The strong rigidity theorem for non-Archimedean ... - CiteSeerX

The strong rigidity theorem for non-Archimedean uniformization Masa-Nori Ishida

and

Fumiharu Kato

Abstract In this paper, we present a purely algebraic proof of the strong rigidity for nonArchimedean uniformization, in case the base ring is of characteristic zero. In the last section, we apply this result to Mumford's construction of fake projective planes. In view of recent result on discrete groups by Cartwright, Mantero, Steger and Zappa, we see that there exist at least three fake projective planes.

Introduction

Let K be a non-Archimedean local eld and R the ring of integers. The Drinfeld upper half space nK is a p-adic analogue of the complex unit ball introduced by Drinfeld in [6]. As a set, nK is the set of all geometric points of the projective space PnK01 which do not lie on any K -rational hyperplanes. Drinfeld proved that the space nK has a natural structure as a rigid analytic space. It has a natural analytic action of PGL(n; K ), and considering the procedure of taking discrete quotients, one gets a good uniformization theory in p-adic analysis. The space nK has two essentially dierent ways of description. One of them is a rigid analytic subspace of PnK01. The other one is a formal scheme b = b nK over the discrete valuation ring R  K . The second description was developed by Kurihara [13] and Musta n [17] independently, and it is sometimes called p-adic unit ball of Kurihara and Musta n. In this paper, we will take up the viewpoint of the second one, because it is related rather directly with a visual combinatorial object called the Bruhat-Tits building. Interesting applications such as [16] were discovered through this viewpoint. Then the procedure of uniformization is presented as follows (cf. [17]): Let 0 be a torsionfree co-compact subgroup of PGL(n; K ). Then one can take a quotient X0 = b =0 in the category of formal schemes over Spf R. It is known that the resulting formal scheme X0 is algebraizable, i.e., the formal completion of a scheme X0 , which is proper and at 0 Mathematical Subject Classi cation. Primary 14J25, secondary 14M17.

Partly supported by the Grants-in-Aids for Scienti c Research, The ministry of Education, Science, Sports and Culture.

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over Spec R. Taking the generic ber, one obtains a nonsingular projective variety X0; over Spec K as the algebraization of the rigid analytic space nK =0. Recall the following result of Musta n on the rigidity of the uniformization by the Drinfeld upper half space.

Theorem 0.1 (cf. [17, x4]) The schemes X01 and X02 are isomorphic over Spec R if and only if 01 and 02 are conjugate in PGL(n; K ). We generalize this theorem as follows in case char K = 0.

Theorem 0.2 Assume that the characteristic of K is zero. Let 01 and 02 be torsionfree co-compact subgroups of PGL(n; K ). Then the schemes X01 ; K K and X02 ; K K are isomorphic over Spec K if and only if 01 and 02 are conjugate in PGL(n; K ), where K denotes an algebraic closure of K . Remark 0.3 In the framework of rigid analytic geometry, Berkovich obtained an equivalent theorem without the assumption on the characteristic of K (cf. [2, Theorem 2]). Since the descriptions of the Drinfeld space are not equal, Berkovich's proof is totally dierent from ours. This theorem shows a strong rigidity property of the Drinfeld upper half space, and will have possible applications to p-adic analysis, number theory and even algebraic geometry. We will prove the theorem in a purely algebraic way. In proving it, we will also clarify several interesting algebro-geometric aspects of the Drinfeld upper half space such as the behavior after base extensions, simultaneous crepant resolution of singularities, etc. A nonsingular complex surface of general type with pg = q = 0 and c21 = 3c2 = 9 is called a fake projective plane (cf. [1, V, Rem.1.2]). In [16], Mumford constructed a torsionfree co-compact subgroup 0 of PGL(3; Q2 ) such that X0; is a fake projective plane for a xed isomorphism Q2 ' C. Recently, Cartwright, Mantero, Steger and Zappa (cf. [4], [5]) looked at certain discrete subgroups of PGL-groups rather systematically, and obtained a complete list of torsionfree co-compact subgroups of PGL(3; Q2) of some kind. Combining this result with our main theorem, we see that there exist at least two more fake projective planes. In x1, we give a breaf summary of the construction of the formal scheme b and the non-Archimedean uniformization basically according to Musta n [17]. In x2 and x3, we observe the base changes b R0 = b R R0 and X0;R0 = X0 R R0 , where R0 is the integer ring of a nite extension of K . The proof of Thorem 0.2 is given in x4. In the last section, we apply it to the existence of new fake projective planes. This work was rstly suggested by Harm Voskuil during his stay in Sendai >from 1993 to 1994. The authors would like to express their deep gratitude to him. The authors also thanks the referee and the editor for nding several mistakes in the rst version of this article. 2

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Non-Archimedean uniformization in general

In this section, we give a breaf summary of the theory of non-Archimedean uniformization since the theory is not so popular. We introduce it basically according to Musta n [17] but in the dual formulations, since we wish to formulate notation and materials as in Mumford [16] and Ishida [9]. Throughout this paper, we x the following notation. Let R be a complete discrete valuation ring. We x a generator  of the maximal ideal of R. Let k = R=R be the residue eld of R and K the fractional eld of R. We assume that the eld k is nite and consists of q elements. We denote by  and 0 the generic point and the closed point of Spec R, respectively. Let n  2 be a natural number. A matrix  = (aij ) 2 GL(n; K ) P 01 de nes a linear automorphism of the vector space V = ni=0 KXi with indeterminates X0 ; . . . ; Xn01 by nX 01 ( ci Xi ) = (X0 ; . . . ; Xn01 ) t (c0 ; . . . ; cn01) i=0 XX = ( aij cj )Xi : i

j

Hence the induced automorphism ^ of P(V ) = Proj K [X0; . . . ; Xn01 ] is given in terms of the homogeneous coordinates (X0 : 1 11 : Xn01 ) by

^ (X0 : 1 11 : Xn01) = (X0 : 11 1 : Xn01): Thus the composite ^  ^ is equal to ( )^. 01 KXi of rank n. We de ne e 0 be the set of all free R-submodules in V = Pni=0 1.1 Let 1 e 0 by an equivalence relation  on 1

M1  M2

, M1 = M2 for some  2 K 2 , M1 = dM2 for some d 2 Z:

e 0= . We write the equivalence class of M 2 1 e 0 by [M ]. The group De ne 10 = 1 PGL(n; K ) acts transitively on the set 10 by [M ] = [M ] for  2 PGL(n; K ) and [M ] 2 10 . Let 31; 32 2 10 . Take a representative M1 2 31. Then, there exists a unique M2 2 32 such that M1  M2 6 M1:

Similarly, there exists a unique e M1 2 31 such that

M2   e M1 6 M2: It is easy to see that the nonnegative integer e does not depend on the choice of M1. De ne d(31 ; 32 ) = e. Then d : 10 2 10 ! Z0 is a metric function on 10. If d(31 ; 32) = 1, i.e., if M1 ) M2 ) M1 , then 31 and 32 are said to be adjacent. 3

De nition 1.2 The Bruhat-Tits building attached to PGL(n; K ), denoted by 1 = 1nK , is a simplicial complex de ned as follows: 1. The set of vertices of 1 is 10 . 2. A subset f30 ; . . . ; 3l g  10 forms an l-simplex if and only if 3i and 3j are adjacent for any i; j with i 6= j .

