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Home , Birational geometry, Caucher Birkar, Fano variety, Flip (mathematics)

P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (61–70)

CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC

C D. H

Abstract On Wednesday, August 1st 2018, was awarded the for his contributions to the and his proof of the bounded- ness of -log canonical Fano varieties. In this note I will discuss some of Birkar’s main achievements. studies the solution sets of systems of polynomial equations. A N complex is a subset of PC defined by homogeneous equations

N X = V (P1;:::;Pr ) P  C where Pi C[x0; : : : ; xN ] are homogeneous polynomials of degree di . Here 2 V (P1;:::;Pr ) is the vanishing set of the polynomials Pi and

N N +1 P := C (0;:::; 0)  /C C n f g is N -dimensional complex . Two non-zero vectors c;¯ c¯ CN +1 0 2 n (0;:::; 0) are equivalent c¯ ∼ c¯ if there is a non-zero constant  C such that f g 0 2  c¯ = c¯0. If c¯ = (c0; : : : ; cN ), then the corresponding equivalence class is denoted N by [c0 : ::: : cN ] P . Typically we assume that X is irreducible (i.e. not the 2 C union of two proper subvarieties) and reduced so that if P C[x0; : : : ; xN ] is an ho- 2 mogeneous polynomial vanishing along X, then P belongs to the ideal (P1;:::;Pr )  C[x0; : : : ; xN ]. We say that a variety is non-singular (of dimension d) at a point x X 2 if locally analytically it is isomorphic to an open subset of Cd . A variety X is smooth if it is non-singular at all of its points. In this case we can view X as a complex manifold. It is natural to try to classify smooth complex projective varieties up to isomorphism. The most natural invariant to consider is the dimension of a given variety. If dim X = 0, then X is a point, so the first interesting case to consider is when dim X = 1. In this case we say that X is a curve. Curves are compact orientable Riemann surfaces which 0 1 are topologically classified by their genus g = dim H (ΩX ). There are 3 main cases to consider:

• g = 0: In this case X P 1 . Š C MSC2010: 14E30. Keywords: Minimal model program, flips, Fano varieties, canonical .

61 62 CHRISTOPHER D. HACON

• g = 1: In this case X is an elliptic curve, belonging to a 1 parameter family 2 2 X = V (x(x z)(x z) zy ) P where  C 0; 1 .  C 2 n f g • g 2: In this case we say that X is a curve of general type. For fixed g there is  a 3g 3-dimensional algebraic family of these curves. Interestingly, the above subdivision in to 3 cases captures many of the properties of curves in a variety of contexts such as topology, differential geometry, and number the- ory. Naturally, one would like to extend these classification results to higher dimensions. One of the main tools used in the study of projective varieties is the canonical

dimX !X = (T _): ^ X 1 0 Note that if dim X = 1, then !X = ΩX and so dim H (!X ) = g recovers the genus of the curve X. In higher dimensions it is better to consider the

0 m R(!X ) := M H (!X˝ ): m 0  0 m For any m > 0 such that H (!˝ ) 0 we obtain a rational map X ¤ N m : X Ü P ; x Ü [s0(x): ::: : sN (x)]

0 m where s0; : : : ; sN is a basis of H (!X˝ ) and the map is defined at every point such that si (x) 0 for some 0 i N . ¤ Ä Ä When dim X = 1 we have that:

• g = 0: !X = !P 1 OP 1 ( 2) so that R(!X ) C. C Š C Š

• g = 1: !X OX so that R(!X ) C[t]. Š Š k • g 2: !X is a line bundle of degree 2g 2 > 0 and in fact !X˝ is very ample  N for k 3 which means that k : X, P is an embedding.  ! C Not surprisingly, in higher dimensions, dim X 2, there are many additional diffi-  culties. One important issue is that given any smooth variety X and any smooth subva- riety Z X of codimension 2, one can define f : X := blZ(X) X the blow up   0 ! of X along Z which is an isomorphism over X Z and n 1 E := f (Z) P(NZX) Z Š ! c is a fiber space whose fibers are projective spaces P of dimension c = codimX (Z) 1. C We say that E is an exceptional divisor, i.e. a codimension 1 subvariety contained in the locus where f is not an isomorphism. Note that X and blZ(X) share many important geometric properties. For example, it is easy to see that they have isomorphic canonical rings R(!X ) R(!X ) and fundamental groups 1(X) 1(X ), however they are Š 0 Š 0 not homeomorphic. For this reason, it is natural to consider the equivalence relation generated by this operation. We say that two varieties X and X 0 are birational if they CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC GEOMETRY 63 have isomorphic (non-empty) open subsets U X and U X such that U U .  0  0 Š 0 Equivalently one sees that X and X 0 are birational if and only if they have isomorphic fields of rational functions C(X) C(X ). The main goal of is to Š 0 classify projective varieties up to birational isomorphism. The strategy is as follows. Starting with a (irreducible and reduced) variety X  P N of dimension d := dim X, by Hironaka’s theorem, there exists X X a finite C 0 ! sequence of blow ups along smooth centers of X, such that X 0 is smooth and birational to X. Thus we may assume that X is smooth and consider the canonical ring R(!X ) = 0 m Lm 0 H (!X˝ ): The of X is given by  Ä(X) := tr:deg: R(!X ) 1 1; 0; 1;:::; dim X : C 2 f g Equivalently, we have that Ä(X) is the maximum dimension of the image of X under the pluricanonical maps k for k N. For example if dim X = 1, then Ä(X) = 1; 0; 2 or 1 is equivalent to g(X) = 0; 1, or 2, which recovers the above subdivision in to  three separate cases. N For a simple higher dimensional example, consider Xk PC a smooth hypersurface  N of degree k. Then !Xk OP N (k N 1) Xk where OP N (l) is the line bundle on P Š C j C C corresponding to homogeneous polynomials of degree l and OP N (l) X is its restriction C j to a subvariety X P N . It is then easy to see that there are three cases depending on  C whether !X is negative, zero or positive.

• k N : we have R(!X ) C so that Ä(X) = 1, Ä k Š • k = N + 1: we have R(!X ) C[t] so that Ä(X) = 0, and k Š

• k > N + 1: then !Xk is very ample so that Ä(X) = dim X. Further examples that include all possible values of the Kodaira dimension Ä(X) 2 1; 0; 1;:::; dim X can be easily constructed by considering the product of two vari- f g eties X Y . If min Ä(X);Ä(Y ) = 1, then Ä(X Y ) = 1. Otherwise, Ä(X Y ) =  f g   Ä(X) + Ä(Y ). One of the most natural and important questions in birational algebraic geometry is, for any birational equivalence class, to produce a distinguished representative with good geometric properties. In dimension 2 this was achieved by the Italian school of algebraic geometry at the beginning of the 20-th century and in dimension 3 it was finally proven by S. Mori and others in the 1980’s Mori [1988]. Around the same time the minimal model program (MMP) was established by work of Y. Kawamata, J. Kollár, S. Mori, M. Reid, V. Shokurov and others. The MMP is a general framework that aims to extend the birational classification results in dimension 2 and 3 to all dimensions. This program has not been completed in full generality, but there has been some recent spectacular progress. Perhaps the most exciting result in this direction is the following result on the finite generation of canonical rings. Theorem 1 (Birkar, Cascini, Hacon, and McKernan [2010], Hacon and McKernan [2010], and Siu [2008]). If X is a smooth complex projective variety, then its canonical ring R(!X ) is finitely generated. 64 CHRISTOPHER D. HACON

Of course this answers one of the most natural questions about the canonical ring, but it also has a number of important consequences that have greatly improved our understanding of birational geometry. For example we have the following. Corollary 2 (Birkar, Cascini, Hacon, and McKernan [2010]). If X is a smooth complex projective variety of general type (so that Ä(X) = dim X), then X has a canonical model Xcan and a minimal model Xmin. The canonical model is a unique distinguished representative of the birational equiv- alence class of X which is given by

Xcan := ProjR(!X ):

One can think of Xcan as being defined by the generators and relations of the canonical ring R(!X ): Unfortunately, there are many examples where Xcan is (mildly) singular. The singu- larities arising in this context are called canonical singularities. It is known that canoni- i i cal singularities are rational so that in particular H (OX ) H (OX ) for i 0. It is Š can  also known that canonical singularities are Q-Gorenstein. This means that even though ! is not necessarily a line bundle, it is a Q-line bundle so that ! k is a line bundle Xcan X˝can for some integer k > 0. Never the less we still abuse notation and refer to !Xcan as the canonical line bundle. For example let V be the quotient of C3 via the involution m (x; y; z) ( x; y; z), then V has canonical singularities and !˝ is a line bundle ! V if and only if m is an even integer. Another important feature is that ! is ample so that ! k is a very ample line Xcan X˝can bundle for any sufficiently big and divisible integer k > 0. This means that k : X N 0 k k ! P = P(H (! )) is an embedding and ! =  O N (1). This gives an easy X˝can X˝can k P geometric interpretation of why R(!X ) is finitely generated. In fact for any sufficiently big and divisible integer k > 0 we may assume that the homomorphism

0 0 kr M H (O N (r)) M H (!˝ ) P ! Xcan r N r N 2 2 0 is surjective. Since C[x0; : : : ; xN ] = Lr N H (OP N (r)) is clearly finitely generated, it follows that its quotient H 0(!2 kr ) is also finitely generated. This ring is Lr N X˝can 2 the ring of Z/kZ invariants in the graded ring R(!Xcan ) and so, by a theorem of E.

