upper triangular matrices,

U=

(aij)_0i,jd∈GLd+1|aii= 1for alli, aij= 0ifi > j .

Forr∈R definer,r ∈Zby requiring rr <r+ 1 andr −1< rr. For a divisorDon an integral schemeX we denote byLX(D)the associated line bundle onX; we will always consider it as a subsheaf of the constant sheaf with value the function field ofX.

K denotes a non-Archimedean local field, OK its ring of integers, π∈ OK a fixed prime element andkthe residue field withqelements,q∈p^N. Letω:K_a^×→Qbe the extension of the discrete valuationω:K^×→Znormalized byω(π) = 1. We fix a totally ramified extension K˙ =K( ˙π)ofKwith ring of integersOK˙ such thatπ˙ ∈ OK˙ satisfiesπ˙^d+1=π.

We writeG= GL_d+1(K). LetT be the torus of diagonal matrices inGand letX_∗(T), resp.

X^∗(T), denote the group of algebraic cocharacters, resp. characters, ofT. For0iddefine the obvious cocharacter ei:Gm→GLd+1, i.e. the one which sends t to the diagonal matrix (ei(t))ij withei(t)ii=t,ei(t)jj = 1 for i=j and ei(t)j1j2 = 0 for j1=j2. Theei form a R-basis ofX_∗(T)⊗R. The pairingX_∗(T)×X^∗(T)→Zwhich sends(x, μ)to the integerμ(x) such that μ(x(y)) =y^μ(x) for anyy∈Gm extends to a duality between theR-vector spaces X_∗(T)⊗RandX^∗(T)⊗R. Let0, . . . , d∈X^∗(T)denote the basis dual toe0, . . . , ed. Let

Φ ={i−j; 0i, jdandi=j} ⊂ X^∗(T).

1. Differential forms on rational varieties in characteristicp >0

The action of GL_d+1(k) = GL(k^d+1) on (k^d+1)^∗= Hom_k(k^d+1, k) defines an action of GL_d+1(k)on the affinek-scheme associated with(k^d+1)^∗, and this action passes to an action of GL_d+1(k)on the projective space

Y₀=P

k^d+1_∗∼=P^dk.

For0jd−1letV₀^jbe the set of allk-rational linear subvarietiesZofY₀withdim(Z) =j, and letV0=d−1

j=0V₀^j. The sequence of projectivek-varieties Y =Yd−1→Yd−2→ · · · →Y0

is defined inductively by letting Y_j+1→Y_j be the blowing up of Y_j in the strict transforms (inYj) of allZ∈ V₀^j. The set

V=the set of all strict transforms inY of elements ofV0

is a set of divisors onY. The action ofGL_d+1(k)onY₀naturally lifts to an action ofGL_d+1(k) on Y. Let Ξ₀, . . . ,Ξ_d be the standard projective coordinate functions on Y₀ and hence on Y corresponding to the canonical basis of(k^d+1)^∗; hence Y₀= Proj(k[Ξ_i; 0id]). Denote byΩ^•_Y the de Rham complex onY with logarithmic poles along the normal crossings divisor

V∈VV onY. Fori, j∈ {0, . . . , d}andg∈GLd+1(k)we call gdlog

Ξi

Ξj

a logarithmic differential1-form onY. We call an exterior product of logarithmic differential 1-forms onY a logarithmic differential form onY.

PROPOSITION 1.1. – For each0sdwe haveH^t(Y,Ω^s_Y) = 0for allt >0. Thek-vector spaceH⁰(Y,Ω^s_Y)is the one generated by all logarithmic differential forms.

Proof. –In [5] we derive this from a general vanishing theorem for higher cohomology of a certain class of line bundles on Y. Note that a corresponding statement over a field F of characteristic zero is shown in [10] Section 3: the de Rham cohomology of the complement of a finite set ofF-rational hyperplanes inP^d_F is generated by (global) logarithmic differential forms.

And the analogous statement for the Monsky–Washnitzer cohomology ofY⁰=Y −

V∈VV was shown in [3]. 2

Remark. – In [5] we give ak-basis forH⁰(Y,Ω^s_Y)consisting of logarithmic differential forms as follows. For a subsetτ⊂ {1, . . . , d}let

U(k)(τ) =

(aij)_0i,jd∈U(k)|aij= 0ifj /∈ {i} ∪τ .

For 0sd denote by Ps the set of subsets of {1, . . . , d} consisting of s elements. The following set is ak-basis ofH⁰(Y,Ω^s_Y):

A.

t∈τ

dlog Ξ_t

Ξ0

τ∈ Ps, A∈U(k)(τ)

.