1.3 It is easy to see that a subset f30 ; . . . ; 3l g  10 forms an l-simplex if and only if, changing indices if necessary, there exist representatives Mi 2 3i for i = 0; . . . ; l such that M0  M1  1 11  Ml  M0 (cf. [17, Lemma 1.1]). Hence, there exists a one-to-one correspondence between the set of l-simplices having a xed vertex 30 = [M0 ] and the set of ags of length l in the vector space M0 =M0 by considering quotients Mi =M0 . In particular, l is at most n 0 1. Moreover, we have the following basic properties of the Bruhat-Tits building 1 attached to PGL(n; K ) (cf. [3] and [17]): 1. 1 is an (n 0 1)-dimensional locally nite simplicial complex. 2. 1 is a chamber complex, i.e., any simplex is a face of some chamber, where we mean by a chamber a simplex of dimension n 0 1. 3. 1 is labelable by Z=nZ, i.e., there exists a map  : 10 ! Z=nZ, called a labeling, in such a way that the vertices of each chamber are mapped bijectively onto Z=nZ ( (3) is called the type of the vertex 3 with respect to the labeling  ). 4. The group PGL(n; K ) acts on 1 by f30 ; . . . ; 3l g = f30; . . . ; 3l g.

A subgroup 0  PGL(n; K ) is discrete, if and only if the stabilizer in 0 of each 3 2 1 is a nite subgroup. 0 is said to be co-compact, if it is discrete and 1 has only nitely many 0-orbits. It is known that a co-compact subgroup 0 acts on 1 freely, if and only if 0 is torsionfree.

Example 1.4 In case n = 2, the Bruhat-Tits building 1 attached to PGL(2; K ) is a tree such that each vertex is an end of q + 1 edges (cf. [14]). Example 1.5 We consider the case n = 3. Let 3 be a vertex of 1. Then 3 is contained in 2(q2 + q + 1) edges of 1, and this set of edges has a natural one-to-one correspondence with the set of k -rational points and k -rational lines of P2k . Let B 2 be the algebraic surface obtained by blowing up P2k along all k-rational points. Then the dual graph of the con guration of exceptional curves and proper transforms of the lines is a onedimensional simplicial complex which is isomorphic to the link of 3 in 1. The dual graph of case q = 2 is in Figure 1 (cf. [16, x1]), where pei (i = 0; . . . ; 6) are the exceptional curves and eli (i = 0; . . . ; 6) are the proper transforms of the lines. 4

1.6 To each 3 = [M ] 2 10, we associate the scheme P(3) = Proj(SymR M ) over Spec R. This de nition is independent of the choice of M . The generic ber of P(3) is equal to the projective space P(V ) for every 3. Hence all these integral R-schemes are canonically birational. f0 and [M1 ]; [M2 ] 2 S imply A subset S of 10 is said to be convex if M1 ; M2 2 1 [M1 + M2] 2 S . Then we denote by 1(S ) the subcomplex of 1 consisting of all simplices in 1 whose vertices are in S . For a subset T of 10 , the convex hull of T is the smallest convex set which contains T . It is equal to the intersetion of all convex sets which contain T. Let S  1 be a nonempty convex subset. In [17] the R-scheme P(1(S )) is constructed as the limit of \joins" of P(3) for 3 2 S . P(1(S )) is an integral scheme locally of nite type over R with the generic ber P(V ). The following characterization of P(1(S )) is convenient. Let D0 be the rational function eld K (X1=X0; . . . ; Xn01 =X0 ), i.e., the rational function eld of P(V ). Then a local ring (A; P ) in D0 with  2 P has a center in P(1(S )) if and only if there exists a simplex in 1(S ) represented by fM0  M1  1 1 1  Ml  M0 g satisfying the following condition: For each i = 0; . . . ; l, there exists nonzero xi 2 Mi such that x0i 1 Mi  A, and Mi+1 is the largest among R-submodules M  Mi with [M ] 2 S and x0i 1 M  P , where we set Ml+1 := M0 . The closed ber P(1(S ))0 is a reduced normal crossing divisor with the dual graph b S )) is de ned as a formal completion of isomorphic to 1(S ). The formal scheme (1( b an integral R-scheme P(1(S )) along the closed ber. b = (1) is the Drinfeld upper half space de ned as a formal scheme. P 01  R Yi ] j 1.7 Let Y0 ; . . . ; Yn01 be a basis of V . Then the set of vertices S = f[ ni=0 i 2 Zg is convex in 10. The subcomplex A(Y0; . . . ; Yn01) := 1(S ) is isomorphic to the triangulation of Rn01 by the Weyl chambers of type An01 . This subcomplex is called an apartment. i

1.8 Each irreducible component of the closed ber of P(1) is isomorphic to the (n01)dimensional smooth k-scheme B n01 which is de ned as follows (cf. Example 1.5): For each integer 0  i  n 0 2, let 6i be the set of i-dimensional k -rational linear subspaces of Pnk 01. Set P0 := Pnk 01 . P1 is de ned to be the blow-up of P0 at all the points belonging to 60 . For 1 < i  n 0 2, Pi is de ned to be the blow-up of Pi01 at the union of proper transforms of the elements of 6i01 . Then we set B n01 = Pn02 . 1.9 Let S be a nonempty convex subset of 10. If a subgroup 0  PGL(n; K ) stabilizes b S )). Furthermore, the action of 0 on S , then 0 acts on P(1(S )) as well as on (1( b S )) is free if and only if that on 1(S ) is free.

(1( b S ))=0 of the Assume that the action is free. Then we can take the quotient (1( formal scheme, but the base becomes an algebraic space in general. In case S = 10 , we 5

b =0 by X0 . In this case, the base of X0 is a scheme since the denote the quotient (1) relative canonical sheaf is ample on each irreducible component.

As for the algebraizability of X0 , Kurihara and Musta n showed the following.

Theorem 1.10 (cf. [13, x2], [17, Thm.4.1]) Let 0 be a torsionfree co-compact subgroup of PGL(n; K ). Then, the formal scheme X0 is algebraizable, i.e., X0 is the completion of a projective scheme X0 over Spec R along its closed ber. Moreover, the algebraization X0 has the following properties: 1. The closed ber X0;0 is a reduced algebraic k-scheme with only normal crossing singularities. The normalization of each irreducible component is isomorphic to B n01. The dual graph of X0;0 is isomorphic to 1=0.

2. The relative canonical sheaf KX0=R of X0 over Spec R is invertible and relatively ample. In particular, the canonical invertible sheaf of the generic ber X0; is ample.

