3-fold log flips. Appendix by Yujiro Kawamata: The minimal discrepancy coefficients of terminal singularities in dimension three.

*(English. Russian original)*Zbl 0785.14023
Russ. Acad. Sci., Izv., Math. 40, No. 1, 95-202 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105-201, Appendix 201-203 (1992).

The author’s abstract (from the meanwhile available English version): “We prove that 3-fold log flips exist. We deduce the existence of log canonical and \(\mathbb{Q}\)-factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and log terminal adjunction.” This paper, published also in the above mentioned substantially corrected (by the author) and translated (by M. Reid) edition, can be recommended especially in this second form, which is enlarged also by an appendix of Y. Kawamata on the discrepancy of a 3- fold terminal singularity and a commentary by M. Reid. This commentary also contains a general review of the contents which is reproduced here:

“The paper proposes a program for constructing 3-fold flips (including Mori’s flips) and log flips, and claims to carry it out. The rough idea is an inductive approach along the following lines: Suppose \(f:X \to Z\) is a flipping contraction, with exceptional curve \(C\). We attempt to construct a partial resolution \(Y \to X\) that either blows up one point on the flipping curve, or blows up \(C\) at the general point, and such that the composite \(Y\to Z\) has \(\rho(Y/Z)=2\), and \(\overline{NE}(Y/Z)\) has two extremal rays (or log rays) \(R_{\text{old}}\) and \(R_{\text{new}}\). One of these gives the old contraction to \(X\), and \(R_{\text{new}}\), if it exists and is divisorial, gives the new contraction to the flipped \(X^ +\). Then \(X\to X^ +\) is the flip.

In carrying out this construction, we need to use auxiliary flips for two purposes:

(1) To establish the model \(Y\): Start from a more-or-less arbitrary resolution of singularities \(\tilde X \to X\) that includes either a blow- up of a point of \(C\) or a blow-up of \(C\) itself, then proceed to climb down from \(\tilde X\) to the controlled model \(Y\) by the minimal model program.

(2) To deal with the possibility that \(R_{\text{new}}\) is not a divisorial contraction.

In order for this to provide a proof of the existence of flips, we need to know that the auxiliary flips can be done. This might be achieved in one of two ways: either (a) by induction, because we can assert that the auxiliary flipping contractions are simpler than \(f:X \to Z\) (for example, some invariant is smaller); or (b) because we know the auxiliary flipping contractions from some other point of view, for example, as fibers of semistable families of surfaces, for which the epic theorem of Tsunoda-Shokurov-Kawamata-Mori is applicable. The author’s attempt at this in the 3-fold case is extremely serious, and it seems to me almost certain that it is correct and complete (after all, he is the master of the spaghetti proof), although the presentation cannot exactly be described as attractive. The Utah seminary [Flips and abundance for algebraic threefolds. A summer seminar at the Univ. Utah 1991, Astér. 211 (1992; Zbl 0782.00073)] seems to guarantee the results (at least in general terms) up to the middle of §8. It seems at least possible to me that we may eventually fully understand the inductive workings of Mori theory, and that we will then be able to make this program work purely by induction, maybe even in higher dimensions when the more concrete approach in terms of classifying singularities seems doomed. In addition to his main theorems, the author introduces several important new ideas, including (1) the LSEPD trick [see example 1.6, (10.5) below, and compare the Utah Seminar (cited above), definition 2.30]; (2) the ideas of §5 on complements of a log divisor and the 1-, 2-, 3-, 4- and 6-complements that are characteristic to dimensions 2 and 3 [compare the Utah Seminar, §19]; (3) the insight in §4 that invariants of log canonical singularities and varieties such as discrepancy, index and so on have “spectral” properties such as a.c.c. [see the Utah Seminar, theorem 1.32, for a discussion]; (4) the ideas and results on “inverting adjunction” of problem 3.3.

