StefanImmervoll Realanalyticprojectiveplaneswithlargeautomorphismgroups

(de Gruyter 2003

Real analytic projective planes with large automorphism groups

Stefan Immervoll

(Communicated by T. Grundho¨fer)

Abstract.We prove that there exist non-classical projective planes whose point space and line space are real analytic (or Nash) manifolds such that the geometric operations of joining points and intersecting lines are real analytic (even Nash) maps on their respective domains. Our examples have the dimensions 2, 4, or 8. These planes are the first examples of non-classical smooth projective planes with large automorphism groups. In dimension 2, they correspond to a class of projective planes discovered by Segre.

Key words.Smooth/real analytic/Nash projective plane, automorphism group.

2000 Mathematics Subject Classification. Primary 51H25, 51A10, Secondary 51H30, 51A35

1 Introduction

In [11], B. Segre constructed examples of non-desarguesian smooth projective planes, whose lines are real algebraic curves in the real projective plane with its usual real algebraic structure. The construction of these planes was motivated by a prize- question posed by Het Wiskundig Genootschap in 1955. However, as mentioned in [10], 75.6, he did ‘not consider the question whether the planes are, for example, real analytic or algebraic planes, that is, whether the geometric operations belong to one of these categories’. In this paper we show that the geometric operations of joining points and intersecting lines are in fact real analytic and even Nash maps. For the definition of Nash functions and maps, see [1], 2.9.3 and 2.9.9, cf. also Section 8.1 in that book. Furthermore, we present the first examples of non-desarguesian projective planes with these properties in dimensions 4 and 8. Recall that by [10], 75.1, or by [7], every holomorphic projective plane is isomorphic toP2Cwith its usual holomorphic structure, and by [12] or [8], every algebraic projective plane over an algebraically closed field is pappian. Our approach also yields a new proof for Segre’s result that the incidence structures constructed by him are projective planes. Note in this context that finite-dimensional, compact, connected projective planes always have dimension 2, 4, 8 or 16, cf. [10], 52.5. It should be possible to prove an analogous result in the 16-dimensional setting by using Veronese coordinates instead of homogeneous coor-

dinates (see [10], 16.1). Homogeneous coordinates cannot be used in this case because of the non-associativity of the octonions.

The projective planes considered in this paper are constructed as follows: the point space P and the line space L are copies of the point space and the line space of P2Kwith their standard smooth, real analytic, and real algebraic structure (K¼R, CorH). Hence, points and lines may be described by means of homogeneous coor- dinates in the usual way. A pointðx;y;zÞ^tAP(wheretdenotes transposition) and a lineða;b;cÞALare calledincidentif

ðjaj²þ jbj²þ jcj²ÞðaxþbyþczÞðjxj²þ jyj²þ jzj²Þ þljcj²czjzj²¼0:

Here,lARis a fixed parameter. Forl¼0 we get the incidence relation of the clas- sical projective planeP2K. The flag space Fl is the set of incident point-line-pairs.

The incidence structures Pl¼ ðP;L;FlÞdefined in this way are self-dual. A polar- ity is given by the map PL!PL:ððx;y;zÞ^t;ða;b;cÞÞ 7! ðða;b;cÞ^t;ðx;y;zÞÞ, where denotes conjugation. Of course, the incidence structures Pl cannot be expected to be projective planes in general. In this paper we prove that they are real analytic and even Nash projective planes for jlj su‰ciently small. To be more pre- cise, our proof yields that jlj<¹₉ is su‰cient. In [11], Segre proves in two di¤erent ways that the planesPlare non-desarguesian forl00 andK¼R. In Section 1 (pp.

36/37) he shows this by a theoretical argument, and in Section 4 (pp. 39/40) he veri- fies directly that Desargues’ theorem fails in Pl forl00 su‰ciently small. A pro- jective planePlwithK¼C;Hhas a 2-dimensional subplane equal to the projective plane constructed by Segre with the same parameterland hence is not desarguesian forl00.

Before we proceed, let us first recall some basic results on automorphisms of com- pact or smooth projective planes. Theautomorphism groupSof a compact (smooth) projective plane P¼ ðP;L;FÞis the group of all automorphisms of P as an inci- dence structure which induce homeomorphisms (di¤eomorphisms) onPandL. These automorphisms are called continuous (smooth) automorphisms. Note that by [4], 4.7, a continuous automorphism of a smooth projective plane is smooth. The automor- phism group S of P is endowed with the compact-open topology derived from its action on PorL, respectively. These two topologies coincide by [10], 44.2. In this way,Sbecomes a locally compact topological group with a countable basis, see [10], 44.3. Hence, the dimension of S is defined, compare [10], 93.5 and 6. By a group of automorphisms ofP we mean a closed subgroup ofSendowed with the induced topology.

The projective planes Pl presented in this paper are the first examples of non- classical smooth projective planes with large automorphism groups. They admit Lie groups of smooth automorphisms of dimension 1, 4 or 13 forl¼1, 2 or 4, respec- tively. By [2], the dimension of the automorphism group of a 2l-dimensional non- classical smooth projective plane is at most 2, 6 or 16 forl¼1, 2 or 4, respectively.