In case n = 3, Mumford proved that if q = 2 and 0 acts transitively on 1, then X0; is a fake projective plane (cf. [16, x1]). 2

Base change and desingularization

We use the following notation in the following three sections. R is a complete discrete valuation ring with the quotient eld K and the residue eld k as in Section 1. Let K 0 be a nite extension of K , and R0 the integral closure of R in K 0 . The residue eld of R0 is denoted by k 0. We assume that the elds k and k0 are nite. We have [K 0 : K ] := ef , where e is the rami cation index and f := [k0 : k ]. R is excellent since it is complete. Hence, R0 is also a complete discrete valuation ring, and is a free R-module of rank ef . Let n  2 be an integer. We denote the Bruhat-Tits buildings attached to PGL(n; K ) and PGL(n; K 0 ) by 1 and 10 , respectively. The sets of their vertices are denoted by 10 and 100 . We regard 10 as a subset of 100 by the correspondence [M ] 7! [M R R0 ]. For a convex subset S of 100, we denote by 10(S ) the subcomplex of 10 generated by S . For each simplex  2 1 of dimension n 0 1, let 0y be the convex hull of the set of vertices of  in 100 , and  y the subcomplex 10(0y ) of 10 .  y is a subdivision of  by Weyl chambers of type An01 . Let 1y0 be the convex hull of 10 in 100, and 1y the subcomplex 10 (1y0 ).

Lemma 2.1 The simplicial complex 1y is equal to the union of y for  particular, 1y is a subdivision of 1.

6

2 1n01 .

In

S The inclusion   y  1y is clear. It is known that any two simplices 1 ; 2 in 1n01 are contained in a common apartment of 1 (cf. [17, Lem.1.2]). Hence if [N1 ] 2 (1 )y0 and [N2 ] 2 (2)y0 , then [N1 + N2 ] is in (3 )y0 for a simplex 3 of the apartment. Hence 1y0 is the union of 0y for  2 1n01 . Let  be a simplex of 1y of dimension d  1. We prove that  is in y for some  by induction on d. Let 3 be a vertex of  , and  0 the complementary (d 0 1)-dimensional face. Then there exist 1 ; 2 in 1n01 with 3 2 (1 )y0 S and  0 2 (2)y . 1 and 2 are contained in a common apartment A of 1. Since 2A y is an apartment of 10 ,  is a simplex of this apartment. Hence  is contained in y for a simplex  of the apartment A. q.e.d. b Let 0  PGL(n; K ) be a torsionfree co-compact subgroup. Set b = (1) and let y y 0 b b

be the formal scheme (1 ) over Spf R (cf. 1.6). The quotient formal schemes of b and b y with respect to 0 are denoted by X0 and X00 , respectively. By Theorem 1.10, the formal scheme X0 is algebraized to a regular scheme X0 over Spec R. Set b R0 = b R R0, X0;R0 = X0 R R0 and X0;R0 = X0 R R0 . Here note that R0 is a at nite R-algebra. The goal of this section is to prove the following propositions:

Proof.

Proposition 2.2 The formal scheme X00 is algebraized to a regular scheme (which we denote by X00 ). Proposition 2.3 There exists a natural 0-equivariant morphism ~ : b y

makes the following diagram commute:

b y ? ? y Spf R0

(1)

~ 0!

! b R0

which

b R0 ? ? y Spf R0 .

=

The morphism ~ descends to a resolution of singularities  : X00 ! X0;R0 which is isomorphic on the generic ber. Moreover, the morphism  has no discrepancy, i.e., !X00 =R0 = 3 !X0 0 =R0 , where !X00 =R0 and !X0 0 =R0 are the relative dualizing sheaves of the R0-schemes X00 and X0;R0 , respectively. ;R

;R

In fact, if e > 1 then X0;R0 is no longer regular. Let us x a generator  (resp.  ) of the maximal ideal of R (resp. R0 ). Clearly, there exists a unit element u 2 (R0 )2 such that  = u e. Since the scheme X0 is etale locally de ned by an equation

z0 1 11 zn01 = ; the scheme X0;R0 is etale locally de ned by

z0 1 11 zn01 = u e ; i.e., it has singularities along the double locus of the closed ber when e > 1. Note that these singularities are locally hypersurfaces in smooth varieties over R0 . In particular, 7

X0;R0 has the relative dualizing invertible sheaf !X0 0 =R0 (cf. [8, Chap.III, x1]). Furthermore, since the locus of the singularity is of codimension two, X0;R0 is normal by Serre's criterion. ;R

2.4 The R0 -scheme P(1y) dominates P(3) for all 3 2 10. By the criterion in 1.6, we see that the local rings of P(1y ) has centers in P(1), i.e., there exists a natural morphism P(1y ) ! P(1). By taking formal completions of these schemes, we get the 0-equivariant commutative diagram (1). The induced morphism e : b y ! b R0 is locally described as follows: Let fZ0 ; . . . ; Zn01g be a basis of the K -linear space V . For each 0  j  n 0 1, let Mi be the R-submodule of V generated by

fZ0 ; . . . ; Zn0j01; Zn0j ; . . . ; Zn01g : Then f[M0 ]; . . . ; [Mn01 ]g form a chamber in 1, and this chamber corresponds to a k valued point in the closed ber of b R0 which is an n-ple intersection of local components. Set z1 = Z1 =Z0 ; z2 = Z2 =Z1 ; . . . ; zn01 = Zn01 =Zn02 ; zn = u eZ0 =Zn01 : Then the singularity of b R0 at this point is de ned by (2)

A = R0 [z1 ; . . . ; zn ]=(z1 1 11 zn 0 u e ):

The restriction of the morphism e : b y ! b R0 at this point is described by the theory of toroidal embeddings. We can show that there exists an ideal I of A such that the restriction of e is equal to the blow-up of Spf A along this ideal. Let U be a suciently small open neighborhood of this point. Set U 0 = (e)01 (U ). Then U00 consists of (e + 1)(e + 2) 1 1 1 (e + n 0 1)=(n 0 1)! components and each component corresponds to an R0 -module (3) R0 a0 Z0 + R0 a1 Z1 + 11 1 + R0 a 01 Zn01 n

with integers 0 = a0  a1  11 1  an01  e(= an ). For a = (a1 ; . . . ; an), we denote by D(a) the assiciated exceptional divisor. Set

C (a) =

n X i=1

(ai 0 ai01 )(e 0 ai + ai01 )

P and D = a C (a)D(a). Then D is an eective divisor with support in the exceptional set of e. We can check that the ideal sheaf OU 0 (0D) is relatively ample, by using the theory of toric varieties over a discrete valuation ring (cf. [12, IV,x3]). The combinatorial part of this singularity is equal to that of [12, III, Expl.2.3]. We review it brie y as follows. We set N = Zn. Then the ring A is de ned by the cone

 = f(x1 ; 11 1 ; xn ) 2 Rn ; x1 ; 1 11 ; xn  0g in NR = Rn and the lattice N 0 , where N 0 is the sublattice of N de ned by

