zbMATH — the first resource for mathematics

Real algebraic threefolds. II: Minimal model program. (English) Zbl 0964.14013
For part I of this article see J. Kollár, Collect. Math. 49, No. 2-3, 335-360 (1998; Zbl 0948.14013).
The article under review exhibits the author’s series of works on topological study of real algebraic varieties, from the viewpoint of the theory of minimal models. Adopting different base coefficient numbers could shift the focus of studies of algebraic varieties; objectivewise the class of real algebraic varieties \(X\) can be naturally seen as a subclass of that of complex ones (by regarding the defining equations having complex unknowns), whereas subjectivewise real algebraic geometry also pursues its unique problems and interest. Nonetheless, real and complex geometries show mutual intimacy; if \(X\) is a complex algebraic variety then its underlying real manifold structure is by itself a real algebraic variety (for e.g., \(z_1 z_2 + z_3^2 = 0\) is in real equations \(u_1 u_2 - v_1 v_2 + u_3^2 - v_3^2 = u_1 v_2 + u_2 v_1 + 2 u_3 v_3 = 0\)). If \(X\) is a real algebraic variety, then methods from complex algebraic geometry apply to the complexification \(X_{\mathbb C}\), which may help to understand the structure of \(X\). In the article under review the author comprehensively extends complex algebraic geometry methods into real algebraic geometry, and aims at a fine structure theory of real algebraic varieties, especially their topologies. Among all, the author adopts the idea of the “minimal model theory,” to perform a series of two types of ‘elementary’ birational transformations (so-called flips and divisorial contractions) starting from the given real algebraic variety \(X\), until it reaches to a model \(X^*\) whose global structure is reduced enough so that the same operations cannot proceed further. In practice such operations are often rather difficult, yet in complex case, at least in dimension \(3\) this is known to completely work out, giving rise to a reasonably understandable \(X^*\). Thus, the author focuses on his study of the 3-dimensional real algebraic varieties, and proposes:
(i) the investigation of the topological effect of such both two types of elementary transformations, and
(ii) the investigation of the topological structure of \(X^*(\mathbb R)\).
In the article under review the author exclusively focuses on the part (i). The following is the main result (theorem 1.2):
Let \(X\) be a smooth, projective real algebraic variety of dimension \(3\), and \(X^*\) the result of the minimal model program over \(\mathbb R\). Assume that the set of real points \(X^* (\mathbb R)\) of \(X^*\) is orientable. Then the “topological normalization” \(\overline{X^* (\mathbb R)}\) is a \(PL\)-manifold and \(X(\mathbb R)\) can be obtained from \(\overline{X^* (\mathbb R)}\) by repeated application of the following operations: (a) throwing away all isolated singular points of \(\overline{X^* (\mathbb R)}\), (b) taking connected sums of connected components, (c) taking the connected sum with \(S^1 \times S^2\), and (d) taking the connected sum with \(\mathbb R \mathbb P^3\).
The author stresses that though eventually one only looks at the real points \(X(\mathbb R)\), the proof essentially requires to keep track of all the complex points \(X (\mathbb C)\). In theorem 1.13 the author completely classifies the possible elementary transforms appearing in the minimal model program:
Let \(X\) be a smooth, projective real algebraic 3-fold such that \(X(\mathbb R)\) satisfies the following conditions: (1) \(X(\mathbb C)\) does not contain a 2-sided \(\mathbb R \mathbb P^2\), (2) \(X(\mathbb C)\) does not contain a 1-sided torus, (3) \(X(\mathbb C)\) does not contain a 1-sided Klein bottle with nonorientable neighborhood. (These are satisfied when \(X\) is orientable.) Let \(f_i \: X_i \rightarrow X_{i+1}\) be any elementary transform in the minimal model program. Then there is a well-defined induced map \(f_i \: X_i(\mathbb R) \to X_{i+1}(\mathbb R)\) which is everywhere defined, and there is a complete list of descriptions for the possibilities.
