Severi asked if there was a complex surface homeomorphic to the projective plane but not biholomorphic to it. Yau (1977) 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 Mumford (1979) 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. Ishida & Kato (1998) found two more examples, using similar methods, and Keum (2006) found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface. Prasad & Yeung (2007), Prasad & Yeung (2010) 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 Cartwright & Steger (2010) 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. Shavel (1978) constructed some "fake quadrics": surfaces of general type with the same Betti numbers as quadrics. Beauville surfaces give further examples.
Higher-dimensional analogues of fake projective surfaces are called fake projective spaces.
The fundamental groupedit
As a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature, see Yau (1977, 1978), any fake projective plane is the quotient of a complex unit ball in 2 dimensions by a discrete subgroup, which is the fundamental group 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. Klingler (2003) and Yeung (2004) 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. Prasad & Yeung (2007), Prasad & Yeung (2010) used the volume formula for arithmetic groups from (Prasad 1989) 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.) Cartwright & Steger (2010) 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 Prasad & Yeung (2007) 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 Prasad & Yeung (2007), 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 Keum (2008) and also by Cartwright & Steger (2010).
List of the 50 fake projective planesedit
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.
Referencesedit
Cartwright, Donald I.; Steger, Tim (2010), "Enumeration of the 50 fake projective planes" (PDF), Comptes Rendus Mathématique, 348 (1): 11–13, doi:10.1016/j.crma.2009.11.016
Ishida, Masa-Nori; Kato, Fumiharu (1998), "The strong rigidity theorem for non-Archimedean uniformization", The Tohoku Mathematical Journal, Second Series, 50 (4): 537–555, doi:10.2748/tmj/1178224897, MR 1653430
Keum, JongHae (2006), "A fake projective plane with an order 7 automorphism", Topology, 45 (5): 919–927, arXiv:math/0505339, doi:10.1016/j.top.2006.06.006, MR 2239523, S2CID 15052978
Klingler, Bruno (2003), "Sur la rigidité de certains groupes fondamentaux, l'arithméticité des réseaux hyperboliques complexes, et les faux plans projectifs", Inventiones Mathematicae, 153 (1): 105–143, Bibcode:2003InMat.153..105K, doi:10.1007/s00222-002-0283-2, MR 1990668, S2CID 120268251
Kulikov, Vik. S.; Kharlamov, V. M. (2002), "On real structures on rigid surfaces", Izvestiya: Mathematics, 66 (1): 133–150, arXiv:math/0101098, Bibcode:2002IzMat..66..133K, doi:10.1070/IM2002v066n01ABEH000374, MR 1917540
Rémy, R. (2007), (PDF), Séminaire Bourbaki, vol. 984, archived from the original (PDF) on 2011-06-09, retrieved 2009-05-08
Shavel, Ira H. (1978), "A class of algebraic surfaces of general type constructed from quaternion algebras", Pacific Journal of Mathematics, 76 (1): 221–245, doi:10.2140/pjm.1978.76.221, MR 0572981
Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I", Communications on Pure and Applied Mathematics, 31 (3): 339–411, doi:10.1002/cpa.3160310304, MR 0480350
Yeung, Sai-Kee (2004), "Integrality and arithmeticity of co-compact lattice corresponding to certain complex two-ball quotients of Picard number one", The Asian Journal of Mathematics, 8 (1): 107–129, doi:10.4310/ajm.2004.v8.n1.a9, MR 2128300
Yeung, Sai-Kee (2010), "Classification of fake projective planes", Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), vol. 13, Int. Press, Somerville, MA, pp. 391–431, MR 2761486
External linksedit
Prasad, Gopal, Fake Projective spaces
January 01, 1970
fake, projective, plane, freedman, example, smoothable, manifold, with, same, homotopy, type, complex, projective, plane, manifold, mathematics, fake, projective, plane, mumford, surface, complex, algebraic, surfaces, that, have, same, betti, numbers, projecti. For Freedman s example of a non smoothable manifold with the same homotopy type as the complex projective plane see 4 manifold In mathematics a fake projective plane or Mumford surface is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane but are not isomorphic to it Such objects are always algebraic surfaces of general type Contents 1 History 2 The fundamental group 3 List of the 50 fake projective planes 4 References 5 External linksHistory editSeveri asked if there was a complex surface homeomorphic to the projective plane but not biholomorphic to it Yau 1977 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 Mumford 1979 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 Ishida amp Kato 1998 found two more examples using similar methods and Keum 2006 found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface Prasad amp Yeung 2007 Prasad amp Yeung 2010 