Fake projective plane
Encyclopedia
In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surface
s that have the same Betti number
s as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
The first example was found by using p-adic uniformization introduced independently by Kurihara and Mustafin.
Mumford also observed that Yau's result together with Weil's theorem on the rigidity of discrete cocompact subgroups of PU(1,2) implies that there are only a finite number of fake projective planes. found two more examples, using similar methods, and found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface
. found a systematic way of classifying all fake projective planes, by showing that there are twenty-eight classes, each of which contains at least an example of
fake projective plane up to isometry, and that there can at most be five more classes which were later shown not to exist. The problem of listing all fake projective planes is reduced to listing all subgroups of appropriate index of an explicitly given lattice associated to each class. By extending these calculations showed that the twenty-eight classes exhaust all possibilities for fake projective planes and that
there are altogether 50 examples determined up to isometry, or 100 fake projective planes up to biholomorphism.
A surface of general type with the same Betti numbers as a minimal surface not of general type must have the Betti numbers of either a
projective plane P2 or a quadric P1×P1. constructed some "fake quadrics": surfaces of general type with the same Betti numbers as quadrics. Beauville surface
s give further examples.
Higher dimensional analogues of fake projective surfaces are called fake projective space
s.
of the fake projective plane. This fundamental group must therefore be a torsion-free
and cocompact discrete subgroup of PU(2,1) of Euler-Poincaré characteristic 3. and showed that this fundamental group must also be an arithmetic group
. Mostow's strong rigidity results imply that the fundamental group determines the fake plane, in the strong sense that any compact surface with the same fundamental group must be isometric to it.
Two fake projective planes are defined to be in the same class if their fundamental groups are both contained in the same maximal arithmetic subgroup of automorphisms of the unit ball. used the volume formula for arithmetic groups from to list 28 non-empty classes of fake projective planes and show that there can at most be five extra classes which are not expected to exist. (See the addendum of the paper where the classification was refined and some errors in the original paper was corrected.)
verified that the five extra classes indeed did not exist and listed all possibilities within
the twenty-eight classes. There are exactly 50 fake projective planes classified up to isometry and hence 100 distinct fake projective planes
classified up to biholomorphism.
The fundamental group of the fake projective plane is an arithmetic subgroup of PU(2,1). Write k for the associated number field (a totally real field) and G for the associated k-form of PU(2,1). If l is the quadratic extension of k over which G is an inner form, then l is a totally imaginary field. There is a division algebra D with center l and degree over l 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of l over k, and a nontrivial Hermitian form on a module over D of dimension 1 or 3 such that G is the special unitary group of this Hermitian form. (As a consequence of and the work of Cartwright and Steger, D has degree 3 over l and the module has dimension 1 over D.) There is one real place of k such that the points of G form a copy of PU(2,1), and over all other real places of k they form the compact group PU(3).
From the result of , the automorphism group of a fake projective plane is either cyclic of order 1, 3, or 7, or the non-cyclic group of order 9, or the non-abelian group of order 21. The quotients of the fake projective planes by these groups were studied by
and also by .
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s that have the same Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
History
Severi asked if there was a complex surface homeomorphic to the projective plane but not biholomorphic to it. showed that there was no such surface, so the closest approximation to the projective plane one can have would be a surface with the same Betti numbers (b0,b1,b2,b3,b4) = (1,0,1,0,1) as the projective plane.The first example was found by using p-adic uniformization introduced independently by Kurihara and Mustafin.
Mumford also observed that Yau's result together with Weil's theorem on the rigidity of discrete cocompact subgroups of PU(1,2) implies that there are only a finite number of fake projective planes. found two more examples, using similar methods, and found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface
Dolgachev surface
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.-Properties:...
. found a systematic way of classifying all fake projective planes, by showing that there are twenty-eight classes, each of which contains at least an example of
fake projective plane up to isometry, and that there can at most be five more classes which were later shown not to exist. The problem of listing all fake projective planes is reduced to listing all subgroups of appropriate index of an explicitly given lattice associated to each class. By extending these calculations showed that the twenty-eight classes exhaust all possibilities for fake projective planes and that
there are altogether 50 examples determined up to isometry, or 100 fake projective planes up to biholomorphism.
