Real algebraic threefolds II. Minimal model program

Kollár, János

doi:10.1090/S0894-0347-99-00286-6

Real algebraic threefolds II. Minimal model program
HTML articles powered by AMS MathViewer

by János Kollár

J. Amer. Math. Soc. 12 (1999), 33-83

DOI: https://doi.org/10.1090/S0894-0347-99-00286-6

PDF | Request permission

Abstract:

This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an orientable 3-manifold (this assumption can be weakened considerably). Then there is a fairly simple description of how the topology of real points changes under the minimal model program. This leads to the solution of the Nash conjecture concerning the topology of real projective varieties which are birational to projective 3-space. Another application is a factorization theorem for birational maps.

References

S. Akbulut and H. King, All knots are algebraic, Comment. Math. Helv. 56 (1981), no. 3, 339–351. MR 639356, DOI 10.1007/BF02566217
Selman Akbulut and Henry King, Topology of real algebraic sets, Mathematical Sciences Research Institute Publications, vol. 25, Springer-Verlag, New York, 1992. MR 1225577, DOI 10.1007/978-1-4613-9739-7
V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR 777682, DOI 10.1007/978-1-4612-5154-5
M. Andreatta and J. Wiśniewski, A survey on contractions of higher dimensional varieties, in Algebraic Geometry, Santa Cruz 1995, Amer. Math. Soc. 1997.
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
Herbert Clemens, János Kollár, and Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989) (English, with French summary). MR 1004926
A. Comessatti, Sulla connessione delle superfizie razionali reali, Annali di Math. 23(3) (1914) 215-283.
Steven Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), no. 3, 521–525. MR 936328, DOI 10.1007/BF01456342
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
William Fulton, Algebraic topology, Graduate Texts in Mathematics, vol. 153, Springer-Verlag, New York, 1995. A first course. MR 1343250, DOI 10.1007/978-1-4612-4180-5
Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR 643101
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
T. Hayakawa, (personal communication).
John Hempel, $3$-Manifolds, Annals of Mathematics Studies, No. 86, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. MR 0415619
A. N. Tjurin, The intermediate Jacobian of three-dimensional varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, 1979, pp. 5–57, 239 (loose errata) (Russian). MR 537684
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
Yujiro Kawamata, Boundedness of $\mathbf Q$-Fano threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 439–445. MR 1175897
Yujiro Kawamata, Divisorial contractions to $3$-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 241–246. MR 1463182
V. Kharlamov, The topological type of non-singular surfaces in $RP^3$ of degree four, Funct. Anal. Appl. 10 (1976) 295-305.
János Kollár, The structure of algebraic threefolds: an introduction to Mori’s program, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 211–273. MR 903730, DOI 10.1090/S0273-0979-1987-15548-0
János Kollár, Minimal models of algebraic threefolds: Mori’s program, Astérisque 177-178 (1989), Exp. No. 712, 303–326. Séminaire Bourbaki, Vol. 1988/89. MR 1040578
J. Kollár, Real Algebraic Surfaces, Notes of the 1997 Trento summer school lectures, (preprint).
J. Kollár, Real Algebraic Threefolds I. Terminal Singularities, Collectanea Math. (to appear).
J. Kollár, Real Algebraic Threefolds III. Conic Bundles (preprint).
J. Kollár, Real Algebraic Threefolds IV. Del Pezzo fibrations (preprint).
J. Kollár, The Nash conjecture for Algebraic Threefolds, ERA of AMS 4 (1998) 63-73.
J. Kollár (with 14 coauthors), Flips and Abundance for Algebraic Threefolds, Astérisque 211 (1992).
János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
János Kollár and Shigefumi Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703. MR 1149195, DOI 10.1090/S0894-0347-1992-1149195-9
J. Kollár - S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, 1998.
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Dimitri Markushevich, Minimal discrepancy for a terminal cDV singularity is $1$, J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 445–456. MR 1424437
T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668–684. MR 171778, DOI 10.2307/2373030
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059, DOI 10.1007/978-1-4612-9906-6
Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
Shigefumi Mori, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. MR 792770, DOI 10.1017/S0027763000021358
Shigefumi Mori, Flip theorem and the existence of minimal models for $3$-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253. MR 924704, DOI 10.1090/S0894-0347-1988-0924704-X
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348
Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963, DOI 10.1090/pspum/046.1/927963
J.-L. Riesler, Construction d’hypersurfaces réelle (Sém. Bourbaki #763), Astérisque, 216 (1993) 69-86.
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
R. I. Shafarevich, Basic Algebraic Geometry (in Russian), Nauka, 1972; Revised English translation: Springer 1994.
Robert Silhol, Real algebraic surfaces with rational or elliptic fiberings, Math. Z. 186 (1984), no. 4, 465–499. MR 744960, DOI 10.1007/BF01162775
Robert Silhol, Real algebraic surfaces, Lecture Notes in Mathematics, vol. 1392, Springer-Verlag, Berlin, 1989. MR 1015720, DOI 10.1007/BFb0088815
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