N 0 = f(c1 ; 11 1 ; cn) 2 Zn ; c1 + 1 11 + cn  0 (mod e) g : 8

For integers i; j with 1  i  n and 0 < j < e, the hyperplane

Hi;j = f(x1 ; 11 1 ; xn ) 2 Rn ; exi = j (x1 + 11 1 + xn)g intersects the interior of  . Let 6 be the fan obtained by dividing  by all these hyperplanes. Then 6 is a nonsingular fan and the resolution of the singularity e corresponds to the morphism of toric varieties associated to this subdivision. The projectivity of the resolution is equivalent to the existence of a real-valued continuous function h on  with the following properties (cf. [18, Chap.2]). (1) h is linear on each cone  2 6. (2) h(x) + h(y)  h(x + y ) for x; y 2  , and the equality holds if and only if x and y are in a common cone of 6. (3) h is zero on the one-dimensional faces of  . This function h is called a strictly convex 6-linear support function. Let qe (x) be the function on [0; e] de ned by qe (x) = (e 0 2j 0 1)x + j (j + 1) if j  x  j + 1 for an integer j . It is easy to see that qe is well-de ned and is upper convex. Note that qe (j ) = j (e 0 j ) for an integer j . An example of h is de ned by

h(x1; 11 1 ; xn ) = qe (x1 ) + 1 11 + qe (xn ) for x = (x1 ; 11 1 ; xn) with x1 + 1 1 1 + xn = e. Actually, hi de ned by hi (x1 ; 11 1 ; xn ) = qe (xi ) is a strictly convex 6i -linear support function for the fan 6i obtained by dividing  by the hyperplanes Hi;1; 1 11 ; Hi;e01 for each i. Hence their sum is strictly convex for 6. Let S = f(c1 ; 1 11 ; cn ) 2 Zn \  ; c1 + 11 1 + cn = eg. The set of one-dimensional cones in 6 is equal to fR0 c ; c 2 S g. Let Dc be the associated prime divisor. Then the divisor X (0h(c))Dc c2S associated to h [18, Chap.2] is relatively ample. This divisor has support in the exceptional set of the resolution by the condition (3) of h. The restriction of this divisor to U 0 is 0D which we described above. Hence 0D is relatively ample. We take a suciently large integer d, so that the sheaf OU 0 (0dD ) is relatively very ample. Set IU = e3OU 0 (0dD)  OU . Then e restricted to U 0 is the blow-up of U by the ideal IU . The ideal IU is a restriction of the ideal I  A de ned as follows: For a vector b = (b1 ; . . . ; bn) 2 (Z0 )n, we de ne n X hb; ai = (ai 0 ai01)bi: i=1

Then I is de ned by

I = hz b  c ; b 2 (Z0 )n; c  0; hb; ai + c  dC (a) for every ai; where z b = z1b1 1 11 znb 2 A. Note that I is invariant even if we replace zi by ui zi for unit elements ui , since it is generated by monomials. By the symmetry of the de nition of n

9

C (a)'s, it is also invariant by permutations of indices. Hence IU 's are patched together to a global ideal sheaf I  O b 0 and e is the blow-up along this ideal. R

Lemma 2.5 We have (e)3! b 0 =R0 = ! b y=R0 , where ! b 0 =R0 and ! by=R0 are the relative b y ! Spf R0 , respectively. b R0 ! Spf R0 and

dualizing sheaves of

R

Proof.

R

By the local expression (2), the dualizing sheaf ! b 0 =R0 is locally generated by R

dz1 dzn01 ^ 1 11 ^ : z1 zn01

(4)

Each component of the exceptional divisor of the resolution e : b y ! b R0 corresponds to an R0 -submodule as (3). Hence b y is etale locally isomorphic to a Zariski open subset of the ane formal scheme Spf R0 [ a1 Z1 =Z0 ;  a2 0a1 Z2=Z1; . . . ;  a 01 0a 02 Zn01=Zn02 ; u e0a 01 Z0 =Zn01 ] = Spf R0 [ a1 z1 ;  a2 0a1 z2 ; . . . ;  a 010a 02 zn01 ;  0a 01 zn] ; n

n

n

n

n

n

and the local generator of the dualizing sheaf ! b y=R0 is again given by the local section (4) since  in the dierentials is a nonzero constant. q.e.d.

2.6 Since ! b 0 =R0 is ample on each component of the closed ber (cf. [17, x4]) and (e)3 ! b 0 =R0 = ! by=R0 , the invertible sheaf ! b ym=R0 (e01 I )O b y is ample on each component of the closed ber of b y for m suciently large. By the invariance, this invertible sheaf descends to the formal scheme X00 . Since it is ample on each component of the closed ber of X00 , this proves Proposition 2.2 by the theory of Grothendieck [EGA3, Thm.5.1.4, Thm.5.4.5]. By [EGA3, Thm.5.4.1], we obtain the following commutative diagram R

R

X00 ?? y Spec R0

 0! X0;R0 0! ?

? y

0! Spec R0 0!

X0 ? ? y Spec R ;

By construction, the R0-morphism  is a desingularization of X0 R R0 . Since  is the descent of ~, it is a blow-up along a closed subscheme whose support is contained in the  X K 0 over Spec K 0. closed ber. In particular, we have an isomorphism X00 ; ! 0; K By Lemma 2.5, we have Proposition 2.3. 3

Automorphisms of formal schemes

b R0 = The goal of this section is to prove the following assertion for the formal scheme

b R R0 :

10

Proposition 3.1 Assume char K = 0. Then the natural homomorphism PGL(n; K ) ! AutR0 ( b R0 ) is an isomorphism.

In case R = R0 , i.e., b = b R0 , this was proved by Musta n (cf. [17, Prop.4.2], [10]). 3.2 For the proof of this proposition, we need to show some lemmas. Let B = B n01 and 6i (0  i  n 0 2) be as in 1.8. Let E  B be the exceptional divisor of the projection p : B ! Pnk 01 and A = An01  B the union of E and the proper transforms of elements of 6n02. By the construction of B , the morphism p de nes a one-to-one correspondence between the set of irreducible components of A and the union 60 [ 1 1 1 [ 6n02 . The intersection D1 \11 1\ Ds of the irreducible components of A is an irreducible subvariety of codimension s if fp(D1); 11 1 ; p(Ds)g with a suitable order is a ag in Pnk 01, and is empty otherwise. Set Bk0 = B k k0 and Ak0 = A k k0 .

Lemma 3.3 The natural homomorphism PGL(n; k) ! Autk0 (Bk0 ; Ak0 ) is an isomorphism, where Aut(Bk0 ; Ak 0 ) denotes the group of k 0-automorphisms of Bk 0 which maps Ak0 to itself.

We prove this lemma by induction on n. Let  be an element of Autk0 (Bk0 ; Ak0 ). If n = 2, then Bk10 = P1k0 and A1k0 is the set of k -rational points. Hence  is a k -rational linear automorphism. For n  3, it suces to show that (E ) = E . Indeed, Bk0 n E is isomorphic to the open subset of Pnk001 whose complement is the union of k-rational linear subspaces of codimension two. Hence Pic(Bk0 n E )  = Z and  induces an automorphism n 0 1 of the homogeneous coordinate ring of Pk0 . Since p(E ) is mapped to itself, it is a k -rational linear automorphism. For n = 3, the components of A2k0 are nonsingular rational curves with the selfintersection numbers 0q or 01, where q is the cardinality of k . Each component is an exceptional divisor if and only if the number is 01. Hence (E ) = E . Assume n > 3. Each point x of Ak0 is said to be i-ple if there exist precisely i irreducible components of Ak0 which contain x. Since Ak0 is a simple normal crossing divisor, it is at most (n 0 1)-ple. For an i-ple point, the i linear subspaces of Pnk 01 corresponding to the components form a ag of length i. Let D be a component of Ak0 with dim p(D) = s. Then the number of (n 0 1)-ple points on D is equal to that of full-length k-rational ags which contain the linear space p(D) as a member. The number of (n 0 1)-ple points on D is calculated easily to be s i+1 20s i+1 Y q 0 1 n0Y q 01 : q01 i=1 q 0 1 i=1 Proof.