LetDbe a divisor onY of the type

D=

V∈V

b_VV

with certainbV ∈Z. We viewLY(D)as a subsheaf of the constant sheafk(Y)with value the function fieldk(Y)ofY; hence we viewΩ^•_Y ⊗OY LY(D)as a subsheaf of the constant sheaf with value the de Rham complex ofk(Y)/k. The differential on the latter provides us with a differential onΩ^•_Y ⊗_OY LY(D).

Consider the open andGLd+1(k)-stable subscheme Y⁰=Y −

V∈V

V

ofY; let us write

ι:Y⁰→Y for the embedding andΩ^•_Y0= Ω^•_Y|Y⁰.

For0sdletL^s_Y be thek-vector subspace ofΩ^s_Y(Y⁰)generated by alls-formsηof the type

η=y₁^m1· · ·y^m_s^sdlog(y₁)∧ · · · ∧dlog(y_s) (3)

with mj ∈Z and y1, . . . , yd ∈ O_Y^×(Y⁰) such that yj =θj/θ0 for a suitable (adapted to η) isomorphism of k-varieties Y₀ ∼= Proj(k[θ_j]_0jd). From Proposition 1.1 it follows that H⁰(Y,Ω^s_Y)is thek-vector subspace ofL^sY generated by alls-formsη of type (3) withm_j= 0 for all1js.

LetL^s_Y, resp.L^s,0_Y , be the constant sheaf onY with valueL^s_Y, resp. with value H⁰(Y,Ω^s_Y).

For a divisorDas above we define

L^s(D) =L^s_Y ∩ LY(D)⊗Ω^s_Y, L^s,0(D) =L^s,0_Y ∩ LY(D)⊗Ω^s_Y, the intersections taking place inside the push forwardι_∗Ω^s_Y0.

THEOREM 1.2. – (a) Suppose bV ∈ {−1,0} for all V. Then the inclusions L^s,0(D) → L^s(D)→ LY(D)⊗Ω^s_Y induce for alljisomorphisms

H^j

Y,L^s,0(D)∼=H^j

Y,L^s(D)∼=H^j

Y,LY(D)⊗Ω^s_Y . (b)LetSbe a non-empty subset ofVsuch thatE=

V∈SV is non-empty. Define the subsheaf L^s_E(0)ofΩ^s_Y ⊗_OY OEas the image of the compositeL^s(0)→Ω^s_Y →Ω^s_Y ⊗_OY OE. Then the inclusion induces for alljan isomorphism

H^j

Y,L^s_E(0)∼=H^j

Y,Ω^s_Y ⊗_OY OE

.

Proof. –(a) First we consider the case D= 0, i.e.bV = 0 for all V. The sheaf L^s,0(0) is constant with value H⁰(Y,Ω^s_Y), hence we get H^j(Y,L^s,0(0)) =H^j(Y,Ω^s_Y) for all j from Proposition 1.1. In order to also compare with H^j(Y,L^s(0)) choose a sequence (ηn)_n1 of elements of L^s_Y of the form (3) such that {ηn; n1} is a k-basis of L^s_Y/H⁰(Y,Ω^s_Y). For n0 let L^s,n_Y be the constant subsheaf of L^sY on Y generated over k by H⁰(Y,Ω^s_Y) and {ηi; ni1}. Letting

L^s,n(0) =L^s,n_Y ∩Ω^s_Y we have

L^s(0) =

n0

L^s,n(0)

and sinceY is quasicompact (so that taking cohomology commutes with direct limits) it suffices to show

H^j

Y,L^s,n(0)

=

H⁰(Y,Ω^s_Y): j= 0, 0: j >0

for alln0. Forn= 0we already did this, forn >0it suffices, by induction, to show H^j

Y, L^s,n(0) L^s,n−1(0) = 0

for allj. LetW ⊂Y be the maximal open subscheme on which the class ofη_n as a section ofL^s,n(0)/L^s,n−1(0)is defined. Thus ifξ:W →Y denotes the open embedding andk_Y the constant sheaf onY with valuekthen sending1∈ktoη_ndefines an isomorphism

ξ!ξ⁻¹k_Y ∼= L^s,n(0) L^s,n−1(0).