These bounds are 2 less than the corresponding bounds in the case of compact pro- jective planes, but it is not known if they are sharp. Our examples show that the bounds found by Bo¨di are not far from the truth. The Lie groups of smooth auto-

morphisms of the projective planesPl mentioned above are in factcompactgroups.

This shows that, in contrast to the automorphism groups of smooth projective planes, the bounds for the dimensions of compact groups of automorphisms of non-classical compact projective planes are the same as those in the smooth setting forlAf1;2;4g, see Theorem 2.9.

2 Proofs and details

LetI¼ ðP;L;FÞbe an incidence structure. The elements ofP,L, andFare called points, lines, and flags, respectively. For pAP,LALwe callP_L¼p_Pðp_L¹ðLÞÞthe point row associated with L andLp¼p_Lðp¹_P ðpÞÞ the line pencil through p, where p_P:F!Pand p_L :F!L denote the canonical projections. A projective plane P¼ ðP;L;FÞis called asmooth projective plane ifPandLare smooth manifolds such that the two geometric operations of joining distinct points and of intersecting distinct lines are smooth, i.e. di¤erentiable in the sense ofC^y.Real analyticorNash projective planesare defined analogously. The next theorem is essential for the proof of the main result of this paper, for a proof see [3], 1.5, or [6], 4.5.

Theorem 2.1.LetI¼ ðP;L;FÞbe an incidence structure which satisfies the following conditions:

(SGP1) There is a positive integer l such that P andLare compact,connected smooth 2l-dimensional manifolds.

(SGP2) The flag space F is a closed smooth 3l-dimensional submanifold of PL, and the canonical projectionsp_Pandp_L are submersions.

(SGP3) Any two distinct lines intersect transversally in P and any two line pencils associated with distinct points intersect transversally inL.

Then there are positive integers m, n such that any two distinct points are joined by exactly m lines and any two distinct lines intersect in exactly n points.

We add some comments on this theorem. The canonical projectionsp_Pandp_L are surjective sinceFis compact, submersions are open maps, andP,Lare connected.

Hence, point rows and line pencils are smoothl-dimensional submanifolds ofPand L, respectively, by (SGP1) and (SGP2), see [3], 1.1, or [6], 4.1. Thus the transversality condition in (SGP3) makes sense. Recall that two linesL,Kof an incidence structure satisfying conditions (SGP1) and (SGP2) are said to intersect transversally in some point p, if the associated point rows PL andPK intersect transversally in p as sub- manifolds of P, i.e. their tangent spaces inp span the tangent space TpP, or, equiva- lently, the intersection of their tangent spaces in p is trivial. They are said to inter- sect transversally if they intersect transversally in each common point. Note that two lines which intersect transversally need not have a common point. Transversal inter- section of line pencils is defined dually.

The proof of the main result of this paper is based on the following corollary (see [3], 1.6, or [6], 4.6).

Corollary 2.2. Assume thatI¼ ðP;L;FÞsatisfies the conditions of Theorem 2.1.If there are two lines whose intersection consists of at most one point,or if there are two points which are joined by at most one line,thenIis a smooth projective plane.

We want to show now that the incidence structuresPlin general (lARarbitrary) admit non-trivial groups of smooth automorphisms which are compact Lie groups.

Lemma 2.3. ForK¼R,the orthogonal groupO₂R acts onPlas a group of smooth automorphisms.ForK¼C,the incidence structurePl admits a group of smooth auto- morphisms isomorphic to the unitary groupU2C.Moreover,also complex conjugation induces a smooth automorphism ofP_l.ForK¼H,the incidence structureP_ladmits a group of smooth automorphisms isomorphic to the product ofSpin₃RandSpin₅Rwith amalgamated centers.

Proof. For K¼R, let G be the subgroup of O3R which fixes ð0;0;1ÞAR³. This subgroup is isomorphic to O2R. The standard action ofGonR³ induces an e¤ective smooth action of Gon the line spaceL. Analogously, we define an e¤ective smooth action of G on P by GP!P:ðg;ðx;y;zÞ^tÞ 7!g¹ðx;y;zÞ^t. By definition of the incidence relation in Pl we see that the induced action of G on PL leaves Fl

invariant, i.e.Gacts onPlas a group of smooth automorphisms.

ForK¼C, an analogous proof shows that the unitary group U₂Cacts onPl as a group of smooth automorphisms. The fact that complex conjugation induces a smooth automorphism ofPlalso follows directly from the definition of the incidence relation.

For K¼H, let G be the subgroup of the unitary group U3H isomorphic to U2HU1H, which acts on the first two components ofðx;y;zÞAH³ as U2Hand on the last component as U1H. Note that U1His isomorphic to Spin₃R and that U₂H is isomorphic to Spin₅R, cf. [10], 95.10. The action of G on H³ induces an e¤ective smooth action of the product of U₂Hand U₁Hwith amalgamated centers on the line space L. As before, we see that this group acts on Pl by smooth auto-

morphisms. r

The preceding lemma will enable us to choose appropriate coordinates in the proof of the main result of this paper.