11

Since this number is invariant under , (D) is a component of Ak0 with dim p((D )) = s or n 0 2 0 s. Since D is in E if and only if dim p(D) < n 0 2, it suces to show that no component D of Ak0 satis es dim p(D) = 0 and dim p((D)) = n 0 2. Suppose that a component D satis es the equalities. Let p1 : D ! E1 ' Pnk002 be the morphism to the exceptional divisor E1 of the blow-up of Pnk001 at the point p(D), and p2 : (D) ! E2 = p((D))  Pnk001 the restriction of p. By the construction of Bk0 , we see that these morphisms are both isomorphic to the morphism Bkn002 ! Pnk002. By the induction hypothesis, the isomorphism D ' (D) is induced by a k -rational isomorphism E1 ' E2. Let l1 and l2 be general lines of E1 and E2 , l10 and l20 their proper transforms in D and (D), respectively. Here, we may replace k 0 by its algebraic closure in order to take suciently general lines. Then l20 intersects the exceptional divisor at (qn01 0 1)=(q 0 1) points while l10 does not intersect the exceptional divisor other than D . Hence the intersection numbers of D 1 l10 and (D ) 1 l20 are 01 and 1 0 (q n01 0 1)=(q 0 1) = 0(q + 11 1 + q n02), respectively. This is a contradiction, since we may assume l20 = (l10 ). q.e.d.

Lemma 3.4 Let 2B (0 log A) be the sheaf of algebraic vector elds with logarithmic zeros along A. Then H0 (2B (0 log A)) = f0g: The restriction  : B n E ! Pnk 01 of p is clearly an open immersion. Let A0 be the union of k-rational hyperplanes of Pnk 01 . Then 2B (0 log A)jB nE is equal to 3 2P 01 (0 log A0 ). We have

Proof.

n k

H0 (2B (0 log A)jB nE ) ' H0 (2P 01 (0 log A0 )) ; n k

since the sheaf 2P 01 (0 log A0 ) is re exive and the complement of the image of  is of codimension two. Hence, it suces to show that H0 (2P 01 (0 log A0)) = f0g. For each 0  i  n 0 1, let Di be the hyperplane Xi = 0. For D = D0 [ 1 11 [ Dn01, 2P 01 (0 log D) is a free sheaf of rank n 0 1 (cf. [18, Prop.3.1]). Let (u1 ; 1 1 1 ; un01) be the coordinate of the ane space (X0 6= 0) de ned by ui = Xi =X0 for i = 1; 1 11 ; n 0 1. Then H0 (2P 01 (0 log D)) is the k-vector space with basis n k

n k

n k

n k

fu1 @u@ ; 1 1 1 ; un01 @u@ g : n0 1

1

The restriction of the hyperplane H = (X0 + 11 1 + Xn01 = 0) to this ane space is de ned by the equation 1 + u1 + 1 11 + un01 = 0. Let @ @  = a1u1 + 11 1 + an01un01 @u1 @un01

be an element of H0 (2P 01 (0 log D )). Since  (1+u1 +11 1+ un01 ) = a1 u1 +1 11+ an01 un01,  has logarithmic zero along H if and only if  = 0. Hence H0(2P 01 (0 log(D + H ))) = f0g. Since D + H  A0 , we have H0(2P 01 (0 log A0 )) = f0g. q.e.d. n k

n k

n k

12

Lemma 3.5 If an R0 -automorphism  of the formal scheme b R0 xes all the irreducible components, then  is the identity. For each 3 2 10 , we denote by B (3) the corresponding component of b R0 with the reduced scheme structure. Set [ B (30)) : A(3) = B (3) \ ( 0 3= 6 3

Proof.

Then the pair (B (3); A(3)) is isomorphic to (Bk0 ; Ak0 ) (cf. 1.6 and 1.8). Hence the automorphism of the pair (B (3); A(3)) induced by  is the pull-back of a linear automorphism of P(3)0 by Lemma 3.3. Since  xes the components neighboring B (3), this k0 -linear automorphism xes all k-rational points of the projective space over k 0. Hence the induced automorphism of B (3) is the identity for an arbitrary 3 2 10 . It suces to prove that the automorphism of

b i = b R (R0 = i+1 ) induced by  is the identity for all nonnegative integers i, since  is their projective limit. b 0 is a reduced We proceed by induction on i. The assertion is clear for i = 0, since

S scheme with support 3210 B (3). Suppose that the automorphism is the identity for i = d 0 1  0. Then  induces an automorphism 3d of the sheaf O b of R0 =( d+1 )-algebras. Then 3d 0 1 de nes a k0 -derivation of O b0 : d

O b0 ! O b 0 R0 ( d)=( d+1) ' O b0 : R

Since b 0 is a normal crossing union of the nonsingular k0 -varieties B (3) for 3 2 10, the sheaf D erk0 (O b0 ) has a natural injective homomorphism to M 2B (3) (0 log A(3)) 3210 (cf. [11, Thm.2.1]). Since H0(2B(3) (0 log A(3))) = f0g by Lemma 3.4, for every 3 2 10, there is no nontrivial k 0-derivation of O b0 . Hence 3d = 1, and the assertion is true for i = d. q.e.d. Proof of Proposition 3.1

injective homomorphism

. For any nite extension K 00 =K 0 , there exists a natural

AutR0 ( b R0 ) ! AutR00 ( b R00 ); where R00 is the integer ring of K 00 . Hence, we may assume that K 0=K is a Galois extension by the condition char K = 0. Let G be the Galois group. Let  be an R0 automorphism of b R0 . For any element  2 G, 01 01  is an R0 -automorphism of

b R0 . Since  xes all components of the formal scheme, this R0 -automorphism also xes 13

them. Hence 01 01  is the identity by Lemma 3.5. This implies that  descends to b We are done since the homomorphism an R-automorphism of the formal scheme . b PGL(n; K ) ! AutR ( )

is an isomorphism by Musta n [17, Prop.4.2]. (A precise proof of this proposition of Musta n is given in [10].) q.e.d. 4

Proof of Theorem 0.2

In this section, we prove the following theorem which is equal to Theorem 0.2.

Theorem 4.1 Assume char K = 0. Let 01 and 02 be torsionfree co-compact subgroups of PGL(n; K ). Let X01 and X02 be the R-schemes obtained by the algebraizations of b =01 and X02 =

b =02 , respectively. Then, X01 ; K K the formal schemes X01 =

and X02 ; K K are isomorphic over Spec K if and only if 01 and 02 are conjugate in PGL(n; K ), where K is the algebraic closure of K . The \if" part of the above theorem is clear (cf. [17, x4]). We are going to prove  X K over Spec K . the other part. Suppose we have an isomorphism X01 ; K K ! 02 ; K 0 There exists a nite Galois extension K =K such that this isomorphism descends to an  X K 0 over K 0 (cf. [EGA4, Cor. 8.8.2.5]). >From now isomorphism X01 ; K K 0 ! 02 ; K on, we use the notation xed in x2. Consider the normal schemes X0 ;R0 = X0 R R0 over Spec R0 for i = 1; 2. Since the generic bers of these schemes are isomorphic, there exists a birational map (5) ' : X01 ;R0 1 11 ! X02 ;R0 over Spec R0 . First, we wish to show that ' is an isomorphism. We look at the canonical rings M 0 (6) H (X0 ;R0 ; !X m 0) 0 0 =R m 0 Proof.

