Smooth projective plane

A smooth projective plane ${\displaystyle {\mathcal {P}}=(P,{\mathfrak {L}})}$ consists of a point space ${\displaystyle P}$ and a line space ${\displaystyle {\mathfrak {L}}}$ that are smooth manifolds and where both geometric operations of joining and intersecting are smooth.

The geometric operations of smooth planes are continuous; hence, each smooth plane is a compact topological plane.[1] Smooth planes exist only with point spaces of dimension 2^m where ${\displaystyle 1\leq m\leq 4}$, because this is true for compact connected projective topological planes.[2][3] These four cases will be treated separately below.

Theorem. The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold.[4]

Automorphisms play a crucial role in the study of smooth planes. A bijection of the point set of a projective plane is called a collineation, if it maps lines onto lines. The continuous collineations of a compact projective plane ${\displaystyle {\mathcal {P}}}$ form the group ${\displaystyle \operatorname {Aut} {\mathcal {P}}}$. This group is taken with the topology of uniform convergence. We have:[5]

Theorem. If ${\displaystyle {\mathcal {P}}=(P,{\mathfrak {L}})}$ is a smooth plane, then each continuous collineation of ${\displaystyle {\mathcal {P}}}$ is smooth; in other words, the group of automorphisms of a smooth plane ${\displaystyle {\mathcal {P}}}$ coincides with ${\displaystyle \operatorname {Aut} {\mathcal {P}}}$. Moreover, ${\displaystyle \operatorname {Aut} {\mathcal {P}}}$ is a smooth Lie transformation group of ${\displaystyle P}$ and of ${\displaystyle {\mathfrak {L}}}$.

The automorphism groups of the four classical planes are simple Lie groups of dimension 8, 16, 35, or 78, respectively. All other smooth planes have much smaller groups. See below.

A projective plane is called a translation plane if its automorphism group has a subgroup that fixes each point on some line ${\displaystyle W}$ and acts sharply transitively on the set of points not on ${\displaystyle W}$.

Theorem. Every smooth projective translation plane ${\displaystyle {\mathcal {P}}}$ is isomorphic to one of the four classical planes.[6]

This shows that there are many compact connected topological projective planes that are not smooth. On the other hand, the following construction yields real analytic non-Desarguesian planes of dimension 2, 4, and 8, with a compact group of automorphisms of dimension 1, 4, and 13, respectively:[7] represent points and lines in the usual way by homogeneous coordinates over the real or complex numbers or the quaternions, say, by vectors of length ${\displaystyle 1}$. Then the incidence of the point ${\displaystyle (x,y,z)}$ and the line ${\displaystyle (a,b,c)}$ is defined by ${\displaystyle ax+by+cz=t|c|^{2}|z|^{2}cz}$, where ${\displaystyle t}$ is a fixed real parameter such that ${\displaystyle |t|<1/9}$. These planes are self-dual.

Compact 2-dimensional projective planes can be described in the following way: the point space is a compact surface ${\displaystyle S}$, each line is a Jordan curve in ${\displaystyle S}$ (a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line. Then ${\displaystyle S}$ is homeomorphic to the point space of the real plane ${\displaystyle {\mathcal {E}}}$, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply Salzmann et al. 1995, §31 to the complement of a line). A familiar family of examples was given by Moulton in 1902.[8][9] These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane.[10] More generally, all non-classical compact 2-dimensional planes ${\displaystyle {\mathcal {P}}}$ such that ${\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\geq 3}$ are known explicitly; none of these is smooth:

Theorem. If ${\displaystyle {\mathcal {P}}}$ is a smooth 2-dimensional plane and if ${\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\geq 3}$, then ${\displaystyle {\mathcal {P}}}$ is the classical real plane ${\displaystyle {\mathcal {E}}}$.[11]

All compact planes ${\displaystyle {\mathcal {P}}}$ with a 4-dimensional point space and ${\displaystyle \operatorname {Aut} {\mathcal {P}}\geq 7}$ have been classified.[12] Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane.[13] According to Bödi (1996, Chap. 10), this shift plane is not smooth. Hence, the result on translation planes implies:

Theorem. A smooth 4-dimensional plane is isomorphic to the classical complex plane, or ${\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\leq 6}$.[14]

Compact 8-dimensional topological planes ${\displaystyle {\mathcal {P}}}$ have been discussed in Salzmann et al. (1995, Chapter 8) and, more recently, in Salzmann (2014). Put ${\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}}$. Either ${\displaystyle {\mathcal {P}}}$ is the classical quaternion plane or ${\displaystyle \dim \Sigma \leq 18}$. If ${\displaystyle \dim \Sigma \geq 17}$, then ${\displaystyle {\mathcal {P}}}$ is a translation plane, or a dual translation plane, or a Hughes plane.[15] The latter can be characterized as follows: ${\displaystyle \Sigma }$ leaves some classical complex subplane ${\displaystyle {\mathcal {C}}}$ invariant and induces on ${\displaystyle {\mathcal {C}}}$ the connected component of its full automorphism group.[16][17] The Hughes planes are not smooth.[18][19] This yields a result similar to the case of 4-dimensional planes:

Theorem. If ${\displaystyle {\mathcal {P}}}$ is a smooth 8-dimensional plane, then ${\displaystyle {\mathcal {P}}}$ is the classical quaternion plane or ${\displaystyle \dim \Sigma \leq 16}$.

Let ${\displaystyle \Sigma }$ denote the automorphism group of a compact 16-dimensional topological projective plane ${\displaystyle {\mathcal {P}}}$. Either ${\displaystyle {\mathcal {P}}}$ is the smooth classical octonion plane or ${\displaystyle \dim \Sigma \leq 40}$. If ${\displaystyle \dim \Sigma =40}$, then ${\displaystyle \Sigma }$ fixes a line ${\displaystyle W}$ and a point ${\displaystyle v\in W}$, and the affine plane ${\displaystyle {\mathcal {P}}\smallsetminus W}$ and its dual are translation planes.[20] If ${\displaystyle \dim \Sigma =39}$, then ${\displaystyle \Sigma }$ also fixes an incident point-line pair, but neither ${\displaystyle {\mathcal {P}}}$ nor ${\displaystyle \Sigma }$ are known explicitly. Nevertheless, none of these planes can be smooth:[21][22][23]

Theorem. If ${\displaystyle {\mathcal {P}}}$ is a 16-dimensional smooth projective plane, then ${\displaystyle {\mathcal {P}}}$ is the classical octonion plane or ${\displaystyle \dim \Sigma \leq 38}$.

The last four results combine to give the following theorem:

If ${\displaystyle c_{m}}$ is the largest value of ${\displaystyle \dim \operatorname {Aut} {\mathcal {P}}}$, where ${\displaystyle {\mathcal {P}}}$ is a non-classical compact 2^m-dimensional topological projective plane, then ${\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\leq c_{m}-2}$ whenever ${\displaystyle {\mathcal {P}}}$ is even smooth.

The condition, that the geometric operations of a projective plane are complex analytic, is very restrictive. In fact, it is satisfied only in the classical complex plane.[24][25]

Theorem. Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure.

• Bödi, R. (1996), "Smooth stable and projective planes", Thesis, Tübingen
• Salzmann, H.; Betten, D.; Grundhöfer, T.; Hähl, H.; Löwen, R.; Stroppel, M. (1995), Compact Projective Planes, W. de Gruyter
• Salzmann, H. (2014), Compact planes, mostly 8-dimensional. A retrospect, arXiv:1402.0304, Bibcode:2014arXiv1402.0304S