|
Which are the simplest algebraic varieties?
Author(s):
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
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
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.
References:
-
- [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
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(2000):
14-01, 14E08, 14E30, 14G05, 14J26, 14P25, 11D25, 11G35, 30F10, 57N10
Retrieve articles in all Journals with MSC
(2000):
14-01, 14E08, 14E30, 14G05, 14J26, 14P25, 11D25, 11G35, 30F10, 57N10
Additional Information:
János
Kollár
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000
Email:
kollar@math.princeton.edu
DOI:
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
Copyright of article:
Copyright
2001,
American Mathematical Society
|