Theorem 2.4. The incidence structures Pl ¼ ðP;L;FlÞ are smooth projective planes forjlj<¹₉.

The next lemma presents the most di‰cult part of the proof of this theorem. In the sequel, we will use the description of the point space Pand the line spaceLof the incidence structure Pl¼ ðP;L;FlÞ by means of the standard charts: for the point spaceP, the corresponding open setsU₁,U₂, andU₃are given byx00, y00, and z00, respectively, and these sets are identified withK²in the usual way. In the latter case, for example, we use the map U₃!K² :ðx;y;zÞ^t 7! ðx=z;y=zÞ. Analogously we define open sets V₁, V₂, and V₃ bya00,b00, andc00, respectively, which

cover the line space L. Sometimes it will be convenient to identify K withR^l by choosing f1g,f1;ig orf1;i;j;kg, respectively, as a basis of KoverR. In this way, left multiplication by some element cAK gives rise to a linear map Lc:R^l!R^l, and right multiplication bycinduces a linear mapRc:R^l !R^l.

In order to avoid cumbersome notation we will sometimes use the same names for di¤erent variables in the following two proofs, if such a choice is natural, facilitates reading, and no confusion is possible.

Lemma 2.5.Letjlj<¹₉.Then the setF¼FVðU3V3Þis a smooth3l-dimensional submanifold of PL. The restrictions of the natural projections p_P and p_L to F are submersions.The sets P_LVU₃ andLpVV₃with pAp_PðFÞand LAp_LðFÞare smooth l-dimensional submanifolds of P and L, respectively. If two distinct lines L;L⁰AV₃ intersect in a point pAU₃ then the submanifolds P_LVU₃ and P_L⁰VU₃ intersect transversally in p.Also the dual statement holds.

Proof. We identify the open subsetsU3JPandV3JLwith two distinct copies of K². By means of these identifications, the setF corresponds to

fðx;y;a;bÞAK²K²j ðjaj²þ jbj²þ1Þðaxþbyþ1Þðjxj²þ jyj²þ1Þ þl¼0g:

For anyða;bÞAV3we define

g_ða;bÞ:K²!K:ðx;yÞ 7! ðjaj²þ jbj²þ1Þðaxþbyþ1Þðjxj²þ jyj²þ1Þ þl:

We want to prove the technical result that the kernels of the di¤erentials D_ðx;yÞg_ða;bÞ and D_ðx;yÞg_ða⁰_;b⁰_Þhave trivial intersection for any two distinct quadruplesðx;y;a;bÞ;

ðx;y;a⁰;b⁰ÞAF. Then the claims above will follow easily. By using transitivity properties of the group of smooth automorphisms of P_l (see Lemma 2.3) we may assume that y¼0,xAR, andbAR. The above incidence relation then shows that axARnf0g (because of jlj<1) and hence that aAR. Analogously we see that a⁰AR. For the sake of simplicity we will assume in the following thatK¼H. Some- times we will identify H with R⁴ and associate to any element wAH a vector ðw1;w2;w3;w4ÞAR⁴. In this way, the di¤erential of the mapy:H!H:t7! jtj²at a point tAHcorresponds to the map Dty:R⁴ !R⁴:ðw1;w2;w3;w4Þ 7! ð2ðw1t1þ w₂t₂þw₃t₃þw₄t₄Þ;0;0;0Þ. Now let ðu;vÞAker D_ðx;0Þg_ða;bÞVker D_ðx;0Þg_ða⁰_;b⁰_Þ and assume thatðu;vÞ0ð0;0Þ. By di¤erentiatingg_ða;bÞatðx;0Þwe get

ðR_x²_þ1LaþLaxþ1DxyÞuþ ðR_x²_þ1LbþLaxþ1D0yÞv

¼ ðx²þ1Þauþ ðaxþ1Þ2xu1þ ðx²þ1Þbv¼0: ð1Þ Here we have considereduandvas elements ofR⁴in the first line and as elements of Hin the second line. Analogously, we get

ðx²þ1Þa⁰uþ ða⁰xþ1Þ2xu1þ ðx²þ1Þb⁰v¼0: ð2Þ

We multiply Equation (1) byb⁰from the left and Equation (2) byb. Subtracting the two equations obtained in this way yields

ðab⁰a⁰bÞðx²þ1Þuþ ððaxþ1Þb⁰ ða⁰xþ1ÞbÞ2xu1 ¼0 and hence

ðab⁰a⁰bÞððx²þ1Þuþ2x²u₁Þ þ ðb⁰bÞ2xu1¼0: ð3Þ As a next step we want to prove that b0b⁰. If we have b⁰¼bAR, then a anda⁰ are zeros of the real polynomial function p:R!R:s7! ðs²þb²þ1Þðsxþ1Þ ðx²þ1Þ þl. Let sAR with p⁰ðsÞ ¼0 (if it exists). Then we have 2sðsxþ1Þ þ ðs²þb²þ1Þx¼0 which implies that

sxþ1¼1 2s²

3s²þb²þ1>1 3:

Hence, we get pðsÞ ¼ ðs²þb²þ1Þðsxþ1Þðx²þ1Þ þl>¹₃þl>0 because of jlj<¹₉. Since pis a real polynomial function of degree 3, this shows that phas pre- cisely one real zero. We conclude thata¼a⁰, a contradiction. So, we haveb0b⁰and therefore alsou00 by Equations (1) and (2). Equation (3) then yields

ðb⁰bÞ¹ðab⁰a⁰bÞ ¼ 2xu1ððx²þ1Þuþ2x²u₁Þ¹ ð4Þ and

jab⁰a⁰bj²

jb⁰bj² ¼ ð2xÞ²u₁²

ð3x²þ1Þ²u₁²þ ðx²þ1Þ²ðu₂²þu²₃þu₄²Þ c ð2xÞ²

ðx²þ1Þ²

u₁² u₁²þu₂²þu₃²þu²₄ c 2x

x²þ1

2

c1:

Thus we get

jðb⁰bÞ¹ðab⁰a⁰bÞjc1: ð5Þ On the other hand, Equation (4) implies that

ðb⁰bÞ¹ðab⁰a⁰bÞxþ1¼12x²u1ððx²þ1Þuþ2x²u1Þ¹

¼ ðx²þ1Þuððx²þ1Þuþ2x²u1Þ¹:

We conclude that

jðb⁰bÞ¹ðab⁰a⁰bÞxþ1j²¼ ðx²þ1Þ²juj²

ð3x²þ1Þ²u²₁þ ðx²þ1Þ²ðu²₂þu₃²þu₄²Þ d ðx²þ1Þ²

ð3x²þ1Þ²

juj²

u₁²þu²₂þu₃²þu₄²:

Because of_3x^x22^þ1þ1>¹₃, we get the inequality

jðb⁰bÞ¹ðab⁰a⁰bÞxþ1j>1

3: ð6Þ

Sinceðx;0;a;bÞAFimpliesða²þb²þ1Þðaxþ1Þðx²þ1Þ þl¼0, we haveaxþ1¼ lða²þb²þ1Þ¹ðx²þ1Þ¹ and, analogously, a⁰xþ1¼ lða⁰²þ jb⁰j²þ1Þ¹ ðx²þ1Þ¹. We multiply the first of these two equations byb⁰ and the second byb.

After subtracting these two equations we obtain ðab⁰a⁰bÞxþ ðb⁰bÞ ¼ l

x²þ1

ða⁰²þ jb⁰j²þ1Þb⁰ ða²þb²þ1Þb ða²þb²þ1Þða⁰²þ jb⁰j²þ1Þ : ð7Þ We want to multiply Equation (7) byðb⁰bÞ¹in order to combine it with (6). We have

ða⁰²þ jb⁰j²þ1Þb⁰ ða²þb²þ1Þb¼ ða⁰²b⁰a²bÞ þ ðjb⁰j²b⁰b³Þ þ ðb⁰bÞ; wherea⁰²b⁰a²b¼ ðb⁰bÞða²þaa⁰þa⁰²Þ ðab⁰a⁰bÞðaþa⁰Þ, andjb⁰j²b⁰b³¼ ðb⁰bÞðjb⁰j²þb⁰bþb²Þ ðb⁰b⁰Þb² with

jb⁰b⁰j²

jb⁰bj² ¼ 4ðb₂⁰²þb₃⁰²þb₄⁰²Þ

ðb₁⁰bÞ²þb₂⁰²þb₃⁰²þb₄⁰² c4:

Hence, we get

jðb⁰bÞ¹ðjb⁰j²b⁰b³Þjcjb⁰j²þ jb⁰bj þb²þ jðb⁰bÞ¹ðb⁰b⁰Þjb²

cjb⁰j²þ jb⁰bj þ3b² and

jðb⁰bÞ¹ða⁰²b⁰a²bÞj

ca²þ jaa⁰j þa⁰²þ jðb⁰bÞ¹ðab⁰a⁰bÞjðjaj þ ja⁰jÞ ca²þ jaa⁰j þa⁰²þ jaj þ ja⁰j

by inequality (5). By combining these inequalities with (6) and (7), we obtain 1

3<jðb⁰bÞ¹ðab⁰a⁰bÞxþ1j

cjlja²þ jaa⁰j þa⁰²þ jaj þ ja⁰j þ jb⁰j²þ jb⁰bj þ3b²þ1

ða²þb²þ1Þða⁰²þ jb⁰j²þ1Þ : ð8Þ Obviously, we have

a²þa⁰²þ jb⁰j²þb²þ1 ða²þb²þ1Þða⁰²þ jb⁰j²þ1Þc1:

Using that s¹₂2

d0, and hencescs²þ¹₄, for everysAR, we get jaj þ ja⁰j þ jaa⁰j þ jbb⁰j þb²ca²þa⁰²þ jaa⁰j²þ jb⁰bj²þ1þb²

cða²þb²þ1Þða⁰²þ jb⁰j²þ1Þ:

Hence, we obtain

ða²þa⁰²þ jb⁰j²þb²þ1Þ þ ðjaj þ ja⁰j þ jaa⁰j þ jb⁰bj þb²Þ þb² c3ða²þb²þ1Þða⁰²þ jb⁰j²þ1Þ;

which shows together with inequality (8) that ¹₃<3jlj, in contradiction to jlj<¹₉. Thus the kernels of the di¤erentials D_ðx;yÞg_ða;bÞand D_ðx;yÞg_ða⁰_;b⁰_Þintersect trivially for any two distinct quadruplesðx;y;a;bÞ;ðx;y;a⁰;b⁰ÞAF.

