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Which are the simplest algebraic varieties?
Author:
János Kollár
Journal:
Bull. Amer. Math. Soc. 38 (2001), 409-433
MSC (2000):
Primary 14-01, 14E08, 14E30, 14G05, 14J26, 14P25; Secondary 11D25, 11G35, 30F10, 57N10
Posted:
June 12, 2001
MathSciNet review:
1848255
Full-text PDF
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Additional Information
Abstract: This paper is a slightly revised version of the notes prepared in connection with the AMS Colloquium Lectures delivered in New Orleans, January 2001.
- [BPV84]
W.
Barth, C.
Peters, and A.
Van de Ven, Compact complex surfaces, Ergebnisse der
Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related
Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
(86c:32026)
- [BenedettiMarin92]
Riccardo
Benedetti and Alexis
Marin, Déchirures de variétés de dimension
trois et la conjecture de Nash de rationalité en dimension
trois, Comment. Math. Helv. 67 (1992), no. 4,
514–545 (French). MR 1185807
(94d:57036), http://dx.doi.org/10.1007/BF02566517
- [BCR87]
J.
Bochnak, M.
Coste, and M.-F.
Roy, Géométrie algébrique réelle,
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in
Mathematics and Related Areas (3)], vol. 12, Springer-Verlag, Berlin,
1987 (French). MR
949442 (90b:14030)
- [Clebsch1866]
A. Clebsch, Die Geometrie auf den Flächen dritter Ordnung, J. f.r.u.a. Math. 65 (1866) 359-380
- [ClemensGriffiths72]
C.
Herbert Clemens and Phillip
A. Griffiths, The intermediate Jacobian of the cubic
threefold, Ann. of Math. (2) 95 (1972),
281–356. MR 0302652
(46 #1796)
- [Comessatti13]
A. Comessatti, Fondamenti per la geometria sopra superfizie razionali dal punto di vista reale, Math. Ann. 73 (1913) 1-72
- [Comessatti14]
A. Comessatti, Sulla connessione delle superfizie razionali reali, Annali di Math. 23(3) (1914) 215-283
- [Dolgachev66]
I. Dolgachev, On Severi's conjecture concerning simply connected algebraic surfaces, Soviet Math. Dokl. 7 (1966) 1169-1171
- [Faltings83]
G.
Faltings, Endlichkeitssätze für abelsche Varietäten
über Zahlkörpern, Invent. Math. 73 (1983),
no. 3, 349–366 (German). MR 718935
(85g:11026a), http://dx.doi.org/10.1007/BF01388432
- [FMT89]
Jens
Franke, Yuri
I. Manin, and Yuri
Tschinkel, Rational points of bounded height on Fano
varieties, Invent. Math. 95 (1989), no. 2,
421–435. MR
974910 (89m:11060), http://dx.doi.org/10.1007/BF01393904
- [Harnack1876]
A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Kurven, Math. Ann. 10 (1876) 189-198
- [HarrisTschinkel00]
J. Harris and Yu. Tschinkel, Rational points on quartics. Duke Math. J. 104 (2000) 477-500 CMP 2001:01
- [Hartshorne77]
Robin
Hartshorne, Algebraic geometry, Springer-Verlag, New York,
1977. Graduate Texts in Mathematics, No. 52. MR 0463157
(57 #3116)
- [Hassett00]
B. Hassett, Special cubic fourfolds. Compositio Math. 120 (2000) 1-23 CMP 2000:07
- [Hempel76]
John
Hempel, 3-Manifolds, Princeton University Press, Princeton, N.
J., 1976. Ann. of Math. Studies, No. 86. MR 0415619
(54 #3702)
- [Iskovskikh67]
V. A. Iskovskikh, Rational surfaces with a pencil of rational curves, Math. USSR Sb. 3 (1967) 563-587
- [Iskovskikh80]
V.
A. Iskovskih, Minimal models of rational surfaces over arbitrary
fields, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979),
no. 1, 19–43, 237 (Russian). MR 525940
(80m:14021)
- [Klein1876]
F. Klein, Über eine neuer Art von Riemannschen Flächen, Math. Ann. 10 (1876) 398-416, Reprinted in : F. Klein, Gesammelte Mathematische Abhandlungen, Springer, 1922, vol. II.