i

i

i

i ;R

of X0 ;R0 for i = 1; 2. Since the sheaves !X0 0 =R0 are ample on X0 for i = 1; 2, it suces to prove that ' induces an isomorphism M 0  M H0(X 0 ; ! m (7) H (X02 ;R0 ; !X 0m 0 =R0 ) 0! 01 ;R X0 0 =R0 ) 1 2 m 0 m 0 of graded R0-algebras. Take the resolutions X00 of X0 ;R0 as in Proposition 2.3 for i = 1; 2. Since these resolutions produce no discrepancy with respect to the relative dualizing sheaves, we have isomorphisms of graded R0 -algebras M 0  M H0(X 0 ; ! m ) H (X0 ;R0 ; !X 0m 0 =R0 ) 0! 0 X00 =R0 m 0 m 0 i

i

i ;R

;R

;R

i

i

i

i

i ;R

14

i

for i = 1; 2. The map (5) induces a birational map '0 : X00 1 1 11 ! X00 2 . Since X00 1 is regular and X00 2 is proper over R, '0 is regular outside a closed subset of codimension not less than two. Let (X00 1 ) be the open subscheme of X00 1 de ned as the intersection of the smooth locus over R0 and the regular locus of the rational map '0 . Clearly X00 1 ;  (X00 1 ), and (X00 1 )0 is open dense in X00 1 ;0 . Let (X00 2 ) be the smooth locus of X00 2 over R0 .

Claim '0((X00 1 ))  (X00 2 ) . Let x be a closed point of (X00 1 ) and let y = '0 (x) 2 X00 2 . Let (Ox ; mx ) and (Oy ; my ) be the local rings of x and y , respectively. Then we have dominations of regular local rings R0  Oy  Ox : Since X00 1 is smooth over R0 at x, Ox = Ox is a regular local ring. Hence  62 m2x . On the other hand, m2y  m2x since my  mx . Hence  62 m2y and Oy = Oy is a regular local ring. Since X00 2 is smooth at all points of (X00 2 )0 which is regular in (X00 2 )0, X00 2 is smooth at y. q.e.d. Since '0 : (X00 1 ) ! (X00 2 ) is a morphism of smooth R0 -schemes, the pull-back homomorphism ('0)3 !(X00 ) =R0 ! !(X00 ) =R0 1 2 L 0  of sheaves on (X01 ) is de ned naturally. For any  2 m0 H0 (X00 2 ; !X m 0 0 ), the pull02 =R back ('0)3  can be expressed as a regular dierential form de ned on a Zariski open set of X00 1 whose complement is of codimension at least two. Hence we can prolong ('0)3  L

m to an element in m0 H0 (X00 1 ; !X 0m 0 0 is invertible and X00 1 is regular. 0 =R0 ), since !X 01 =R 1 Thus, we have an injective homomorphism M 0 0 m M 0 0 m H (X02 ; !X 0 =R0 ) 0! H (X01 ; !X 0 =R0 ) 02 01 m 0 m 0 Proof.

of graded R-algebras. This is an isomorphism, since the inverse is obtained similarly by 01 . Hence, we have the isomorphism (7). This implies that we can prolong (5) to an  X 0 over Spec R0 . isomorphism ' : X01 ;R0 ! 02 ;R Then we lift the isomorphism ' to an automorphism e of the universal covering b R0 . Then we have e01 e01 = 02 . By Theorem 3.1, e comes from an formal scheme

element of PGL(n; K ). Hence the groups 01 and 02 are conjugate in PGL(n; K ) as desired. q.e.d. 5

Fake projective planes

In this section, we consider fake projective planes as an application of our main theorem. A fake projective plane is, by de nition, a nonsingular projective complex surface of 15

general type with (8)

pg = q = 0 ; c21 = 3c2 = 9:

The underlying topological space of a fake projective plane has the same Betti numbers as the projective plane CP2 (cf. [1, V, Rem.1.2]). In [16], Mumford constructed a fake projective planes as follows. He showed that, in case n = 3, the invariants of the surface X0; satis es (8) if K = Q2 and 0 acts on 10 transitively. Then, for a xed isomorphism  C of elds, X C is a fake projective plane. Mumford [16] constructed one Q2 ! 0; Q2 example of torsionfree co-compact subgroup 0 of PGL(3; Q2 ) which acts transitively on 10 . Theorem 4.1 implies that, if we nd another such 0 which is not conjugate to that of Mumford, we get another example of fake projective planes.

5.1 Here, we refer to the recent work of Cartwright, Mantero, Steger, and Zappa on discrete subgroups of PGL(3; K ) (cf. [4], [5]). Let 1 be the Bruhat-Tits building attached to PGL(3; K ). Let 30 = [M0 ] be a vertex. The labeling  : 10 ! Z=3Z is de ned as follows. For 3 2 1, we choose a representative M of 3 contained in M0 . Then  (3) is de ned to be the length modulo 3 of the R-module M0 =M . It is easy to see that this de nition does not depend on the choice of M . We denote by N0 the set of all adjacent vertices of 30 . We set P := f3 2 N0 j  (3) = 1g and L := f3 2 N0 j  (3) = 2g. P and L correspond to the sets of k -rational points and lines of P(M0 =M0 ) ' P2k , respectively. Let 0 be a discrete subgroup of PGL(3; K ) which may not be torsionfree. We assume that 0 acts simply transitively on 10. Then, for each 3 2 10, there exists a unique element g3 2 0 such that g3 (30 ) = 3. The correspondence 3 7! (3) := g301 (30) de nes a bijection  : P ! L, which is called a point{line correspondence (cf. [4, x1]). Set F = f(3; 30 ; 300 ) 2 P 3 j g3g30 g300 = 1g: Then it was shown that the group 0 is generated by fg3 j 3 2 P g with the fundamental relation fg3 g30 g300 = 1 j (3; 30; 300 ) 2 Fg (cf. [4, Thm.3.1]). The combinatorial data (; F ) is called the triangle presentation of 0. In the rest of this section, we assume q = 2. Then there exist exactly eight combinatorial possibilities of triangle presentations for P2k up to projective transformations and correlations (cf. [5, Thm.1]). These were named A1, A2, A3, A4, B 1, B 2, B 3 and C 1 in [5].

Remark 5.2 In [5, x2], the classi cation of triangle presentations was done up to \correlations" of the projective plane. Hence a triangle presentation is identi ed with its reverse. This identi cation is not convenient for us, since the group associated to the reverse is not isomorphic but anti-isomorphic to the original group, in general. Among the eight triangle presentations, A1, A2 and A3 are not reverse symmtric, while the others are reverse symmetric. In the table of [5, p.207], \N" (not reverse symmetric) 16

for A4 might be a misprint. In this section, we distinguish A1, A2 and A3 with their reverses which we write A10 , A20 and A30 , respectively.