Noether on the finite generation of the integral closure, one sees that R(!Xcan ) is finitely generated. The existence of canonical models plays a pivotal role in our understanding of va- rieties of general type. For example it is crucial in the construction of a projective moduli space parametrizing canonically polarized varieties (with canonical singulari- ties) and their natural degenerations (see eg. Kollár and Shepherd-Barron [1988], Kol- lár [2013a,b], Hacon and Xu [2013], Hacon, McKernan, and Xu [2018a,b], and Kovács and Patakfalvi [2017]). When 0 Ä(X) < dim X, X can not have a canonical model, however it is still Ä expected that X has a minimal model Xmin which has mild (terminal) singularities, such that !Xmin is a nef Q-line bundle. This means that the intersection numbers with any CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC GEOMETRY 65

m curve C X are non-negative c1(!X ) C 0. In fact, it is expected that !˝ is base  min   Xmin point free for any m > 0 sufficiently divisible. Thus that m : X Z := Proj(R(!X )) ! is a morphism (i.e. the map is everywhere defined on X). If F is a general fiber of m m, then it is known that Ä(F ) = 0 and in fact we have !F˝ = OF . Therefore, to understand the geometry of varieties of intermediate Kodaira dimension, it is important to understand minimal models of varieties of Kodaira dimension 0 and their moduli spaces. This is a very active area of research but unluckily our knowledge of these varieties is still incomplete. It is conjectured that minimal models can be obtained by a finite sequence of elemen- tary birational maps

X = X1 Ü X2 Ü ::: Ü Xn = Xmin: These elementary birational maps are called flips and divisorial contractions. Divisorial contractions are birational morphisms f : X X of normal varieties ! 0 such that dim Ex(f ) = dim X 1, c1(!X ) C < 0 for any f -vertical curve and  (X/X 0) = 1. Recall that codimension 1 subvarieties are called divisors. It is easy to see that X 0 also has mild singularities. In dimension 2 the picture is very explicit. If 1 !X is not nef, then there exists a 1 curve E X such that E is rational E P and  Š c1(!X ) E = 1. By a theorem of Castelnuovo, these curves can always be contracted  by a morphism f : X X1 such that X1 is smooth and if P = f (E) X1, then !  X = blP (X1). Flipping contractions are birational morphisms f : X Z of normal varieties ! such that dim Ex(f ) < dim X 1, c1(!X ) C < 0 for any f -vertical curve and  (X/Z) = 1. It is easy to see that Z does not have mild singularities, in fact !Z is not even a Q-line bundle and so we can not replace X by Z. In this case it is necessary to replace X by the f + : X + Z of f : X Z. This operation is given + ! ! by X = ProjZ(R(!X /Z)). It is well defined because, by the relative version of m Theorem 1, R(!X /Z) = m 0f OX (!X˝ ) is finitely generated over Z. (Note that ˚   the question is local over Z and so we may assume that Z is affine. In this case we can m 0 m + identify f OX (!X˝ ) with H (!X˝ ).) It is easy to see that the singularities of X are also mild and thus we may replace X by X +. Repeating this procedure, we obtain a sequence of flips and divisorial contractions X = X1 Ü X2 Ü :::. It is hoped that after finitely many flips and divisorial contrac- tions we obtain a minimal model Xmin = Xn (recall that we are assuming Ä(X) 0).  The most important feature of flips and divisorial contractions is that they improve the geometry of X by removing some !X negative curves and replacing them with !X positive curves. If X X is a divisorial contraction, then it is easy to see that ! 0 B2(X 0) = B2(X) 1 and since B2(X) N, any sequence of divisorial contractions is Ü 2 finite. On the other hand if X X 0 is a flip, then B2(X) = B2(X 0) and so it is not clear that any sequence of flips is finite and in fact this turns out to be one of the most important and difficult open questions of the minimal model program. When X is of general type (Ä(X) = dim X), we are unable to show that arbitrary sequences of flips are always finite, however we can show that certain sequences of flips (for the minimal model program with scaling) are always finite and this implies the existence of minimal models in this case (cf. Corollary 2). 66 CHRISTOPHER D. HACON

Birkar has proven some of the strongest known results on the finiteness of flips.