- [Knutson71]
Donald
Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol.
203, Springer-Verlag, Berlin, 1971. MR 0302647
(46 #1791)
- [KodairaSpencer58]
K.
Kodaira and D.
C. Spencer, On deformations of complex analytic structures. I,
II, Ann. of Math. (2) 67 (1958), 328–466. MR 0112154
(22 #3009)
- [Kollár91]
János
Kollár, Flips, flops, minimal models, etc, Surveys in
differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA,
1991, pp. 113–199. MR 1144527
(93b:14059)
- [Kollár95]
János
Kollár, Nonrational hypersurfaces, J. Amer. Math. Soc. 8 (1995), no. 1, 241–249. MR 1273416
(95f:14025), http://dx.doi.org/10.1090/S0894-0347-1995-1273416-8
- [Kollár96]
János
Kollár, Rational curves on algebraic varieties,
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
Modern Surveys in Mathematics [Results in Mathematics and Related Areas.
3rd Series. A Series of Modern Surveys in Mathematics], vol. 32,
Springer-Verlag, Berlin, 1996. MR 1440180
(98c:14001)
- [Kollár98a]
J. Kollár, Low degree polynomial equations, in: European Congress of Math. Birkhäuser, 1998, 255-288 MR 98m:14075
- [Kollár98b]
János
Kollár, Real algebraic threefolds. I. Terminal
singularities, Collect. Math. 49 (1998),
no. 2-3, 335–360. Dedicated to the memory of Fernando Serrano.
MR
1677128 (2000c:14078)
- [Kollár98c]
J. Kollár, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63-73
- [Kollár99a]
János
Kollár, Real algebraic threefolds. II. Minimal
model program, J. Amer. Math. Soc.
12 (1999), no. 1,
33–83. MR
1639616 (2000c:14079), http://dx.doi.org/10.1090/S0894-0347-99-00286-6
- [Kollár99b]
János
Kollár, Real algebraic threefolds. III. Conic bundles,
J. Math. Sci. (New York) 94 (1999), no. 1,
996–1020. Algebraic geometry, 9. MR 1703903
(2000h:14049), http://dx.doi.org/10.1007/BF02367244
- [Kollár00a]
János
Kollár, Real algebraic threefolds. IV. Del Pezzo
fibrations, Complex analysis and algebraic geometry, de Gruyter,
Berlin, 2000, pp. 317–346. MR 1760882
(2001c:14087)
- [Kollár00b]
J. Kollár, Unirationality of cubic hypersurfaces, preprint, alg-geom/0005146
- [Kollár00c]
J. Kollár, The Nash conjecture for nonprojective threefolds, preprint, alg-geom/0009108
- [Kollár-Mori98]
János
Kollár and Shigefumi
Mori, Birational geometry of algebraic varieties, Cambridge
Tracts in Mathematics, vol. 134, Cambridge University Press,
Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti;
Translated from the 1998 Japanese original. MR 1658959
(2000b:14018)
- [KoMiMo92]
János
Kollár, Yoichi
Miyaoka, and Shigefumi
Mori, Rationally connected varieties, J. Algebraic Geom.
1 (1992), no. 3, 429–448. MR 1158625
(93i:14014)
- [Kollár-Smith97]
J. Kollár and K. Smith, Rational and Non-rational Algebraic Varieties (e-prints: alg-geom/9707013)
- [Manin66]
Yu. I. Manin, Rational surfaces over perfect fields (in Russian), Publ. Math. IHES 30 (1966) 55-114
- [Manin72]
Yu.
I. Manin, Cubic forms: algebra, geometry, arithmetic,
North-Holland Publishing Co., Amsterdam, 1974. Translated from the Russian
by M. Hazewinkel; North-Holland Mathematical Library, Vol. 4. MR 0460349
(57 #343)
Yu.
I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical
Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986.
Algebra, geometry, arithmetic; Translated from the Russian by M.
Hazewinkel. MR
833513 (87d:11037)
- [Mikhalkin97]
G. Mikhalkin, Blow up equivalence of smooth closed manifolds, Topology, 36 (1997) 287-299
- [Milnor63]
J.