We want to show next that there are infinitely many lines inV₃ through any point ðx;yÞAp_PðFÞ. Using the transitivity properties of the automorphism group ofPl

we may arrange again that ðx;yÞ ¼ ðx;0Þ with xAR. Because of ðx;0ÞAp_PðFÞ there is a line ða⁰;b⁰ÞAV₃ incident with the point ðx;0Þ. We then have ðja⁰j²þ jb⁰j²þ1Þða⁰xþ1Þðx²þ1Þ þl¼0, which shows thatx00. Hence the real polyno- mial function qb :R!R:s7! ðs²þb²þ1Þðsxþ1Þðx²þ1Þ þl has degree 3 for everybAR. Thus, for anybARthere existsaARsuch thatða²þb²þ1Þðaxþ1Þ ðx²þ1Þ þl¼0, i.e. such thatðx;0;a;bÞAF.

Byðx;yÞwe denote again an arbitrary point ofpPðFÞ. Choose two distinct lines ða;bÞ;ða⁰;b⁰ÞAV3throughðx;yÞ. By definition ofgða;bÞandgða⁰;b⁰Þ, the dimensions of the kernels of the two di¤erentials Dðx;yÞgða;bÞ and Dðx;yÞgða⁰;b⁰Þ are at leastl. Since they intersect trivially, their dimension is preciselyl and hence these di¤erentials are surjective. In particular, also the total di¤erential of the map

f :K²K²!K:ðx;y;a;bÞ 7! ðjaj²þ jbj²þ1Þðaxþbyþ1Þðjxj²þ jyj²þ1Þ þl is surjective at every point ofF. ThereforeF is a 3l-dimensional submanifold of U₃V₃and hence ofPL.

Now we want to show that the restriction of the natural projectionp_L toF is a submersion. Choose ðx;y;a;bÞAF arbitrarily. An element of ker D_ðx;y;a;bÞp_L has the form ðu;v;0;0Þwithðu;vÞAK². By definition ofF we have Dðx;yÞgða;bÞðu;vÞ ¼ Dðx;y;a;bÞfðu;v;0;0Þ ¼0. Since the kernel of Dðx;yÞgða;bÞisl-dimensional, we conclude that the dimension of ker Dðx;y;a;bÞp_L is at mostl. Thus the di¤erential Dðx;y;a;bÞp_L is surjective. Hence the restriction ofpLtoFand, for reasons of symmetry, also the restriction ofpPtoFare submersions. By [3], 1.1, or [6], 4.1, it follows that the sets PLVU3 and LpVV3 are smooth l-dimensional submanifolds of P and L, respec- tively, for any pApPðFÞ,LApLðFÞ.

It remains to show that any two distinct lines L¼ ða;bÞ andL⁰¼ ða⁰;b⁰Þin V₃ which intersect in a point p¼ ðx;yÞAU₃intersect transversally in p. The dual state- ment then follows by symmetry. Choose ðu;vÞ in the intersection of the tangent spaces of the point rowsP_L andP_L⁰ in p. Sinceg_ða;bÞ vanishes onP_LVU₃, we con- clude that D_ðx;yÞg_ða;bÞðu;vÞ ¼0. Analogously, we get D_ðx;yÞg_ða⁰_;b⁰_Þðu;vÞ ¼0 and hence ðu;vÞ ¼ ð0;0Þ, since the kernels of Dðx;yÞg_ða;bÞðu;vÞ ¼0 and D_ðx;yÞg_ða;bÞðu;vÞ ¼0 have

trivial intersection. This completes the proof. r

Proof of Theorem2.4. As in the classical projective planeP0¼P2K, the point rows of the linesð1;0;0Þ;ð0;1;0ÞALintersect precisely in the pointð0;0;1Þ^tAP. Hence, by Corollary 2.2, it su‰ces to verify the conditions of Theorem 2.1. We first show that the flag space Fl is a 3l-dimensional submanifold of PL and that p_L is a submersion. Then also the natural projectionp_Pis a submersion for reasons of sym- metry. By the previous lemma, it remains to prove these properties in neighbour- hoods of flags ðp;LÞin PL, where the last coordinate of p or L is 0. By using transitivity properties of the group of smooth automorphisms ofP_l(see Lemma 2.3), we see that it is su‰cient to consider the following cases:

(F1) p¼ ðx;y;1Þ^t,L¼ ð1;0;0Þ, (F2) p¼ ðx;1;0Þ^t,L¼ ð1;0;0Þ, (F3) p¼ ð1;0;0Þ^t,L¼ ða;b;1Þ.