Theorem 5.3 (cf. [5, x3]) (1) There exist discrete subgroups of PGL(3; F2 ((X ))) with triangle presentations A1, A2, A3 and A4 which act simply transitively on 10, where F2 ((X )) is the quotient eld of F2 [[X ]]. (2) There exist discrete subgroups of PGL(3; Q2) with triangle presentations B 1, B 2, B 3 and C 1 which act simply transitively on 10 . We denote the group associated to A1 by 0A1 , and so on. Then the transpose groups of 0A1 , 0A2 and 0A3 in PGL(3; F2 ((X ))) have the triangle presentations A10, A20 and A30 , respectively.

5.4 Since the actions of the groups above preserve the orientation of 1 de ned by the labeling, the restrictions of the actions to 11 are also free. Hence, for each of these groups, the action is free on 1 if and only if it is free on 12 . For each triangle presentation, the number of rows in the table [5, p.212] is equal to the number of 0-orbits in 12 . If the action of 0 on 12 is free, then the twenty-one F2 -rational points of the rational surface B 2 are identi ed to seven triple points of X0;0 (cf. [9, x1]). Otherwise, it has more than seven 0-orbits. Hence the action of 0 on 1 is free if and only if the number of the 0-orbits in 12 is equal to seven. (1) By the table on [5, p.212], we see that 0A1 and 0A2 act freely on 1, while 0A3 and 0A4 are not free on 12 . (2) The group 0C 1 is equal to the group found by Mumford which is embedded in p PGL(3; Q( 07))  PGL(3; Q2 ). The groups 0B1 , 0B2 and 0B3 are embedded in p PGL(3; Q( 015))  PGL(3; Q2 ) (cf. [5, x3]). By the same table, we see that 0B 3 is not free on 12 , while other three groups are free on 1. 5.5 Since mutually conjugate groups in PGL(3; Q2 ) de ne equivalent triangle presentations, 0B1 , 0B 2 and 0C 1 are not conjugate to each other. Hence, by Theorem 4.1, the fake projective planes obtained by these groups are mutually distinct. Hence there exist at least three fake projective planes. The Figures 2, 3 and 4 are the the con gurations of the double curves of X0;0 for these groups. Note that the numbers of double curves with nodes for these three surfaces are mutually distinct.

17

el6

pe6



.......................................................................................... ........ .... ........ ... ........ ........ ... . . . . . ... . . ........ ... .... ........ .... ... ........ . . . . . . . ........ . . . . ... . . . . . . ........ . . . . ... ... . . . . .... . . ... ... ... . . . .. ... ... ... . . . . . .. .... ...... . . . . . ... . . .... .. . .. ... .. ... ..... .. ... ... .. ... ... .. ... . . .... . . . .. . ... . . . . . . . ... .. . ... . . . . . . . ... .. . ... . . . . . . . ... .. ... . . . . . . . ... . . . ... . . . . . . . . ........................................................................................................................................................................................................................................................................ . . . .. .. . ... . . . . . . ... . .. .. . ... ... .. .. .. .. .. .. ... ... .. .. . . . . .. . . . . . . . . . . .. . . . .. . . . . . . .. . . ... .. . . . . . . . . .. ... . . . . . . . .. ... ... . .. . . .. ... ... .. .. . . ..... .. ..... . . .... ..... . .. . . ... ..... .. . . . . . .. . .. . .. . . . . . .. .. . ... . . . . . ... .. .. .. . ........... . . . . . . . ... .. .. ..... . . . . .. ......... . . . . . . . . .. ... .. .. ........ . . . . .. . . . . . . . . ........ .. ... .. . . . . . .. . . . . . . . . . ........ .. ... ......... .. .. .......... .. .. ....... .. .. .. .. .. ......... .. .......... .. .. .. ........ ..... . . . .. ... ............... . . . . . . . . . . . ... ........ .. . .. ........... .. ... ........ ... .. ... .......... ... .. ........... .. .. ... ......... ... .. .......... . .. ... . . . . . . . . . . . . . . . .. ... ... .. .. ........... ...... . .. ... ........ ... .. .. ........ ... ........ ..... ..... ........ .. .. ........ ........ ... . . .. . . . . . . . . . . ... ........ ........ ... . . . . . . .......... ... ... .. . . . . . . . . . ... . . .. ......... .. . . . . . . . . ... . . . ........ . .. .... ........ .. ... .. .. ........ ........ ... .... ... .. ............... ......... .... ... ...... . . . . . . . . . . . . . . ... . ..... ..... . .. ... .. ............... ........ .... ... ... ........ ........ .. .. .... ... ........ ........ .. . .. . . . .................. . .......... .. ........ .. .. ..... ........ .. ........ .. ........ .. ........ . . . ........ . . . . . . . . . ........ .. ........ ........ .... .. ........... ........ . ........................................................................................

 pe2

el2

el3

 pe3

 pe5 el5

el0

 pe0

 pe1

el4



el1

pe4

Figure 1 The dual graph of the 14 rational curves on B 2

18

.......................................... ............................................. ........ .......... .......... ....... ........ ....... ....... ....... ....... ...... ....... . ...... . . . . . . . . . ...... .. ..... .. . . . . . . . . ..... ..... .. .. . . . . . . . .... .... .... . . . ... . ... .. . . .. . . ......... . ... . . . . . . . . . . . . . . . . . . . ...... .... ................. . . . . .. . . . . .. . . . . ......... .. ... . . . . . .. . . . . . . . . ........ ... . . .. . . . . . . . . . . . ...... .. .. .. . . . . . . . . . . ...... . .. .. . . . . . . . . . ..... .. .. .. . . . . . . . . . . .. ..... .. . . . . . .. . ... . . . .. . . .. . . . .. . . .. . . . .. . . . . .. . .. . . . . . . .. . . . . . . . . .. ..... .............. ...... ............. . . . . . . . . . . . . .. . . . . . . . . .. ...... ...... .. .. . . . . . . . . .. . . . . . . . . . ..... ..... .. .. . . . . . . . . . . . . . . . . .. ..... .... .. .. . . ..... . . . . . . .. . ... .. ... . . . . . . . . .. ... . . . . ... . . . . . . . .. .. ... .. . ... . .. . . . ..... ... ... ... . . . . . . . .................................................................................................................................................................................................... . . . ................... . . . . . . . . . . . ... .. . .. .. . . .. . . . . ... . .. . . ... . . .. . . .. . . . . . . . . . . . . .. .. . . .. .. .. .... .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. . . . .. . . .. . . . . . .. . .. .. . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . .. . .. .. .. .. .. .. .. .. .. .. .. . .. . .. .. . . . . . .. .. .. . .. .. .. .. .. .. .. ... .. .... .. ... .. .. .. ... .. . .. . . . .... .. ................................................................................................................................................................................................................................................................................................................................................................... .. . .. .... ...... ... ... . .. ..... . . . . . . . .... ..... . .. ... ... ..... ..... ..... ..... .. .. ...... ...... ...... ...... .. .. ....... ....... ...... ...... . . . . . .. .. . . . . . . . . . . . . . ........ ........ .. .. . . . . . .. . . . . . . . . . . . . . . . ............. ............ .. .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ ................ .. . . .. . . .. . . .. . . .. ..

