Real algebraic threefolds. II: Minimal model program.

*(English)*Zbl 0964.14013For 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)],

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)],

Reviewer: Yasuyuki Kachi (Knoxville)

##### MSC:

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 |

##### Keywords:

MMP; real algebraic threefolds; minimal model; flips; divisorial contractions; factorization problem of birational morphisms##### References:

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