Termination of log flips for algebraic 3-folds.

*(English)*Zbl 0814.14016We shall prove that there exists no infinite sequence of successive log flips for algebraic 3-folds. Let \(X\) be a normal \(\mathbb{Q}\)-factorial variety and \(B\) a \(\mathbb{Q}\)-divisor such that the pair \((X,B)\) has only weak log terminal singularities. A log flip for \((X,B)\) is a diagram \(X \overset \varphi {} Y \overset {\varphi^ +} \leftarrow X^ +\) consisting of projective birational morphisms between normal varieties such that

(1) \(\rho(X/Y) = 1\), i.e., \(\varphi\) is not an isomorphism and for any two curves \(C\) and \(C'\) which are mapped to points by \(\varphi\), there exists a positive number \(\alpha\) such that \(C \sim_{\text{num}} \alpha C'\),

(2) \(\rho(X^ +/Y) = 1\),

(3) \(\text{codim Exc}(\varphi) \geq 2\), where Exc denotes the exceptional locus,

(4) \(\text{codim Exc}(\varphi^ +) \geq 2\),

(5) \(-(K_ X + B)\) is \(\varphi\)-ample,

(6) \(K_{X^ +} + B^ +\) is \(\varphi^ +\)-ample, where \(B^ +\) is the strict transform of \(B\) on \(X^ +\).

The resulting pair \((X^ +,B^ +)\) is automatically \(\mathbb{Q}\)-factorial and weak log terminal. The main result of this paper is the following:

Theorem 1. Let \(X\) be a 3-dimensional normal \(\mathbb{Q}\)-factorial variety and \(B\) a \(\mathbb{Q}\)-divisor such that the pair \((X,B)\) has only log terminal singularities. Then there exists no infinite sequence of successive log flips such as \[ X = X_ 0 @>\varphi_ 0>> Y_ 0 @<\varphi^ +_ 0<< X_ 1 @>\varphi_ 1>> Y_ 1 @<\varphi^ +_ 2<< X_ 2 @>\varphi_ 2>> Y_ 2 @<\varphi^ +_ 2<< X_ 3 @>\varphi_ 3>> \dots \] where the first pair of arrows is a log flip for \((X_ 0,B_ 0)\) with \(B_ 0 = B\), the second is for \((X_ 1,B_ 1)\) with the strict transform \(B_ 1\) of \(B_ 0\), the third is for \((X_ 2,B_ 2)\) with the strict transform \(B_ 2\) of \(B_ 1\), and so on.

By the log minimal model program [see the author, K. Matsuda, K. Matsuki in Algebraic Geometry, Proc. Sympos., Sendai 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006)], the above results combined yield the following existence theorem of log minimal models:

Theorem 2. Let \(X_ 0\) be a 3-dimensional nonsingular projective variety and \(B_ 0 = \sum_ i b_ i S_ i\) a \(\mathbb{Q}\)-divisor such that \(0<b_ i \leq 1\) and the \(S_ i\) are mutually distinct nonsingular prime divisors with normal crossings. Then there exist a pair \((X,B)\) of a normal projective \(\mathbb{Q}\)-factorial variety and a \(\mathbb{Q}\)-divisor having only weak log terminal singularities, and a birational map \(f:X_ 0 \to X\) which is surjective in codimension 1, i.e., the image of the domain of \(f\) contains all the points of codimension 1, such that one of the following holds: \(B\) being the image of \(B_ 0\),

(1) there exists a surjective morphism \(\varphi : X \to Y\) with connected fibers to a normal projective variety \(Y\) such that \(\dim Y < \dim X\) and \(-(K_ X + B)\) is \(\varphi\)-ample,

(2) \(K_ X + B\) is nef.

[See also the following review.].

(1) \(\rho(X/Y) = 1\), i.e., \(\varphi\) is not an isomorphism and for any two curves \(C\) and \(C'\) which are mapped to points by \(\varphi\), there exists a positive number \(\alpha\) such that \(C \sim_{\text{num}} \alpha C'\),

(2) \(\rho(X^ +/Y) = 1\),

(3) \(\text{codim Exc}(\varphi) \geq 2\), where Exc denotes the exceptional locus,

(4) \(\text{codim Exc}(\varphi^ +) \geq 2\),

(5) \(-(K_ X + B)\) is \(\varphi\)-ample,

(6) \(K_{X^ +} + B^ +\) is \(\varphi^ +\)-ample, where \(B^ +\) is the strict transform of \(B\) on \(X^ +\).

The resulting pair \((X^ +,B^ +)\) is automatically \(\mathbb{Q}\)-factorial and weak log terminal. The main result of this paper is the following:

Theorem 1. Let \(X\) be a 3-dimensional normal \(\mathbb{Q}\)-factorial variety and \(B\) a \(\mathbb{Q}\)-divisor such that the pair \((X,B)\) has only log terminal singularities. Then there exists no infinite sequence of successive log flips such as \[ X = X_ 0 @>\varphi_ 0>> Y_ 0 @<\varphi^ +_ 0<< X_ 1 @>\varphi_ 1>> Y_ 1 @<\varphi^ +_ 2<< X_ 2 @>\varphi_ 2>> Y_ 2 @<\varphi^ +_ 2<< X_ 3 @>\varphi_ 3>> \dots \] where the first pair of arrows is a log flip for \((X_ 0,B_ 0)\) with \(B_ 0 = B\), the second is for \((X_ 1,B_ 1)\) with the strict transform \(B_ 1\) of \(B_ 0\), the third is for \((X_ 2,B_ 2)\) with the strict transform \(B_ 2\) of \(B_ 1\), and so on.

By the log minimal model program [see the author, K. Matsuda, K. Matsuki in Algebraic Geometry, Proc. Sympos., Sendai 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006)], the above results combined yield the following existence theorem of log minimal models:

Theorem 2. Let \(X_ 0\) be a 3-dimensional nonsingular projective variety and \(B_ 0 = \sum_ i b_ i S_ i\) a \(\mathbb{Q}\)-divisor such that \(0<b_ i \leq 1\) and the \(S_ i\) are mutually distinct nonsingular prime divisors with normal crossings. Then there exist a pair \((X,B)\) of a normal projective \(\mathbb{Q}\)-factorial variety and a \(\mathbb{Q}\)-divisor having only weak log terminal singularities, and a birational map \(f:X_ 0 \to X\) which is surjective in codimension 1, i.e., the image of the domain of \(f\) contains all the points of codimension 1, such that one of the following holds: \(B\) being the image of \(B_ 0\),

(1) there exists a surjective morphism \(\varphi : X \to Y\) with connected fibers to a normal projective variety \(Y\) such that \(\dim Y < \dim X\) and \(-(K_ X + B)\) is \(\varphi\)-ample,

(2) \(K_ X + B\) is nef.

[See also the following review.].

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14E05 | Rational and birational maps |

14J30 | \(3\)-folds |

14J17 | Singularities of surfaces or higher-dimensional varieties |