A surface of general type with the same Betti numbers as a minimal surface not of general type must have the Betti numbers of either a
projective plane P2 or a quadric P1×P1. constructed some "fake quadrics": surfaces of general type with the same Betti numbers as quadrics. Beauville surface
Beauville surface
In mathematics, a Beauville surface is one of the surfaces of general type introduced by . They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.-Construction:...
s give further examples.
Higher dimensional analogues of fake projective surfaces are called fake projective space
Fake projective space
In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it....
s.
The fundamental group
As a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature, see , any fake projective plane is the quotient of a complex unit ball in 2 dimensions by a discrete subgroup, which is the fundamental groupFundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the fake projective plane. This fundamental group must therefore be a torsion-free
Torsion-free
In mathematics, the term torsion-free may refer to several unrelated notions:* In abstract algebra, a group is torsion-free if the only element of finite order is the identity....
and cocompact discrete subgroup of PU(2,1) of Euler-Poincaré characteristic 3. and showed that this fundamental group must also be an arithmetic group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
. Mostow's strong rigidity results imply that the fundamental group determines the fake plane, in the strong sense that any compact surface with the same fundamental group must be isometric to it.
Two fake projective planes are defined to be in the same class if their fundamental groups are both contained in the same maximal arithmetic subgroup of automorphisms of the unit ball. used the volume formula for arithmetic groups from to list 28 non-empty classes of fake projective planes and show that there can at most be five extra classes which are not expected to exist. (See the addendum of the paper where the classification was refined and some errors in the original paper was corrected.)
verified that the five extra classes indeed did not exist and listed all possibilities within
the twenty-eight classes. There are exactly 50 fake projective planes classified up to isometry and hence 100 distinct fake projective planes
classified up to biholomorphism.
The fundamental group of the fake projective plane is an arithmetic subgroup of PU(2,1). Write k for the associated number field (a totally real field) and G for the associated k-form of PU(2,1). If l is the quadratic extension of k over which G is an inner form, then l is a totally imaginary field. There is a division algebra D with center l and degree over l 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of l over k, and a nontrivial Hermitian form on a module over D of dimension 1 or 3 such that G is the special unitary group of this Hermitian form. (As a consequence of and the work of Cartwright and Steger, D has degree 3 over l and the module has dimension 1 over D.) There is one real place of k such that the points of G form a copy of PU(2,1), and over all other real places of k they form the compact group PU(3).
From the result of , the automorphism group of a fake projective plane is either cyclic of order 1, 3, or 7, or the non-cyclic group of order 9, or the non-abelian group of order 21. The quotients of the fake projective planes by these groups were studied by
and also by .
List of the 50 fake projective planes
| k | l | T | index | Fake projective planes |
|---|---|---|---|---|
| Q | Q(√−1 ) | 5 | 3 | 3 fake planes in 3 classes |
| Q(√−2 ) | 3 | 3 | 3 fake planes in 3 classes | |
| Q(√−7 ) | 2 | 21 | 7 fake planes in 2 classes. One of these classes contains the examples of Mumford and Keum. | |
| 2, 3 | 3 | 4 fake planes in 2 classes | ||
| 2, 5 | 1 | 2 fake planes in 2 classes | ||
| Q(√−15 ) | 2 | 3 | 10 fake planes in 4 classes, including the examples founded by Ishida and Kato. | |
| Q(√−23 ) | 2 | 1 | 2 fake planes in 2 classes | |
| Q(√2) | Q(√(−7+4√2)) | 2 | 3 | 2 fake planes in 2 classes |
| Q(√5) | Q(√5, ζ3) | 2 | 9 | 7 fake planes in 2 classes |
| Q(√6) | Q(√6,ζ3 ) | 2 or 2,3 | 1 or 3 or 9 | 5 fake planes in 3 classes |
| Q(√7) | Q(√7,ζ4 ) | 2 or 3,3 | 21 or 3,3 | 5 fake planes in 3 classes |
- k is a totally real field.
- l is a totally imaginary quadratic extension of k, and ζ3 is a cube root of 1.
- T is a set of primes of k where a certain local subgroup is not hyperspecial.
- index is the index of the fundamental group in a certain arithmetic group.