Flip and Flop Janos Kollâr Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Dedicated to my teacher Teruhisa Matsusaka The study of smooth projective varieties naturally breaks into two parts. One part is to study the relationship between birationally equivalent varieties and the other part is to study the different birational equivalence classes. The aim of this talk is to review some interesting features of birational equivalence for three dimensional varieties over C For smooth projective curves birational equivalence implies isomorphism, hence the first part of the problem is not interesting. For smooth projective surfaces the simplest birational transformation is the blowing up. One removes a point of a surface S and replaces it with a copy of P 1 corresponding to the local complex directions at the removed point. Any birational equivalence is a composite of blowing ups and their inverses. There are no other kinds of birational transformations. In dimension three one can blow up points and smooth curves. It is conjectured that any birational transformation is a composite of blowing ups and their inverses, though at the moment no one seems to know how to prove this. However it became clear in the last decade that even if the above factorisation is possible it may not be the best way of factoring birational transformations. A large theory has been developped by several people, I refer to the talk of (Mori 1990) in the same volume for a general overview. My aim in this talk is to concentrate on certain special birational transformations between threefolds, called flip and flop. For technical reasons it is natural to investigate these transformations not only for smooth threefolds but also for certain singular ones. The smallest class of singularities that allows the theory to work well is given by the following definition : Definition 1. Let X be a normal variety and let / : X' -> X be a resolution of singularities with exceptional divisors,- c X f, Assume that (V(mKx) is locally free for some m > 0. The smallest such m will be called the index of X. One can write Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990
710 Jânos Kollâr (9(mK x >) S f(9(mk x ) <E> 0( «%) > for some integers a t. We say that X has terminal (resp. canonical) singularities if at > 0 (resp. a,- > 0) for every i. The survey paper (Reid 1987) is the best source of information concerning terminal (and canonical) singularities. The informal definition of flip or flop is the following. Let X be a threefold and let C cz X be a compact curve. Remove C from X and try to replace it with another compact curve C + in such a way that C + U (X C) becomes a threefold X +. To avoid the unexciting possibility of C = C + we require that X and X + be nonisomorphic. Such an X + need not exist at all and it also need not be unique. It turns out that the nature of the transformation X > X + depends on the sign of C K x. The C K x = 0 case is called flop, the C K x < 0 case called flip. Very little is known about the case C K x > 0 and I will not consider it at all. The formal definitions are the following: Definition 2. (2.1) A three dimensional curve neighborhood is a pair C cz X where C is a proper connected curve and X is the germ of a normal threefold along C. One can think of X as an analytic representative of the germ. (2.2) A three dimensional curve neighborhood C cz X is called contractible if there is a morphism f'.cax-^pey satisfying the following properties : (i) Y is the germ of a normal singularity around the point P ; (ii) f(c)=p', (iii) / : X C > Y P is an isomorphism. / and Y are uniquely determined by C cz X. f is called the contraction morphism of C cz X. (2.3) Two three dimensional curve neighborhoods Q cz Xi are called bimeromorphic if there is an isomorphism (X\ C\) = (X2 C2). (2.4) Let C\ cz X\ be a three dimensional curve neighborhood. Let H\ be a line bundle such that iff l \C\ is ample. A three dimensional curve neighborhood C2 cz X2 bimeromorphic to C\ cz X\ is called the opposite of C{ cz Xi with respect to H\ if there is a line bundle #2 on X2 such that Ü2IC2 is ample and (X 2 - C 2,H?\X 2 - C 2 ) s (Xi - Ci, J^l*! - d), for some positive integers m,n. By (Kollâr 1991, 2.1.6) the opposite is unique, though it need not exist. Definition 3. Let /:Ccl->Pe7bea three dimensional contractible curve neighborhood. Assume that X has terminal singularities and that K X \C = 0. Let if be a line bundle on X such that ff _1 C is ample. The opposite of C cz X (if it exists) will be called the flop of C cz X. (It turns out that the flop is independent of the choice of H.) Definition 4. Let /:CczX»PeYbea three dimensional contractible curve neighborhood. Assume that X has terminal singularities and that K^ C is ample.