Milnor, Morse theory, Based on lecture notes by M. Spivak and
R. Wells. Annals of Mathematics Studies, No. 51, Princeton University
Press, Princeton, N.J., 1963. MR 0163331
(29 #634)
- [Milnor64]
J.
Milnor, On the Betti numbers of real
varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 0161339
(28 #4547), http://dx.doi.org/10.1090/S0002-9939-1964-0161339-9
- [Milnor-Stasheff74]
John
W. Milnor and James
D. Stasheff, Characteristic classes, Princeton University
Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76. MR 0440554
(55 #13428)
- [Moishezon67]
B. Moishezon, On
-dimensional compact varieties with algebraically independent meromorphic functions, Amer. Math. Soc. Transl. 63 (1967) 51-177
- [Mori82]
Shigefumi
Mori, Threefolds whose canonical bundles are not numerically
effective, Ann. of Math. (2) 116 (1982), no. 1,
133–176. MR
662120 (84e:14032), http://dx.doi.org/10.2307/2007050
- [Morin40]
U. Morin, Sull' unirazionalità dell'ipersuperficie algebrica di qualunque ordine e dimensione sufficientemente alta, in: Atti dell II Congresso Unione Math. Ital. pp. 298-302, 1940
- [Nash52]
John
Nash, Real algebraic manifolds, Ann. of Math. (2)
56 (1952), 405–421. MR 0050928
(14,403b)
- [Nikulin79]
V.
V. Nikulin, Integer symmetric bilinear forms and some of their
geometric applications, Izv. Akad. Nauk SSSR Ser. Mat.
43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
(80j:10031)
- [Plücker1839]
J. Plücker, Theorie der algebraischen Kurven, Bonn, 1839
- [Rolfsen76]
Dale
Rolfsen, Knots and links, Publish or Perish Inc., Berkeley,
Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288
(58 #24236)
- [Schläfli1863]
L. Schläfli, On the distribution of surfaces of the third order into species, Phil. Trans. Roy. Soc. London, 153(1863)193-241. Reprinted in : L. Schläfli, Gesammelte Mathematische Abhandlungen, Birkhäuser, 1953, vol. II.
- [Scott83]
Peter
Scott, The geometries of 3-manifolds, Bull. London Math. Soc.
15 (1983), no. 5, 401–487. MR 705527
(84m:57009), http://dx.doi.org/10.1112/blms/15.5.401
- [Segre42]
B.
Segre, The Non-singular Cubic Surfaces, Oxford University
Press, Oxford, 1942. MR 0008171
(4,254b)
- [Segre51]
Beniamino
Segre, On the rational solutions of homogeneous cubic equations in
four variables, Math. Notae 11 (1951), 1–68. MR 0046064
(13,678d)
- [Serre79]
Jean-Pierre
Serre, Local fields, Graduate Texts in Mathematics,
vol. 67, Springer-Verlag, New York, 1979. Translated from the French
by Marvin Jay Greenberg. MR 554237
(82e:12016)
- [Shafarevich72]
I.
R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New
York, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren
der mathematischen Wissenschaften, Band 213. MR 0366917
(51 #3163)
- [Tognoli73]
A.
Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup.
Pisa (3) 27 (1973), 167–185. MR 0396571
(53 #434)
- [Viro90]
O.
Ya. Viro, Real plane algebraic curves: constructions with
controlled topology, Algebra i Analiz 1 (1989),
no. 5, 1–73 (Russian); English transl., Leningrad Math. J.