The author’s theorem on log flips already has very substantial applications in the literature, most notably Kawamata’s solution of the abundance conjecture [Y. Kawamata, Invent. Math. 108, No. 2, 229- 246 (1992; Zbl 0777.14011) and the Utah Seminar (cited above), chapters 10-15].”

The reviewer refuses any further comments and refers the interested reader to M. Reid’s paragraph 10 of the English version, containing also historical, philosophical and terminological remarks.

“The paper proposes a program for constructing 3-fold flips (including Mori’s flips) and log flips, and claims to carry it out. The rough idea is an inductive approach along the following lines: Suppose \(f:X \to Z\) is a flipping contraction, with exceptional curve \(C\). We attempt to construct a partial resolution \(Y \to X\) that either blows up one point on the flipping curve, or blows up \(C\) at the general point, and such that the composite \(Y\to Z\) has \(\rho(Y/Z)=2\), and \(\overline{NE}(Y/Z)\) has two extremal rays (or log rays) \(R_{\text{old}}\) and \(R_{\text{new}}\). One of these gives the old contraction to \(X\), and \(R_{\text{new}}\), if it exists and is divisorial, gives the new contraction to the flipped \(X^ +\). Then \(X\to X^ +\) is the flip.

In carrying out this construction, we need to use auxiliary flips for two purposes:

(1) To establish the model \(Y\): Start from a more-or-less arbitrary resolution of singularities \(\tilde X \to X\) that includes either a blow- up of a point of \(C\) or a blow-up of \(C\) itself, then proceed to climb down from \(\tilde X\) to the controlled model \(Y\) by the minimal model program.

(2) To deal with the possibility that \(R_{\text{new}}\) is not a divisorial contraction.

In order for this to provide a proof of the existence of flips, we need to know that the auxiliary flips can be done. This might be achieved in one of two ways: either (a) by induction, because we can assert that the auxiliary flipping contractions are simpler than \(f:X \to Z\) (for example, some invariant is smaller); or (b) because we know the auxiliary flipping contractions from some other point of view, for example, as fibers of semistable families of surfaces, for which the epic theorem of Tsunoda-Shokurov-Kawamata-Mori is applicable. The author’s attempt at this in the 3-fold case is extremely serious, and it seems to me almost certain that it is correct and complete (after all, he is the master of the spaghetti proof), although the presentation cannot exactly be described as attractive. The Utah seminary [Flips and abundance for algebraic threefolds. A summer seminar at the Univ. Utah 1991, Astér. 211 (1992; Zbl 0782.00073)] seems to guarantee the results (at least in general terms) up to the middle of §8. It seems at least possible to me that we may eventually fully understand the inductive workings of Mori theory, and that we will then be able to make this program work purely by induction, maybe even in higher dimensions when the more concrete approach in terms of classifying singularities seems doomed. In addition to his main theorems, the author introduces several important new ideas, including (1) the LSEPD trick [see example 1.6, (10.5) below, and compare the Utah Seminar (cited above), definition 2.30]; (2) the ideas of §5 on complements of a log divisor and the 1-, 2-, 3-, 4- and 6-complements that are characteristic to dimensions 2 and 3 [compare the Utah Seminar, §19]; (3) the insight in §4 that invariants of log canonical singularities and varieties such as discrepancy, index and so on have “spectral” properties such as a.c.c. [see the Utah Seminar, theorem 1.32, for a discussion]; (4) the ideas and results on “inverting adjunction” of problem 3.3.

The author’s theorem on log flips already has very substantial applications in the literature, most notably Kawamata’s solution of the abundance conjecture [Y. Kawamata, Invent. Math. 108, No. 2, 229- 246 (1992; Zbl 0777.14011) and the Utah Seminar (cited above), chapters 10-15].”

The reviewer refuses any further comments and refers the interested reader to M. Reid’s paragraph 10 of the English version, containing also historical, philosophical and terminological remarks.

Reviewer: M.Roczen (Berlin)

##### MSC:

14J30 | \(3\)-folds |

14E05 | Rational and birational maps |

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