The author also gives an application of his result to the factorization problem of birational morphisms defined over \(\mathbb R\). Except for the classically known case of dimension 2, it is a hard question whether it is possible to express an arbitrary birational morphism \(f \: X \to Y\) between smooth projective varieties (over any field) as a composite of those ‘elementary blow ups’ \(f_i \_: X_i \to X_{i-1}\). Depending on how one interprets the meaning of ‘elementary blow ups’ it transfers to different types of problems in various degrees of difficulties. If one allows those \(f_i\)’s to be any elementary transforms in the sense of minimal model theory, then the answer follows from minimal model theory, thus it is affirmative in dimension 3. A drawback is that then \(X_i\) in general possesses singularities. One ideal form of the factorization problem would be thus to allow \(X_i\) to be only smooth varieties, and \(f_i\) to be blow ups with smooth (and reduced) centers. This conjectural statement was perceived by Abhyankar already back in 1950’s. A weaker version of the statement in the complex case is due to S. D. Cutkosky [Adv. Math. 132, No. 2, 167-315 (1997; Zbl 0934.14006) and “Local monomialization and factorization of morphisms”, Astérisque 260 (1999; Zbl 0941.14001)], after contributions by Shannon and Christensen, for the local case, and then recent works by Abramovich–Karu–Matsuki–Włodarczyk appeared for the global case [D. Abramovich, K. Matsuki and S. Rashid, Tohoku Math. J., II. Ser. 51, No., 489-537 (1999)]. The author obtains a theorem of factorization for real manifolds which is slightly weaker than the above ideal version, in theorem 2.2:
Let \(f:Y\to X\) be a birational morphism between smoooth and projective threefolds over \(\mathbb R\). Assume that the topological conditions (1-3) stated above hold for \(Y(\mathbb R)\). Then \(f\) can be factored as \(f = f_1 \circ \cdots \circ f_n\), where each \(f_i \: X_i \to X_{i-1}\) is such that \(X_i\) has only ‘compound \(A_1\)’-type singularities at real points and \(f_i\) must be either the blow-up along a reduced (smooth or singular) point in \(X_{i-1}\), the blow-up along a curve in \(X_{i-1}\) which contains only finitely many real points, or is trivial along a Zariski neighborhood of the set of real points. The author notes that it are those \(\mathbb R\)-trivial ones which involve full complexity in carrying out the steps.
The contents (verbatim): 1. Introduction. 2. Applications and speculations. 3. The minimal model program over \(\mathbb R\). 4. The topology of real points and the MMP. 5. The topology of divisorial contractions. 6. The gateway method. 7. Small and divisor-to-curve contractions. 8. Proof of the main theorems. 9. \(cAx\) and \(cD\)-type points. 10. \(cA\)-type points. 11. \(cE\)-type points. 12. Hyperbolic 3-manifolds. Acknowledgements, References.
Also see the review of part III of this paper [J. Kollár, J. Math. Sci., New York 94, No. 1, 996-1020 (1999); see the following review Zbl 0964.14014)],

14E30 Minimal model program (Mori theory, extremal rays)
14P25 Topology of real algebraic varieties
14E05 Rational and birational maps
14M20 Rational and unirational varieties
57N10 Topology of general \(3\)-manifolds (MSC2010)
14J30 \(3\)-folds
Full Text: DOI arXiv
[1] S. Akbulut and H. King, All knots are algebraic, Comment. Math. Helv. 56 (1981), no. 3, 339 – 351. · Zbl 0477.57018 · doi:10.1007/BF02566217 · doi.org
[2] Selman Akbulut and Henry King, Topology of real algebraic sets, Mathematical Sciences Research Institute Publications, vol. 25, Springer-Verlag, New York, 1992. · Zbl 0808.14045
[3] V. I. Arnol\(^{\prime}\)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. V. I. Arnol\(^{\prime}\)d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi.
[4] M. Andreatta and J. Wiśniewski, A survey on contractions of higher dimensional varieties, in Algebraic Geometry, Santa Cruz 1995, Amer. Math. Soc. 1997. CMP 98:07
[5] 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). · Zbl 0633.14016
[6] Herbert Clemens, János Kollár, and Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989) (English, with French summary). · Zbl 0689.14016
[7] A. Comessatti, Sulla connessione delle superfizie razionali reali, Annali di Math. 23(3) (1914) 215-283. · JFM 45.0889.02
[8] Steven Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), no. 3, 521 – 525. · Zbl 0616.14003 · doi:10.1007/BF01456342 · doi.org
[9] 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. · Zbl 0813.14039
[10] William Fulton, Algebraic topology, Graduate Texts in Mathematics, vol. 153, Springer-Verlag, New York, 1995. A first course. · Zbl 0852.55001
[11] 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. · Zbl 0498.55001
[12] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[13] T. Hayakawa, (personal communication).