1 (1990), no. 5, 1059–1134. MR 1036837
(91b:14078)
- [Viterbo98]
C. Viterbo, Symplectic real algebraic geometry, to appear
- [Zeuthen1874]
H.G. Zeuthen, Sur les différentes formes des courbes du quatrième ordre, Math. Ann. 7(1874) 410-432
- [BPV84]
- W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer, 1984 MR 86c:32026
- [BenedettiMarin92]
- R. Benedetti and A. Marin, Déchirures de variétés de dimension trois, Comm. Math. Helv. 67 (1992) 514-545 MR 94d:57036
- [BCR87]
- J. Bochnak, M. Coste and M-F. Roy, Géométrie algébrique réelle, Springer, 1987; revised English translation: Real algebraic geometry, Springer, 1999 MR 90b:14030
- [Clebsch1866]
- A. Clebsch, Die Geometrie auf den Flächen dritter Ordnung, J. f.r.u.a. Math. 65 (1866) 359-380
- [ClemensGriffiths72]
- C. H. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. Math. 95 (1972) 281-356 MR 46:1796
- [Comessatti13]
- A. Comessatti, Fondamenti per la geometria sopra superfizie razionali dal punto di vista reale, Math. Ann. 73 (1913) 1-72
- [Comessatti14]
- A. Comessatti, Sulla connessione delle superfizie razionali reali, Annali di Math. 23(3) (1914) 215-283
- [Dolgachev66]
- I. Dolgachev, On Severi's conjecture concerning simply connected algebraic surfaces, Soviet Math. Dokl. 7 (1966) 1169-1171
- [Faltings83]
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983) 349-366 MR 85g:11026a
- [FMT89]
- J. Franke, Yu. I. Manin and Yu. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989) 421-435. MR 89m:11060
- [Harnack1876]
- A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Kurven, Math. Ann. 10 (1876) 189-198
- [HarrisTschinkel00]
- J. Harris and Yu. Tschinkel, Rational points on quartics. Duke Math. J. 104 (2000) 477-500 CMP 2001:01
- [Hartshorne77]
- R. Hartshorne, Algebraic Geometry, Springer, 1977 MR 57:3116
- [Hassett00]
- B. Hassett, Special cubic fourfolds. Compositio Math. 120 (2000) 1-23 CMP 2000:07
- [Hempel76]
- J. Hempel, 3-manifolds, Princeton Univ. Press, 1976 MR 54:3702
- [Iskovskikh67]
- V. A. Iskovskikh, Rational surfaces with a pencil of rational curves, Math. USSR Sb. 3 (1967) 563-587
- [Iskovskikh80]
- V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR Izv. 14 (1980) 17-39 MR 80m:14021
- [Klein1876]
- F. Klein, Über eine neuer Art von Riemannschen Flächen, Math. Ann. 10 (1876) 398-416, Reprinted in : F. Klein, Gesammelte Mathematische Abhandlungen, Springer, 1922, vol. II.
- [Knutson71]
- D. Knutson, Algebraic spaces, Springer Lecture Notes vol. 203, 1971 MR 46:1791
- [KodairaSpencer58]
- K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II. Ann. of Math. 67 (1958) 328-466 MR 22:3009
- [Kollár91]
- J. Kollár, Flips, Flops, Minimal Models, etc., Surv. in Diff. Geom. 1 (1991) 113-199 MR 93b:14059
- [Kollár95]
- J. Kollár, Nonrational hypersurfaces, Jour. AMS 8 (1995) 241-249 MR 95f:14025
- [Kollár96]
- J. Kollár, Rational Curves on Algebraic Varieties, Springer Verlag, Ergebnisse der Math. vol. 32, 1996 MR 98c:14001
- [Kollár98a]
- J. Kollár, Low degree polynomial equations, in: European Congress of Math. Birkhäuser, 1998, 255-288 MR 98m:14075
- [Kollár98b]
- J. Kollár, Real Algebraic Threefolds I. Terminal Singularities, Collectanea Math. (FERRAN SERRANO, 1957-1997) 49 (1998) 335-360 MR 2000c:14078
- [Kollár98c]
- J. Kollár, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63-73
- [Kollár99a]
- J. Kollár, Real Algebraic Threefolds II. Minimal Model Program, Jour. AMS 12 (1999) 33-83 MR 2000c:14079
- [Kollár99b]
- J. Kollár, Real Algebraic Threefolds III. Conic Bundles, J. Math. Sci. (New York) 94 (1999) 996-1020 MR 2000h:14049
- [Kollár00a]
- J. Kollár, Real Algebraic Threefolds IV. Del Pezzo Fibrations, in: Complex analysis and algebraic geometry, de Gruyter, Berlin, 2000, 317-346 MR 2001c:14087
- [Kollár00b]