Note that the point ð1;0;0Þ^t and the line ð1;0;0Þ are not incident. Moreover, the condition thatðp;LÞis a flag implies thatx¼0 in the first two cases and that a¼0 in (F3). As in the proof of the previous lemma we introduce appropriate inhomo- geneous coordinates. In the Case (F1) we identify U₃ and V₁ with two copies of K². In this way, the point p corresponds toð0;yÞAK² and the lineLcorresponds to ð0;0ÞAK². The setFl ðU3V₁Þ is then given byðf^ð1ÞÞ¹ðf0gÞ, where f^ð1Þ is defined by

f^ð1Þ:K²K² !K:

ðx;y;b;cÞ 7! ð1þ jbj²þ jcj²ÞðxþbyþcÞðjxj²þ jyj²þ1Þ þljcj²c:

For any ðb;cÞAK² we define g^ð1Þ_ðb;cÞ:K²!K:ðx;yÞ 7!f^ð1Þðx;y;b;cÞ. We have D_ð0;yÞg^ð1Þ_ð0;0Þ:K²!K:ðu;vÞ 7! ðjyj²þ1Þu, which shows that Dð0;yÞg^ð1Þ_ð0;0Þis surjective.

Hence, the total di¤erential of f^ð1Þ in ð0;y;0;0Þis also surjective. Thus there exists an open neighbourhoodW ofðp;LÞinPLsuch thatFlVW is a 3l-dimensional submanifold ofW. Moreover, we see as in the proof of Lemma 2.5 that the restric- tion of the natural projection p_L to FlVW is a submersion if the neighbourhood W ofðp;LÞis small enough so that the di¤erential ofg^ð1Þ_ð0;0Þ is surjective at all points ofW.

For (F2) we identify U2 and V1 with K² such that ðp;LÞ corresponds to ð0;0;0;0ÞAK²K². We define

f^ð2Þ:K²K²!K:

ðx;z;b;cÞ 7! ð1þ jbj²þ jcj²ÞðxþbþczÞðjxj²þ1þ jzj²Þ þljcj²czjzj² such thatFlVðU2V₁Þis identified with the setðf^ð2ÞÞ¹ðf0gÞ. The di¤erential of

g^ð2Þ_ð0;0Þ:K²!K:ðx;zÞ 7!xðjxj²þ1þ jzj²Þ;

defined analogously to g^ð1Þ_ðb;cÞ, at ð0;0Þ is given by D_ð0;0Þg^ð2Þ_ð0;0Þ:K²!K:ðu;vÞ 7!u and hence is surjective. As in the previous case we conclude that there is an open neighbourhoodW ofðp;LÞinPLsuch thatFlVW is a submanifold ofW and p_L restricted toFlVW is a submersion.

In (F3) we identify U₁V₃ withK²K² such that the flagðp;LÞcorresponds toð0;0;0;bÞAK²K². The setFlVðU1V₃Þis then identified withðf^ð3ÞÞ¹ðf0gÞ, where

f^ð3Þ:K²K²!K:

ðy;z;a;bÞ 7! ðjaj²þ jbj²þ1ÞðaþbyþzÞð1þ jyj²þ jzj²Þ þlzjzj²: We defineg^ð3Þ_ð0;bÞ:K² !K:ðy;zÞ 7!f^ð3Þðy;z;0;bÞ. Then we have

D_ð0;0Þg^ð3Þ_ð0;bÞ:K²!K:ðu;vÞ 7! ðjbj²þ1ÞðbuþvÞ;

which shows that Dð0;0Þg^ð3Þ_ð0;bÞ is surjective. The Case (F3) is then completed as the previous two cases above. Hence,F_l is a 3l-dimensional submanifold ofPLand the natural projections pPandp_L are submersions. Note thatFis obviously closed in PL. ThuspP andp_L are surjective sinceFis compact, submersions are open maps, and P, L are connected. It follows that point rows and line pencils are l- dimensional submanifolds ofPandL, respectively, see [3], 1.1, or [6], 4.1.

In order to complete this proof, it su‰ces for reasons of symmetry to show that any two distinct lines L, L⁰ intersect transversally. By using transitivity properties of the group of smooth automorphisms of Pl, the di¤erent possibilities of pairs ðL;L⁰ÞALLreduce to the following three cases:

(L1) L¼ ða;b;1Þ,L⁰¼ ða⁰;b⁰;1Þ,

(L2) L¼ ða;b;1Þ,L⁰¼ ð1;0;0Þ, (L3) L¼ ða;1;0Þ,L⁰¼ ð1;0;0Þ.