If we had W =Y then the induction hypothesis H¹(Y,L^s,n−1(0)) = 0 and the long exact cohomology sequence associated with

0→L^s,n−1(0)→L^s,n(0)→ L^s,n(0) L^s,n−1(0)→0 would imply that there existeda1, . . . , a_n−1∈ksuch thatηn+n−1

i=1 aiηiis a global section of L^s,n(0), in particular ofΩ^s_Y. But this would contradict the fact that{ηn;n1}is ak-basis of

L^s_Y/H⁰(Y,Ω^s_Y). HenceW=Y. On the other hand we may writeηn=y₁^m1· · ·y_s^msdlog(y1)∧

· · · ∧dlog(ys)withyj=θj/θ0 as in (3) and it is clear thatC=Y −W is the pull back under Y →Y0of a union of some hyperplanes V(θi)⊂Y0. In particularCis connected. Denote by γ:C→Y the closed embedding. The long exact cohomology sequence associated with

0→ξ!ξ⁻¹k_Y →k_Y →γ_∗γ⁻¹k_Y →0

showsH^j(Y, ξ_!ξ⁻¹k_Y) = 0for alljbecauseCis non-empty and connected. The induction and thus the discussion of the caseD= 0is finished.

To treat arbitrary D with b_V ∈ {−1,0} for all V we induct on dim(Y) and on r(D) =

V∈V|bV|. We will only showH^j(Y,L^s(D)) =H^j(Y,LY(D)⊗_OY Ω^s_Y)(which is the relevant statement for the subsequent sections), the proof ofH^j(Y,L^s,0(D)) =H^j(Y,LY(D)⊗_OY Ω^s_Y) is literally the same (replace each occurrence ofL^s(D)withL^s,0(D)).

AssumebV =−1for someV. LetD=D+V. We want to compare the exact sequences 0→L^s(D)→L^s(D)→L^sV(D)→0

(the sheafL^s_V(D)being defined such that this is an exact sequence) and

0→ LY(D)⊗Ω^s_Y → LY(D)⊗Ω^s_Y → LY(D)⊗Ω^s_Y ⊗ OV →0.

Sincer(D)< r(D)the induction hypothesis says that the map between the respective second terms induces isomorphisms in cohomology. It will be enough to prove the same for the respective third terms. From [11] (see also [5]) it follows thatV decomposes as

V =Y¹×Y²

such that bothY^tare successive blowing ups of projective spaces of dimensions smaller thand in allk-rational linear subvarieties, just asY is the successive blowing up of projective space of dimensiondin allk-rational linear subvarieties. Denote byV^tthe corresponding set of divisors onY^t(like the setVof divisors onY) and letΩ^•_Yt denote the logarithmic de Rham complex on Y^twith logarithmic poles alongV^t. LetΩ^•_V denote the logarithmic de Rham complex onV with logarithmic poles along all divisors which are pullbacks of elements ofV¹orV². Then

Ω^•_V = Ω^•_Y1⊗kΩ^•_Y2.

Let DV be the divisor on V induced byD. More precisely,DV =

bW(W ∩V), the sum ranging over allW∈ V which intersectV transversally. It also follows from [11] (and [5]) that DV is of the formD_Y^V1+D^V_Y2 whereD^V_Yt fort= 1,2is the pullback toV, via the projection V →Y^t, of a divisorDY^t onY^twhich is the sum, with multiplicities in{0,−1}, of elements ofV^t. The above then generalizes as

LV(DV)⊗Ω^•_V =

LY¹(D_Y¹)⊗Ω^•_Y1

⊗k(LY²(D_Y²)⊗Ω^•_Y2).

(4)

Choose an isomorphism Y₀ ∼= Proj(k[θ_j]_0jd) and elements 0 j₁ =j₂ d such that y=θ_j1/θ_j2∈ OY(U)is an equation forV∩Uin a suitable open subsetUofY withV∩U=∅. We have an exact sequence

0→ LV(DV)⊗Ω^s_V^{−1∧dlog(y)}−→ LY(D)⊗Ω^s_Y ⊗ OV → LV(DV)⊗Ω^s_V →0 (5)

where by (4) the extreme terms (takes=sands=s−1) decompose as LV(D_V)⊗Ω^s_V∼=

s1+s2=s

LY¹(D_Y¹)⊗Ω^s_Y¹1

⊗k

LY²(D_Y²)⊗Ω^s_Y²2

. (6)

On the other hand, define fort= 1,2the sheavesL^•(DY^t)onY^tjust as we defined the sheaves L^•(.)onY (and use the same name for their push forward toY). Then using the decomposition (6) we may view the sheaf

s1+s2=s

L^s1(D_Y¹)⊗kL^s2(D_Y²)

as a subsheaf ofLV(DV)⊗Ω^s_V and a local consideration shows that (5) restricts to an exact sequence

0→

s1+s2=s−1

L^s1(D_Y¹)⊗kL^s2(D_Y²)^∧dlog(y)−→ L^sV(D) (7)

→

s1+s2=s

L^s1(D_Y¹)⊗kL^s2(D_Y²)→0.