- J. Kollár, Unirationality of cubic hypersurfaces, preprint, alg-geom/0005146
- [Kollár00c]
- J. Kollár, The Nash conjecture for nonprojective threefolds, preprint, alg-geom/0009108
- [Kollár-Mori98]
- J. Kollár and S. Mori, Birational geometry of algebraic varieties, English edition: Cambridge Univ. Press, 1998; Japanese edition: Iwanami Shoten, 1998 MR 2000b:14018
- [KoMiMo92]
- J. Kollár - Y. Miyaoka - S. Mori, Rationally Connected Varieties, J. Alg. Geom. 1 (1992) 429-448 MR 93i:14014
- [Kollár-Smith97]
- J. Kollár and K. Smith, Rational and Non-rational Algebraic Varieties (e-prints: alg-geom/9707013)
- [Manin66]
- Yu. I. Manin, Rational surfaces over perfect fields (in Russian), Publ. Math. IHES 30 (1966) 55-114
- [Manin72]
- Yu. I. Manin, Cubic forms (in Russian), Nauka, 1972. English translation: North-Holland, 1974, second expanded edition, 1986 MR 57:343; MR 87d:11037
- [Mikhalkin97]
- G. Mikhalkin, Blow up equivalence of smooth closed manifolds, Topology, 36 (1997) 287-299
- [Milnor63]
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J., 1963 MR 29:634
- [Milnor64]
- J. Milnor, On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 (1964) 275-280 MR 28:4547
- [Milnor-Stasheff74]
- J. Milnor and J. Stasheff, Characteristic classes, Princeton Univ. Press, 1974 MR 55:13428
- [Moishezon67]
- B. Moishezon, On
-dimensional compact varieties with algebraically independent meromorphic functions, Amer. Math. Soc. Transl. 63 (1967) 51-177
- [Mori82]
- S. Mori, Threefolds whose Canonical Bundles are not Numerically Effective, Ann. of Math. 116 (1982) 133-176 MR 84e:14032
- [Morin40]
- U. Morin, Sull' unirazionalità dell'ipersuperficie algebrica di qualunque ordine e dimensione sufficientemente alta, in: Atti dell II Congresso Unione Math. Ital. pp. 298-302, 1940
- [Nash52]
- J. Nash, Real algebraic manifolds, Ann. Math. 56 (1952) 405-421 MR 14:403b
- [Nikulin79]
- V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111-177 MR 80j:10031
- [Plücker1839]
- J. Plücker, Theorie der algebraischen Kurven, Bonn, 1839
- [Rolfsen76]
- D. Rolfsen, Knots and links, Publish or Perish, 1976 MR 58:24236
- [Schläfli1863]
- L. Schläfli, On the distribution of surfaces of the third order into species, Phil. Trans. Roy. Soc. London, 153(1863)193-241. Reprinted in : L. Schläfli, Gesammelte Mathematische Abhandlungen, Birkhäuser, 1953, vol. II.
- [Scott83]
- P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983) 401-487 MR 84m:57009
- [Segre42]
- B. Segre, The non-singular cubic surfaces, Clarendon Press, 1942 MR 4:254b
- [Segre51]
- B. Segre, The rational solutions of homogeneous cubic equations in four variables, Notae Univ. Rosario 11 (1951) 1-68 MR 13:678d
- [Serre79]
- J.-P. Serre, Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979 MR 82e:12016
- [Shafarevich72]
- I. R. Shafarevich, Basic Algebraic Geometry (in Russian), Nauka, 1972. English translation: Springer, 1977, second expanded edition, 1994 MR 51:3163; MR 95m:14001/14002
- [Tognoli73]
- A. Tognoli, Su una congettura di Nash, Ann. Sci. Norm. Sup. Pisa 27 (1973) 167-185 MR 53:434
- [Viro90]
- O. Ya. Viro, Real plane algebraic curves, Leningrad Math. J. 1 (1990) 1059-1134 MR 91b:14078
- [Viterbo98]
- C. Viterbo, Symplectic real algebraic geometry, to appear
- [Zeuthen1874]
- H.G. Zeuthen, Sur les différentes formes des courbes du quatrième ordre, Math. Ann. 7(1874) 410-432
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Additional Information
János Kollár
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000
Email:
kollar@math.princeton.edu
DOI:
http://dx.doi.org/10.1090/S0273-0979-01-00917-X
PII:
S 0273-0979(01)00917-X
Received by editor(s):
February 7, 2001
Posted:
June 12, 2001
Article copyright:
© Copyright 2001 American Mathematical Society
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