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The Nash conjecture for threefolds

Author(s): János Kollár
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 63-73.
MSC (1991): Primary 14P25
Posted: September 15, 1998
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Abstract: Nash conjectured in 1952 that every compact differentiable manifold can be realized as the set of real points of a real algebraic variety which is birational to projective space. This paper announces the negative solution of this conjecture in dimension 3. The proof shows that in fact very few 3-manifolds can be realized this way.

References:

[Akbulut-King91]
S. Akbulut and H. King, Rational structures on $3$-manifolds, Pacific J. Math. 150 (1991), 201-204. MR 93c:57028

[AGV85]
V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps I-II, Birkhäuser 1985, 1988. MR 86f:58018; MR 89g:58024

[Benedetti-Marin92]
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. MR 90b:14030

[CKM88]
H. Clemens, J. Kollár and S. Mori, Higher Dimensional Complex Geometry, Astérisque 166 (1988). MR 90j:14046
[Comessatti14]
A. Comessatti, Sulla connessione delle superfizie razionali reali, Annali di Math. 23(3) (1914), 215-283.

[Cutkosky88]
S. D. Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), 521-525. MR 89k:14070

[Hartshorne77]
R. Hartshorne, Algebraic Geometry, Springer, 1977. MR 57:3116

[Hempel76]
J. Hempel, $3$-manifolds, Princeton Univ. Press, 1976. MR 54:3702

[Iskovskikh80]
V. A. Iskovskikh, Anticanonical models of three-dimensional algebraic varieties, J. Soviet Math. 13 (1980), 745-814. MR 81i:14026b

[Kawamata92]
Y. Kawamata, Boundedness of $\Bbb Q$-Fano threefolds, Proc. Int. Conf. Algebra, Contemp. Math. vol. 131 (1992), 439-445. MR 93g:14047

[Kawamata96]
Y. Kawamata, Divisorial contractions to $3$-dimensional terminal quotient singularities, in Higher dimensional complex varieties (Trento, 1994), de Gruyter, (1996), 241-246. MR 98g:14005

[Keel-McKernan98]
S. Keel and J. McKernan, Rational curves on quasi-projective varieties, Mem. AMS (to appear). CMP 98:09

[Kharlamov76]
V. Kharlamov, The topological type of non-singular surfaces in $RP^3$ of degree four, Funct. Anal. Appl. 10 (1976), 295-305. MR 56:8584

[Kollár87]
J. Kollár, The structure of algebraic threefolds - an introduction to Mori's program, Bull. AMS 17 (1987), 211-273. MR 88i:14030

[Kollár93]
J. Kollár, Effective Base Point Freeness, Math. Ann. 296 (1993), 595-605. MR 94f:14004

[Kollár96]
J. Kollár, Rational Curves on Algebraic Varieties, Springer-Verlag, Ergebnisse der Math. vol. 32, 1996. MR 98c:14001

[Kollár97]
J. Kollár, Real Algebraic Threefolds I. Terminal Singularities, Collectanea Math. (to appear).

[Kollár98a]
J. Kollár, Real Algebraic Threefolds II. Minimal Model Program, J. AMS (to appear).

[Kollár98b]
J. Kollár, Real Algebraic Threefolds III. Conic Bundles (preprint).

[Kollár98c]
J. Kollár, Real Algebraic Threefolds IV. Del Pezzo Fibrations (in preparation).

[KoMiMo92]
J. Kollár, Y. Miyaoka and S. Mori, Rationally Connected Varieties, J. Alg. Geom. 1 (1992), 429-448. MR 93i:14014

[Kollár-Mori98]
J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, 1998 (to appear).

[Manetti91]
M. Manetti, Normal degenerations of the complex projective plane, J. f.r.u.a. Math. 419 (1991), 89-118. MR 92f:14028
[Manetti93]
M. Manetti, Normal projective surfaces with $\rho=1, P_{-1}\geq 5$, Rend. Sem. Mat. Univ. Padova 89 (1993), 195-205. MR 94k:14027

[Mikhalkin97]
G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology, 36 (1997), 287-299. MR 98f:57046

[Mori82]
S. Mori, Threefolds whose Canonical Bundles are not Numerically Effective, Ann. of Math. 116 (1982), 133-176. MR 84e:14032

[Nash52]
J. Nash, Real algebraic manifolds, Ann. Math. 56 (1952), 405-421. MR 14:403b

[Reid85]
M. Reid, Young person's guide to canonical singularities, in Algebraic Geometry, Proc. Symp. Pure Math. vol. 46, pp. 345-414. MR 89b:14016

[Reid94]
M. Reid, Nonnormal Del Pezzo surfaces, Publ. RIMS Kyoto Univ. 30 (1994), 695-728. MR 96a:14042

[Rolfsen76]
D. Rolfsen, Knots and links, Publish or Perish, 1976. MR 58:24236, MR 95c:57018

[Scott83]
P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. MR 84m:57009

[Shafarevich72]
R. I. Shafarevich, Basic Algebraic Geometry (in Russian), Nauka, 1972. English translation: Springer, 1977, $2$nd edition, 1994. MR 51:3162; MR 56:5538; MR 95m:14001; MR 95m:14002

[Tognoli73]
A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa 27 (1973), 167-185. MR 53:434

[Viterbo98]
C. Viterbo, (personal communication).

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Additional Information:

János Kollár
Affiliation: University of Utah, Salt Lake City, UT 84112
Email: kollar@math.utah.edu

DOI: 10.1090/S1079-6762-98-00049-3
PII: S 1079-6762(98)00049-3
Received by editor(s): July 17, 1998
Posted: September 15, 1998
Communicated by: Robert Lazarsfeld
Copyright of article: Copyright 1998, American Mathematical Society

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