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 Cartwright amp Steger 2010 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 Shavel 1978 constructed some fake quadrics surfaces of general type with the same Betti numbers as quadrics Beauville surfaces give further examples Higher dimensional analogues of fake projective surfaces are called fake projective spaces The fundamental group editAs a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature see Yau 1977 1978 any fake projective plane is the quotient of a complex unit ball in 2 dimensions by a discrete subgroup which is the fundamental group of the fake projective plane This fundamental group must therefore be a torsion free and cocompact discrete subgroup of PU 2 1 of Euler Poincare characteristic 3 Klingler 2003 and Yeung 2004 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 Prasad amp Yeung 2007 Prasad amp Yeung 2010 used the volume formula for arithmetic groups from Prasad 1989 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 Cartwright amp Steger 2010 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 Prasad amp Yeung 2007 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 Prasad amp Yeung 2007 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 Keum 2008 and also by Cartwright amp Steger 2010 List of the 50 fake projective planes editk 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 z3 2 9 7 fake planes in 2 classes Q 6 Q 6 z3 2 or 2 3 1 or 3 or 9 5 fake planes in 3 classes Q 7 Q 7 z4 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 z3 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 References editCartwright Donald I Steger Tim 2010 Enumeration of the 50 fake projective planes PDF Comptes Rendus Mathematique 348 1 11 13 doi 10 1016 j crma 2009 11 016 Ishida Masa Nori Kato Fumiharu 1998 The strong rigidity theorem for non Archimedean uniformization The Tohoku Mathematical Journal Second Series 50 4 537 555 doi 10 2748 tmj 1178224897 MR 1653430 Keum JongHae 2006 A fake projective plane with an order 7 automorphism Topology 45 5 919 927 arXiv math 0505339 doi 10 1016 j top 2006 06 006 MR 2239523 S2CID 15052978 Keum JongHae 2008 Quotients of fake projective planes Geometry amp Topology 12 4 2497 2515 arXiv 0802 3435 doi 10 2140 gt 2008 12 2497 MR 2443971 S2CID 14476192 Klingler Bruno 2003 Sur la rigidite de certains groupes fondamentaux l arithmeticite des reseaux hyperboliques complexes et les faux plans projectifs Inventiones Mathematicae 153 1 105 143 Bibcode 2003InMat 153 105K doi 10 1007 s00222 002 0283 2 MR 1990668 S2CID 120268251 Kulikov Vik S Kharlamov V M 2002 On real structures on rigid surfaces Izvestiya Mathematics 66 1 133 150 arXiv math 0101098 Bibcode 2002IzMat 66 133K doi 10 1070 IM2002v066n01ABEH000374 MR 1917540 Mumford David 1979 An algebraic surface with K ample K2 9 pg q 0 PDF American Journal of Mathematics 101 1 233 244 doi 10 2307 2373947 JSTOR 2373947 MR 0527834 Prasad Gopal 1989 Volumes of S arithmetic quotients of semi simple groups Publications Mathematiques de l IHES 69 69 91 117 doi 10 1007 BF02698841 MR 1019962 S2CID 53556391 Prasad Gopal Yeung Sai Kee 2007 Fake projective planes Inventiones Mathematicae 168 2 321 370 arXiv math 0512115 Bibcode 2007InMat 168 321P doi 10 1007 s00222 007 0034 5 MR 2289867 S2CID 1990160 Prasad Gopal Yeung Sai Kee 2010 Addendum to Fake projective planes Inventiones Mathematicae 182 1 213 227 arXiv 0906 4932 Bibcode 2010InMat 182 213P doi 10 1007 s00222 010 0259 6 MR 2672284 S2CID 17216453 Remy R 2007 Covolume des groupes S arithmetiques et faux plans projectifs d apres Mumford Prasad Klingler Yeung Prasad Yeung PDF Seminaire Bourbaki vol 984 archived from the original PDF on 2011 06 09 retrieved 2009 05 08 Shavel Ira H 1978 A class of algebraic surfaces of general type constructed from quaternion algebras Pacific Journal of Mathematics 76 1 221 245 doi 10 2140 pjm 1978 76 221 MR 0572981 Yau Shing Tung 1977 Calabi s conjecture and some new results in algebraic geometry Proceedings of the National Academy of Sciences of the United States of America 74 5 1798 1799 Bibcode 1977PNAS 74 1798Y doi 10 1073 pnas 74 5 1798 JSTOR 67110 MR 0451180 PMC 431004 PMID 16592394 Yau Shing Tung 1978 On the Ricci curvature of a compact Kahler manifold and the complex Monge Ampere equation I Communications on Pure and Applied Mathematics 31 3 339 411 doi 10 1002 cpa 3160310304 MR 0480350 Yeung Sai Kee 2004 Integrality and arithmeticity of co compact lattice corresponding to certain complex two ball quotients of Picard number one The Asian Journal of Mathematics 8 1 107 129 doi 10 4310 ajm 2004 v8 n1 a9 MR 2128300 Yeung Sai Kee 2010 Classification of fake projective planes Handbook of geometric analysis No 2 Adv Lect Math ALM vol 13 Int Press Somerville MA pp 391 431 MR 2761486External links editPrasad Gopal Fake Projective spaces Retrieved from https en wikipedia org w index php title Fake projective plane amp oldid 1203777205, wikipedia, wiki, book, books, library,