Flip and Flop 711 The opposite of C cz X with respect to K x (if it exists) will be called the flip of C cz X. The simplest example of flops is the following: Example 5. Let V be the total space of the line bundle 0( 1, 1) over P 1 x P 1, Both of the projections m : P 1 X P 1 -> P 1 can be extended to morphisms Pi : (P 1 x P 1 cz V) - (Ci S P 1 cz Xi). It is easy to see that Xj is smooth and the normal bundle of C, cz X,- is 0( 1) + (9( 1). P2 Pf * -Xi > X2 is bimeromorphic but it is not an isomorphism. This example was first noticed by (Atiyah 1958). It was used in a systematic way by (Kulikov 1977) to study the birational transformations of threefolds that have a basepoint-free pencil of K3 surfaces. Examples of flips are much more difficult to present since X cannot be smooth. A fairly exhaustive list is given in (Kollâr-Mori 1991). From the logical point of view (which is the reverse of the historical order) the theory of flips and flops has three large parts. I. Prove that flops and flips exist. II. Describe C + cz X + in terms of C cz X as precisely as possible. III. Applications. The answer to part I is very simple to state : Theorem 6 (Reid 1983). Flops exist. Theorem 7 (Mori 1988). Flips exist. The existence of flips is one of the most difficult results in three dimensional geometry; see (Kollâr 1990a) for an introduction. Previously an important special case was settled by (Tsunoda 1987; Shokurov 1985; Mori 1985; Kawamata 1988) using different methods. (The order corresponds to the order in which the proofs were announced. The only complete published version is the last one.) It is interesting to note that the above proofs provide very little information beyond the existence of X +. Thus the answer to part II proceeds along very different lines. For flops there is a detailed structure theory. This is based on the following observation: Theorem 8 (Mori 1987). Let f : C cz X - P e Y be a three dimensional contractible curve neighborhood such that K X \C 0. Assume in addition that X is smooth (or that it has only index one terminal singularities). Then Y embedds into (C 4 and in suitable local coordinates its equation can be written as x 2 + h(y,z,t)
712 Jânos Kollâr for some power series h. Let x : Y > Y be the involution (x,y,z,t) i-> ( x,y,z,t). Then (f + :X +^Y)^(Tof:X-+Y). In particular, X + is smooth iff X is smooth. If X has higher index points then the symmetry breaks down in some cases. However in these cases another explicit description is possible, see (Kollâr 1989; Kollâr 1991, 2.2.2). There are always many similarities between X and X + : Theorem 9. Let X\ and X2 be projective threefolds with terminal singularities only. Assume that X2 is obtained from X\ by a sequence of flops. The following objects do not depend on j (up to isomorphism): (9.1) The intersection homology groups IH l (Xj,<D) together with their Hodge structures (Kollâr 1989, 4.12). (9.2) The collection of analytic singularities of Xj (Kollâr 1989, 4.11). (9.3) The miniversal deformation space DeîXj (Kollâr-Mori 1991, 12.6). (9.4) The integral cohomology groups H\X h i) (Kollâr 1991, 3.2.2). (9.5) PicXj cz WeilX, (Kollâr 1991, 3.2.2). (9.6) h (Xj, (9(D)) for every Weil divisor D. (9.7) h\x, (9(mK X] )) for every i and m. The isomorphisms are canonical except possibly for (9.2). Describing C + cz X + in terms of C cz X is harder for flips. The first breakthrough was a result of (Shokurov 1985) which showed that the singularities of X + are less "difficult" than the singularities of X in a complicated technical sense. For a given singularity the "difficulty" is not easy to compute but the definition is very well suited to inductive proofs. It turns out that the situation is very complicated. The problem is that the singularities of X + depend not only on the singularities of X but also on the global structure of X. If X is given by coordinate patches and transition functions along C = P 1 then the singularities of X + do not depend continuously on the transition functions. This makes any description very subtle. Several special cases are worked out in (Kollâr-Mori 1991); many more - but not all - cases are treated in unpublished works of Mori. I would like to mention one patholgy that seems interesting: Proposition 10 (Kollâr-Mori 1991, 13.5, 13.7). Let C+ cz X + be the flip ofcax. Then the number of irreducible components of C + is less than or equal to the number of irreducible components of C. For every m > 0 there are examples where C + is irreducible and C has m irreducible components. Let us now turn to some of the applications. The most important application of flops is the following result which shows that birational maps between certain threefolds can be factored into a sequence of flops. This factorisation is much more useful than a factorisation into blowing ups and downs would be. Its