Comparing the long exact cohomology sequences associated with (5) and (8) we conclude that to show that

H^j

Y,L^sV(D)

→H^j

Y,LY(D)⊗Ω^s_Y ⊗ OV

is an isomorphism for anyj, it suffices to show that H^j

Y,

s1+s2=s

L^s1(D_Y¹)⊗kL^s2(D_Y²)

→H^j

Y,

s1+s2=s

LY¹(D_Y¹)⊗Ω^s_Y¹1

⊗k

LY²(D_Y²)⊗Ω^s_Y²2

is an isomorphism, fors=sands=s−1. By the Künneth formula this reduces to showing that

H^j

Y^t,L^s(D_Y^t)

→H^j Y^t,

LY^t(D_Y^t)⊗Ω^s_Yt

is an isomorphism, for any s andt∈ {1,2}. But this follows from our induction hypothesis since the dimension ofY^tis smaller than that ofY.

(b) We have an exact sequence 0→ LY

−

V∈S

V →

T⊂S

|T|=|S|−1

LY

−

V∈T

V → · · · →

V∈S

LY(−V)

→ OY → OE→0

which yields a similar exact sequence by tensoring withΩ^s_Y. A local consideration shows that the latter exact sequence restricts to an exact sequence

0→L^s

−

V∈S

V →

T⊂S

|T|=|S|−1

L^s

−

V∈T

V → · · · →

V∈S

L^s(−V)

→L^s(0)→L^sE(0)→0.

It follows that it is enough to show that for all subsetsT⊂Sand anyjthe map H^j

Y,L^s

−

V∈T

V

→H^j

Y,LY

−

V∈T

V ⊗OY Ω^s_Y

is an isomorphism. But this follows from part (a). 2

Remark (not needed in the sequel). – If −1 bV p−1 for all V then the inclusion L^•,0(D)→ LY(D)⊗Ω^•_Y induces isomorphisms

H^j

Y,L^•,0(D)∼=H^j

Y,LY(D)⊗Ω^•_Y . To see this letD=

V∈Vb_VV withb_V = min{0, bV}. Then Theorem 1.2 applies toD. Now note that on the one handL^•,0(D) =L^•,0(D)(logarithmic differential forms have pole orders at most one) and on the other handLY(D)⊗Ω^•_Y → LY(D)⊗Ω^•_Y is a quasiisomorphism (use that anybV >0is invertible ink).

2. Reduction of rationalG-representations LetT=T /K^×. Forμ=d

j=0ajj∈X^∗(T)let μ=

1 d+ 1

d j=0

a_{j d}

j=0

_j

−μ, (8)

an element of the subspaceX^∗(T)⊗_d+1¹ .ZofX^∗(T)⊗_d+1¹ .Z. If for0jdwe let

aj(μ) =(

i=ja_i)−da_j d+ 1 (9)

then

μ= d j=0

aj(μ)j.

LetM be an irreducibleK-rational representation ofG. For a weightμ∈X^∗(T)letMμbe the maximal subspace ofM on whichTacts throughμ.

LEMMA 2.1. – The number

|M|= d i=0

a_i forμ=d

i=0aii∈X^∗(T)such thatMμ= 0is independent of the choice of such aμ;for such μwe haveμ∈X^∗(T)if and only if|M| ∈(d+ 1).Z, if and only if there is ah∈Zsuch that the center ofGacts trivially onM⊗Kdet^h.

Proof. –This is clear since allμwithMμ= 0differ by linear combinations of elements ofΦ (see [12] II.2.2). 2

We fix aGLd+1/OK-invariantOK-latticeM⁰inM (see [12] I.10.4).

LEMMA 2.2. – We haveM⁰=

μ∈X^∗(T)M_μ⁰withM_μ⁰=M⁰∩Mμ.

Proof. –We reproduce a proof from notes of Schneider and Teitelbaum. Fix μ∈X^∗(T).

It suffices to construct an element Πμ in the algebra of distributions Dist(GLd+1/Z) (i.e.

defined overZ) which on M acts as a projector ontoMμ. For 0idletHi= (dei)(1)∈ Lie(GLd+1/Z); then dμ(Hi)∈Z (inside Lie(Gm/Z)) for any μ ∈X^∗(T). According to [8] Lemma 27.1 we therefore find a polynomialΠ∈Q[X₀, . . . , X_d] such thatΠ(Z^d+1)⊂Z, Π(dμ(H₀), . . . , dμ(H_d)) = 1andΠ(dμ(H₀), . . . , dμ(H_d)) = 0for anyμ∈X^∗(T)such that μ =μ and Mμ = 0. Moreover [8] Lemma 26.1 says that Π is a Z-linear combination of polynomials of the form

X0

b0 · · · Xd

bd with integersb0, . . . , bd0.