Figure 2 Con guration of the double curves in the closed ber of the uniformized scheme for 0B 1

19

.. .. ... .. .. .. .. .. .... .. .. .... ........................................................................ . . . . . . . . . . . . .......... ... . . . .. . . . . . . . ......... . ... .... ....... .. ........ .. ....... .. ....... .. . . . . ...... . . . . .. .. . ...... . . . . . . . . . ..... . . . . . . . . ..... .. . . . . . . . . ..... ... ... . . . . ..... . .. . . . . . . .... .. .. .. . ... . . . .. . .. . .. . . . .. . .... . . . . . . . . . . . . . . . . . . .. . . . ... . . . . . .... .. . . .... . .. . . .. .. . . . .. . . . .. .. . . .. . . . . .. . . . . .. . . . . .... . . . .. . . . . . . . . . . . . . . . ....... .. ....... ... . . . .. . . . . . . . . . . ....... .. . .. . . .. . . . . . . . . . . . . . . . . . . ............. .. ....... ....... .. .. .. . . . . . . . . . . . . .......... ...................... .. . . . . . . .................. .. . . . . . . . ................ . . . . . . .. ... ............... ......... .. . . . . . . . . . . . . . . . . . . . .. .. . . . ... .......... .. .. . . . . . . . .. .. . ....... . . .. .. . . .. . . . . . . . .. ....... . . . . . . . . .. . . . .. . . . . ....... ... . . . . . .. . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . . ............. ... . . . .. . . . . . .. ............. . .. . . . .. ....... . ....... . . . . ... . . .. . . . . . ....... . .. .. . . . . . . . . . . . . . .. ....... . .. ... . . . . . . . . . . . . . . . . ... .. .. . . . . . . . . . .. .. . . .. . .. .. ... . .. ... . .. .. . .. .. . .. .. . . . . . . .. .. .... . .. .. . ... .. ..... . .. .. . .. .. ..... . .. .... .. .... ..... . . . . .. .. . . . .. .. . .. .. .......... . ............................................................................................................................................... .................................................................................................................................................. .. .. . . . .. . . . . . .. . ... .... . . . .. ... . . . . . . . ... . ... . . . . . . . . . .... .. . .... . . .. .... .... ..... .. . .. .... ... . .... . .. . . . . . . . . . . . . . . . ... .. . . . . . . . . .... . . . . . . ................... ..... ...... ....... ....... ......... ........ . . . .......... . . . . . ............. ...... .................................................................

Figure 3 Con guration of the double curves in the closed ber of the uniformized scheme for 0B 2

20

.. .. ... .. .. .. .. ... .. . .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................ . . . . . . . . . . . . . . . ... . . . . . . .. .... . .. . . . . .. . . . .... .. . .. .. ... .. .. .. . . .. . .. . . ... . . . . .. . . . .. . .. . .. . . . .. . . . . . .. . . . . . .. ... . ... . . . . . .. . . . . . .. . .. . . . . . . . .. ... . . . . . .. .. ... . . . .. .. .. .... . . ...................................................................... . . ... . . . . .. .. . . .. ... . .. .. . . .. ... . . . .. . . .. . .. . .. . . . ... . . . . .. . . .. . .. . . .. ... . . . . .. . .. .. . .. ... . .. .. . .. . .. . . . . . . . . .. ... .. . .. .. .... ..... .. .... .. .. ...... . ........ ...................................................................................................................................................................................................... .. .... .... ... ... .. .. .. .... .. .... . . . ... . ... . .. .. .. .. .. .. .. .. ... .. .. . .. . .. . .. . . .. . . . .. . . .. . . .. . .. . . . .. .. . .. . . . . . . .. .. .. . . . . . . ... ... .. .. . . . . .. .. .. . .. . . . .. . . . . . .. ... .. .. ... .. .... .. .. .. . .. . .. . . .. ............................................................................................................................................................................................................................................................... . . . ... . . .. ... . . .... . . . . ..... . . . . . ...... .. . ... . ...... .. ...... ....... ...... ....... .. .. ....... ......... ........ ........... .. .. .......................................... .................................. ... .. . .. ..

Figure 4 Con guration of the double curves in the closed ber of the uniformized scheme for 0C 1

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References

[1] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, SpringerVerlag, Berlin, Heidelberg,1984. [2] V. G. Berkovich, The automorphism group of the Drinfeld half-plane, C. R. Acad. Sci. Paris, Ser. I. Math. 321, 1127-1132 . [3] K. S. Brown, Buildings, Springer-Verlag, New York, 1989. [4] D. I. Cartwright, A. M. Mantero, T. Steger and A. Zappa, Group acting simply transitively on the vertices of a building of type Ae2 , I, Geometriae Dedicata 47 (1993), 143{166. [5] D. I. Cartwright, A. M. Mantero, T. Steger and A. Zappa, Group acting simply transitively on the vertices of a building of type Ae2, II | the cases q = 2 and q = 3, Geometriae Dedicata 47 (1993), 167{223. [6] V. G. Drinfeld, Elliptic modules, Math. USSR Sbornik 23 (1974), 561{592. [7] V. G. Drinfeld, Covering of p-adic symmetric regions, Funct. Anal. and Appl. 10 (1976), 29{40. [8] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer-Verlag, Berlin, Heidelberg, New York, 1966. [9] M.-N. Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math. J. 40 (1988), 367{396. [10] M.-N. Ishida, Convex sets in the p-adic open ball, preprint. [11] M.-N. Ishida and T. Oda, Torus embeddings and tangent complexes, Tohoku Math. J. 33 (1981), 337{381. [12] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Math. 339, Springer-Verlag, Berlin, Heidelberg, New York, 1973. [13] A. Kurihara, Construction of p-adic unit balls and the Hirzebruch proportionality, Amer. J. Math. 102 (1980), 565{648. [14] D. Mumford, An analytic construction of degenerating curves over complete local rings, Compositio Math. 24 (1972), 129{174. [15] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239{272. [16] D. Mumford, An algebraic surface with K ample, (K 2 ) = 9; pg = q = 0, Amer. J. Math. 101, (1979), 233{244. 22

[17] G. A. Musta n, Nonarchimedean uniformization, Math. USSR Sbornik, 34 (1978), 187{214. [18] T. Oda, Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties, Ergebnisse der Math. (3), 15, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1988. [19] M. Rapoport, Non-Archimedean period domains, Procedings ICM 1994 Zurich (1994). [20] M. Rapoport and Th. Zink, Period spaces for p-divisible groups, Bergische Universitat Wuppertal and Universitat Bielefeld (1994). [21] M. Raynaud, Geometrie analytic rigide, d'apres Tate, Kiehl, Bull. Soc. Math. France, Memoir 39{40 (1974), 319{327. [22] P. Schneider and U. Stuhler, The cohomology of p-adic symmetric spaces, Invent. Math. 105 (1991), 47{122. [23] H. Voskuil, Ultrametric uniformization and Symmetric Spaces, Thesis, Rijksuniversiteit Groningen, 1990. [EGA3] A. Grothendieck and J. Dieudonne, Elements de Geometrie Algebrique III, Publ. Math. IHES 11 (1961), 17 (1963). [EGA4] A. Grothendieck and J. Dieudonne, Elements de Geometrie Algebrique IV, Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966) and 32 (1967).

Masa-Nori Ishida

Mathematical Institute Tohoku University Sendai 980-77 Japan e-mail : [email protected] Fumiharu Kato

Graduate School of Mathematics Kyushu University 33 Fukuoka 812, Japan e-mail : [email protected]

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