In the first case, we may use the group of smooth automorphisms ofPl in order to choose appropriate coordinates for possible intersection points ofLandL⁰. We may assume that these two lines intersect in the point ð1;0;0Þ^t or in a point ðx;y;1Þ^t. Since the second case has been treated already in Lemma 2.5, we assume that the intersection point of LandL⁰ isð1;0;0Þ^t. Then we havea;a⁰¼0 and hence b0b⁰ since Land L⁰ are distinct. We identify the open sets U1 andV3 with two disjoint copies ofK²such thatLandL⁰are identified withð0;bÞandð0;b⁰Þ, respectively, and ð1;0;0Þ^t is identified with ð0;0Þ. The map g^ð3Þ_ð0;bÞ of the previous paragraph vanishes onP_LVU₁. Thus the di¤erential

D_ð0;0Þg^ð3Þ_ð0;bÞ:K²!K:ðu;vÞ 7! ðjbj²þ1ÞðbuþvÞ

vanishes on the tangent space ofPLVU1inð0;0Þ, and an analogous statement holds for the line L⁰. Since the kernels of the di¤erentials D_ð0;0Þg^ð3Þ_ð0;bÞand D_ð0;0Þg^ð3Þ_ð0;b0Þ have trivial intersection, we conclude thatLandL⁰intersect transversally inð1;0;0Þ^t.

In the Case (L2), let ðx;y;zÞ^t denote an intersection point of LandL⁰. Then we havex¼0 and henceðx;y;zÞ^t ¼ ð0;y;1Þ^torðx;y;zÞ^t¼ ð0;1;0Þ^t. Let us first assume that ð0;y;1Þ^t is an intersection point of Land L⁰. By using the transitivity proper- ties of the group of smooth automorphisms acting onPl we may assume that yAR.

After identifying U3 with K², the intersection point corresponds to ð0;yÞ and the submanifoldsPLVU3 andP_L⁰ VU3correspond tog¹_ða;bÞð0Þandf0g Kwithgða;bÞas in the proof of Lemma 2.5. Chooseðu;vÞin the intersection of the tangent spaces of P_LVU₃andP_L⁰ VU₃ inð0;yÞ. Then we get

ðy²þ1Þauþ ðy²þ1Þbvþ ðbyþ1Þ2yv1¼0

by di¤erentiatingg_ða;bÞ(compare the proof of Lemma 2.5) andu¼0. Thus we have ðy²þ1Þbvþ ðbyþ1Þ2yv1¼0 and hence

jbj ¼2jbyþ1j jyj y²þ1

jv1j jvj

provided thatv00. We obtain that

jbyjc2jbyþ1j y²

y²þ1 c2jbyþ1j:

Because of

g_ða;bÞð0;yÞ ¼ ðjaj²þ jbj²þ1Þðbyþ1Þðy²þ1Þ þl¼0

we havejbyþ1jcjlj, which implies that 1cjbyj þ jbyþ1jc3jlj, a contradiction.

Thus we havev¼0. This proves the transversal intersection ofLandL⁰inð0;y;1Þ^t. Let us now consider the case thatLandL⁰intersect in the point ð0;1;0Þ^t. Then we have b¼0. We identify U2 withK² such thatð0;1;0Þ^t corresponds to ð0;0ÞAK². The submanifolds PLVU3 and PL⁰VU3 are identified with the submanifolds ðg^ð4Þ_ða;0ÞÞ¹ðf0gÞandf0g K, respectively, where

g^ð4Þ_ða;0Þ:K²!K:ðx;zÞ 7! ðjaj²þ1ÞðaxþzÞðjxj²þ1þ jzj²Þ þlzjzj²: The di¤erential Dð0;0Þg^ð4Þ_ða;0Þ:ðu;vÞ 7! ðjaj²þ1ÞðauþvÞvanishes on the tangent space ofPLVU3 inð0;0Þ. This proves the transversal intersection ofLandL⁰inð0;1;0Þ^t, since ker Dð0;0Þg^ð4Þ_ða;0Þandf0g Khave trivial intersection.

In the third case, both point rowsPLandPL⁰ are equal to point rows of the clas- sical projective planeP0¼P2K. Hence they intersect transversally. r For the projective planes Pl, where jlj<¹₉, the join map4and the intersection map5are not only smooth but real analytic and even Nash maps, i.e. thePlare real analytic or Nash projective planes, respectively. This will be obtained from the fol- lowing general fact:

Proposition 2.6. LetP¼ ðP;L;FÞbe a projective plane which satisfies the following conditions:

(APP1) There is a positive integer l such that P andLare real analytic(or Nash) 2l- dimensional manifolds.

(APP2) The flag spaceF is a real analytic(or Nash) 3l-dimensional submanifold of PL,and the canonical projectionsp_Pandp_L are submersions.

Suppose, moreover, that any two distinct point rows and any two distinct line pencils intersect transversally. Then the join map4and the intersection map5are real ana- lytic(or Nash maps,respectively).

This proposition can be proved by simply copying the proof of [3], 1.4, or [6], 4.4, and using a real analytic or Nash version, respectively, of the implicit function theo- rem, see, e.g., [9], 1.8.3, and [1], 2.9.8.