Thus [12] II.1.12 implies that

Πμ= Π(H0, . . . , Hd)

lies in Dist(GLd+1/Z). By construction it acts onM as a projector ontoMμ. 2

We return to the setting from Section 1. For∅ =σ{0, . . . , d}denote byV_σ⁰the common zero set inY0of allΞjwithj∈σ, and letVσ∈ V be its strict transform underY →Y0. Denote byY the open subscheme ofY obtained by deleting all divisorsV ∈ V which arenotof the particular formV =V_σfor some∅ =σ{0, . . . , d}. ThenY⁰⊂Y⊂Y andU(k)acts onY and onY⁰ and moreover U(k).Y=Y (theU(k)-translates ofY coverY). For eachV ∈ V there is a unique∅ =σ{0, . . . , d}such that there exists ag∈U(k)withgVσ=V, see [5].

LetM=M⁰/π.M⁰. The decomposition from Lemma 2.2 induces a corresponding decom- position

M=

μ∈X^∗(T)

M_μ.

Denote again byMthe constant sheaf onY with valueM. We define a subsheafM[OY]of M⊗kι_∗OY⁰|Y onYby

M[O Y] =

μ∈X^∗(T)

M_μ⊗kLY

∅=σ{0,...,d}

−

j∈σ

a_j(μ)

(V_σ∩Y) . (10)

LEMMA 2.3. – M[O Y]extends uniquely to aGLd+1(k)-stable subsheafM[O Y]ofM⊗k

ι_∗OY⁰.

Proof. –This can be checked directly, an easier variant of the proof of Theorem 3.3 below.

However, it is even aconsequenceof Theorem 3.3: explicitly, M[OY] =(M⁰_O˙

K⊗_OKOX)⊗_OX˙ OY

OY-torsion in the notations used there. 2

DEFINITION. – We say that the weights ofM are small if for anyμ∈X^∗(T)withMμ= 0 and for any∅ =σ{0, . . . , d}we have

0

j∈σ

aj(μ)

1.

(11)

LEMMA 2.4. – The weights ofM are small if and only if, when regarded as a representation ofSLd+1(K), it is one of the following:the trivial representation, the standard representation K^d+1, or the dual(K^d+1)^∗of the standard representation ofSL_d+1(K).

Proof. –One easily checks thatμ=_d

i=0aii∈X^∗(T)satisfies inequality (11) for allσif and only if all coefficientsaiare the same (case (i)) or if there is precisely one0idwith ai=aj+ 1for allj=i(case (ii)) or withai=aj−1for allj=i(case (iii)). IfM|SLd+1(K)=K the only weight occurring is as in case (i), ifM|SLd+1(K)=K^d+1the weights occurring are as in case (ii), ifM|SLd+1(K)= (K^d+1)^∗the weights occurring are as in case (iii). 2

For0sdconsider the following sheafM[L ^s_Y]onY:

M[L ^sY] =M⊗kL^s_Y ∩M[O Y]⊗OY Ω^s_Y, the intersection taking place insideι_∗(M⊗kΩ^s_Y|Y⁰).

THEOREM 2.5. – If the weights ofM are small then the inclusionM[L^s_Y]→M[OY]⊗OY

Ω^s_Y of sheaves onY induces isomorphisms H^∗

Y,M[L ^s_Y]∼=H^∗

Y,M[O Y]⊗_OY Ω^s_Y . Proof. –Consider the following ordering onX^∗(T): define

d i=0

aii>

d i=0

a_ii

(12)

if and only if there exists a0i0dsuch thatai=a_i for alli < i0, andai0 > a_i0. By [12]

II.1.19 the filtration(F^μM)_μ∈X∗(T)ofM defined by F^μM =

μ∈X^∗(T) μμMμ

(13)

is stable for the action ofU(K). Hence the filtration(F^μM⁰)_μ∈X∗(T)ofM⁰defined by F^μM⁰=

μ∈X^∗(T) μμ

M_μ⁰=F^μM∩M⁰

is stable for the action ofU(OK), and the induced filtrations(F^μM[O Y])_μ∈X∗(T)ofM[O Y] and(F^μM[L ^s_Y])_μ∈X∗(T)ofM[L ^s_Y]areU(k)-stable. We denote by

Gr^μ(.) = F^μ(.)

μ>μF^μ(.)