Flip and Flop 713 importance is clear only in light of the minimal model theory of threefolds (see e.g. (Mori 1990)). Theorem 11 (Reid 1983; Kawamata 1988; Kollâr 1989). Let X x and X 2 be normal projective threefolds with terminal singularities only. Assume that they have Q-factorial singularities. Assume furthermore that K Xi has nonnegative intersection number with any curve C cz X\ for i = 1,2. Then any birational map f : X\ > X2 can be obtained as a composite of flops, Another application is to compact nonprojective threefolds. Under certain conditions there is a very economical way of finding a birationally equivalent model which is projective. Theorem 12 (Kollâr 1991, 5.2.3). Let X be a proper algebraic threefold with Q- factorial terminal singularities. Assume that K x has nonnegative intersection number with any curve C cz X. Then after finitely many flops one obtains a proper algebraic threefold X + with Q-factorial terminal singularities such that X+ admits a birational morphism g : X + Y onto a projective variety Y. Y has terminal singularities and g contracts only finitely many curves. Flips are one step of Mori's program (also called minimal model program) for threefolds, in fact the most difficult step. Their main importance is thus derived from the importance and applications of the program. See (Kollâr 1991) or (Mori 1990) for details. There are many problems concerning flips even in dimension three. The most interesting open question about flips is the following. Conjecture 13. Reid's Conjecture on General Elephants (Reid 1987; Kollâr- Mori 1991). The contraction map provides a one-to-one correspondence between the following two sets: Extremal neighborhoods: (Three dimensional contractible curve neighborhoods C cz X such that") """ \x has canonical singularities and K X \C is ample. J Flipping singularities: (Three. ^,^ din dimensional normal singularities P e Y such that Ky is not], '~ [Q-Cartiei ^~rtier and some D e \ Ky has a Du Val singularity at P. I Reid's original hope was that this equivalence can be used to obtain a proof of the existence of flips. To do this one needs to produce a member of Ky with Du Val singularity and then to use this member to construct X +. It is still to be seen whether either of these steps can be done in the spirit envisaged by Reid. A proof along these lines would considerably enhance our understanding of flips.
714 Jânos Kollâr (Kollâr-Mori 1991, 3.1) shows that for every flipping singularity there is a corresponding extremal neighborhood. The opposite direction is also proved for extremal neighborhoods with terminal singularities and irreducible C in (Kollâr- Mori 1991, 1.7). The general case probably requires different methods. There is also the possibility that the correspondence is more complicated. The conjecture implies that every non-gorenstein singularity on an extremal neighborhood is pseudo-terminal. I do not see any a priori reason why this should be so. On the other hand, even certain terminal singularities cannot occur on extremal neighborhoods, thus unexpected restrictions are possible. The higher dimensional problems are discussed in the talk of (Kawamata 1990). Acknowledgement. Partialfinancialsupport was provided by the NSF under grant numbers DMS-8707320 and DMS-8946082 and by an A. P. Sloan Research Fellowship. References Atiyah, M. (1958) : On analytic surfaces with double points. Proc. Roy. Soc. 247, 237-244 Kawamata, Y. (1988) : The crêpant blowing-up of 3-dimensional canonical singularities and its application to the degeneration of surfaces. Ann. Math 127, 93-163 Kawamata, Y. (1990) : Canonical singularities and minimal models of algebraic varieties. These Proceedings, p. 699 Kollâr, J. (1989): Flops. Nagoya Math. J. 113, 14-36 Kollâr, J. (1990): Minimal models of algebraic threefolds: Mori's Program. Astérisque 177-178, 303-326 Kollâr, J. (1991) : Flips, flops, minimal models etc. J. Diff. Geom. (to appear) Kollâr, J., Mori, S. (1991): Soon to be written up Kulikov, V. (1977) : Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11, 957-989 Mori, S. (1985) : Minimal models for semistable degenerations of surfaces. Lectures at Columbia University. Unpublished Mori, S. (1987) : Personal communication Mori, S. (1988) : Flip theorem and the existence of minimal models for 3-folds. Journal AMS 1, 117-253 Mori, S. (1990) : Birational Classification of algebraic threefolds. These Proceedings, p. 235 Reid, M. (1983) : Minimal models of canonical threefolds, Algebraic Varieties and Analytic Varieties. Adv. Stud. Pure Math. vol. 1 (Iitaka, S., ed.). Kinokuniya and North-Holland, pp. 131-180 Reid, M. (1987): Young person's guide to canonical singularities. Algebraic Geometry Bowdoin 1985, Proc. Symp. Pure Math. vol. 46, 345-416 Shokurov, V. (1985a): The nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat. 49, 635-651 Shokurov, V (1985b) : Letter to M. Reid Tsunoda, S. (1987) : Degenerations of surfaces. Algebraic Geometry, Sendai, Adv. Stud. Pure Math. vol. 10 (Oda, T., ed.). Kinokuniya and North-Holland, pp. 755-764