It remains to check the conditions of the above proposition for the projective planes Pl with jlj<¹₉. For simplicity we concentrate on the Nash setting in the sequel. First, the point spacePand the line spaceLare copies of the point space and the line space of the classical projective plane P2Kwith their usual algebraic struc- ture. Hence,PandLare Nash manifolds. In the proofs of Theorem 2.4 and Lemma 2.5 we have shown that for each flag there is an open neighbourhood W inPL (identified with an open subset of K²K²) and a real polynomial submersion f_W :W !Ksuch thatFlVW¼ f_W¹ð0Þ. By a Nash version of the standard result on preimages of regular values we conclude thatFlis a Nash submanifold ofPL, cf. [6], the end of Chapter 3, or [5], 5.1–5.9, [9], 1.8.1, [1], 2.9.7. The other conditions required in Proposition 2.6 have already been verified above. Hence, the join map4

and the intersection map5are Nash maps and, in particular, they are real analytic.

So, we have proved the following

Theorem 2.7.Forjlj<¹₉,the incidence structuresPlare Nash projective planes and,in particular,real analytic projective planes.

The following theorem contains results on the dimensions of the automorphism groups of the planesPl, which are direct consequences of Lemma 2.3.

Theorem 2.8.The smooth projective planesP_ladmit groups of smooth automorphisms which are compact Lie groups of dimension1, 4or13for l¼1, 2or4,respectively.

By the main result of [2], the dimension of the automorphism group of a 2l- dimensional, non-classical smooth projective plane is at most 2, 6 or 16 forl¼1, 2 or 4, respectively. By Theorem 2.8, the smooth projective planesPladmit Lie groups of smooth automorphisms whose dimensions are close to these bounds. The dimensions of automorphism groups of non-classical compact projective planes of dimension 2l can be higher than in the smooth case, see [10], Section 65. The maximal dimen- sions ofcompactgroups of automorphisms of non-classical compact projective planes (with l¼1, 2 or 4), however, are the same as the dimensions of the Lie groups in Theorem 2.8, i.e. in this respect there is no di¤erence between compact projective planes and smooth projective planes. Indeed, by [10], 32.21 and 22 a compact group of automorphisms of a 2-dimensional, non-classical compact projective plane is a Lie group of dimension at most 1. In the 4-dimensional case, 71.9 and 72.6 in [10] imply that the dimension of a compact group of automorphisms acting on a non-classical compact projective plane is at most 4. Finally, in dimension 8 a compact group of automorphisms acting on a non-classical compact projective plane is at most 13- dimensional, see [10], 84.9. Even more, the identity connected component of such a group is necessarily isomorphic to SO₂R forl¼1, to U₂Cforl¼2, and to the product of Spin₃R and Spin₅R with amalgamated centers for l¼4. The following theorem summarizes the general information obtained in this way.

Theorem 2.9. The maximal dimensions of compact groups of automorphisms of 2l- dimensional,non-classical smooth projective planes are the same as those in the case of 2l-dimensional,non-classical compact projective planes for lAf1;2;4g.

References

[1] J. Bochnak, M. Coste, M.-F. Roy,Real algebraic geometry. Springer 1998.

MR 2000a:14067 Zbl 0912.14023

[2] R. Bo¨di, Smooth stable and projective planes. Habilitationsschrift, Tu¨bingen 1996.

[3] R. Bo¨di, S. Immervoll, Implicit characterizations of smooth incidence geometries.Geom.

Dedicata83(2000), 63–76. MR 2001j:51018 Zbl 0977.51007

[4] R. Bo¨di, L. Kramer, On homomorphisms between generalized polygons.Geom. Dedicata 58(1995), 1–14. MR 96k:51017 Zbl 0836.51001

[5] T. Bro¨cker, K. Ja¨nich,Introduction to di¤erential topology. Cambridge Univ. Press 1987.

MR 83i:58001 Zbl 0486.57001

[6] S. Immervoll, Smooth projective planes, smooth generalized quadrangles, and iso- parametric hypersurfaces. Dissertation, Tu¨bingen 2001.

[7] L. Kramer, Holomorphic polygons.Math. Z.223(1996), 333–341. MR 97k:51009 Zbl 0871.51007

[8] L. Kramer, K. Tent, Algebraic polygons.J. Algebra182(1996), 435–447. MR 97d:51005 Zbl 0866.51005

[9] S. G. Krantz, H. R. Parks,A primer of real analytic functions. Birkha¨user 1992.

MR 93j:26013 Zbl 0767.26001

[10] H. Salzmann, D. Betten, T. Grundho¨fer, H. Ha¨hl, R. Lo¨wen, M. Stroppel, Compact projective planes. de Gruyter 1995. MR 97b:51009 Zbl 0851.51003

[11] B. Segre, Plans graphiques alge´briques re´els non desargue´siens et correspondances cre´- moniennes topologiques.Rev. Math. Pures Appl.1(1956), 35–50. MR 20 #6424 Zbl 0063.09087

[12] K. Strambach, Algebraische Geometrien. Rend. Sem. Mat. Univ. Padova 53 (1975), 165–210. MR 54 #2649 Zbl 0339.14002

Received 8 February, 2002

S. Immervoll, Mathematisches Institut, Universita¨t Tu¨bingen, Auf der Morgenstelle 10, 72076 Tu¨bingen, Germany

Email: [email protected]