[14] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. · Zbl 0345.57001
[15] 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). V. A. Iskovskih, Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, 1979, pp. 59 – 157, 239 (loose errata) (Russian). V. A. Iskovskih, Birational automorphisms of three-dimensional algebraic varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, 1979, pp. 159 – 236, 239 (loose errata) (Russian).
[16] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. · Zbl 0412.57007
[17] Yujiro Kawamata, Boundedness of \?-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. · Zbl 0785.14024
[18] Yujiro Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 241 – 246. · Zbl 0894.14019
[19] V. Kharlamov, The topological type of non-singular surfaces in \(RP^3\) of degree four, Funct. Anal. Appl. 10 (1976) 295-305. · Zbl 0362.14013
[20] 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. · Zbl 0649.14022
[21] János Kollár, Minimal models of algebraic threefolds: Mori’s program, Astérisque 177-178 (1989), Exp. No. 712, 303 – 326. Séminaire Bourbaki, Vol. 1988/89. · Zbl 0711.14008
[22] J. Kollár, Real Algebraic Surfaces, Notes of the 1997 Trento summer school lectures, (preprint).
[23] J. Kollár, Real Algebraic Threefolds I. Terminal Singularities, Collectanea Math. (to appear).
[24] J. Kollár, Real Algebraic Threefolds III. Conic Bundles (preprint). · Zbl 0964.14014
[25] J. Kollár, Real Algebraic Threefolds IV. Del Pezzo fibrations (preprint). · Zbl 1078.14088
[26] J. Kollár, The Nash conjecture for Algebraic Threefolds, ERA of AMS 4 (1998) 63-73. · Zbl 0896.14030
[27] J. Kollár (with 14 coauthors), Flips and Abundance for Algebraic Threefolds, Astérisque 211 (1992).
[28] János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429 – 448. · Zbl 0780.14026
[29] János Kollár and Shigefumi Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533 – 703. · Zbl 0773.14004
[30] J. Kollár - S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, 1998.
[31] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023
[32] Dimitri Markushevich, Minimal discrepancy for a terminal cDV singularity is 1, J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 445 – 456. · Zbl 0871.14003
[33] T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668 – 684. , https://doi.org/10.2307/2373030 T. Matsusaka and D. Mumford, Correction to: ”Two fundamental theorems on deformations of polarized varieties”, Amer. J. Math. 91 (1969), 851. · Zbl 0128.15505 · doi:10.2307/2373355 · doi.org
[34] Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 47. · Zbl 0349.57001
[35] Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133 – 176. · Zbl 0557.14021 · doi:10.2307/2007050 · doi.org
[36] Shigefumi Mori, On 3-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43 – 66. · Zbl 0589.14005
[37] Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117 – 253. · Zbl 0649.14023
[38] J. Nash, Real algebraic manifolds, Ann. Math. 56 (1952) 405-421. · Zbl 0048.38501
[39] 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.
[40] 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. · Zbl 0634.14003
[41] J.-L. Riesler, Construction d’hypersurfaces réelle (Sém. Bourbaki #763), Astérisque, 216 (1993) 69-86.
[42] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original.
[43] Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint.
[44] Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 · doi.org
[45] B. Segre, The non-singular cubic surfaces, Clarendon Press, 1942. · JFM 68.0358.01
[46] R. I. Shafarevich, Basic Algebraic Geometry (in Russian), Nauka, 1972; Revised English translation: Springer 1994.
[47] Robert Silhol, Real algebraic surfaces with rational or elliptic fiberings, Math. Z. 186 (1984), no. 4, 465 – 499. · Zbl 0558.14022 · doi:10.1007/BF01162775 · doi.org
[48] Robert Silhol, Real algebraic surfaces, Lecture Notes in Mathematics, vol. 1392, Springer-Verlag, Berlin, 1989. · Zbl 0691.14010
[49] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.