Theorem 3 (Birkar [2011]). Assume the termination of flips for all (d 1)-dimensional varieties. If dim X = d and Ä(X) 0, then there are no infinite sequences of KX -flips.  We refer the reader to Birkar [ibid.] for further details and more precise statements. We also note that if one just considers the minimal model program with scaling, then there is a much stronger result (cf. Birkar and Hu [2014]): Assume the non-vanishing conjecture so that if X is non-uniruled, then Ä(X) 0.  Then there are no infinite sequences of flips with scaling. Recall that X is uniruled if it is covered by rational curves so that there is a dominant (surjective on to an open subset) rational map P 1 W Ü X. We may assume that C  ( 1 ) = 0( m ) 0( m) dim PC W dim X. It is then easy to see that H !P˝1 W H !X˝ for  C  m 0  m 0 = 1 ( 2 ) ( ) = 0 m . Since !P˝1 OP m , it follows easily that H !P˝1 W and hence  C 0 m  also H (!X˝ ) = 0 for all m > 0. The converse statement that non-uniruled varieties 0 m have non-negative Kodaira dimension so that H (!˝ ) 0 for some m > 0 is the X ¤ non-vanishing conjecture. This is a very difficult and important open question closely related to the famous . Thus, conjecturally, the case of negative Kodaira dimension Ä(X) = 1 corresponds to the case of uniruled varieties. There is a very precise structure theorem for these varieties.

Theorem 4 (Birkar, Cascini, Hacon, and McKernan [2010]). Let X be a uniruled vari- Ü ety. Then there is a finite sequence of flips and divisorial contractions X X 0 and a morphism f : X Z such that dim X > dim Z, (X /Z) = 1, c1(!X ) C < 0 for 0 ! 0 0 0  any curve C contained in a fiber of f .

The morphism f : X 0 Z is called a Mori fiber space. The fibers F of f are ! 1 Fano varieties with terminal singularities so that !F is an ample Q-line bundle. Fano varieties are well understood. They play an important role in algebraic geometry and many related subjects. For example it is known that they are simply connected. By the above theorem we may think of Fano varieties as the building blocks for uniruled varieties. Therefore, to understand the geometry of uniruled varieties, it is important to understand the behaviour of families of Fano varieties. The most basic question is if Fano varieties with terminal singularities of a given dimension are bounded and hence are described by finitely many parameters. More precisely, do d-dimensional Fano varieties with terminal singularities appear as fibers of finitely many algebraic families of projective varieties? This is a very important intensely studied long standing question which is known to have many applications in birational geometry and other related areas. The boundedness of Fano varieties is known in many interesting cases, including the following cases.

1. In dimension 1, the only is P 1.

2. In dimension 2, terminal singularities are smooth and it is known that there are 10 families of smooth Fano varieties. CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC GEOMETRY 67

3. In dimension 3, it is known by work of Mori and Mukai and Iskovskih that there are 105 families of smooth Fano 3-folds (Iskovskih [1977] and Mori and Mukai [1981/82]) and that there are many more families of terminal Fano threefolds. 4. Smooth Fano varieties in any dimension are bounded by results of Nadel, Cam- pana, Kollár, Mori and Miyaoka in the early 1990’s (Nadel [1991], Campana [1991], and Kollár, Miyaoka, and Mori [1992]). 5. The boundedness of terminal Fano 3-folds was shown by Kawamata [1992] and for canonical Fano 3-folds by Kollár, Miyaoka, Mori, and Takagi [2000]. 6. The boundedness of -log terminal toric Fano varieties was shown by A. A. Borisov and L. A. Borisov [1992]. 7. The boundedness of -log terminal Fano surfaces was shown by Alexeev [1994]. 8. The boundedness of -log terminal Fano varieties of bounded Cartier index, con- jectured by Batyrev, was shown by Hacon, McKernan, and Xu [2014]. Based on their results, V. Alexeev, A. Borisov and L. Borisov conjectured that the set of -log terminal Fano varieties of dimension d is bounded. This conjecture is known as the BAB conjecture and it was probably the most important and influential conjec- ture in birational geometry concerning Fano varieties. Using many of the recent results related to the minimal model program and the existence of a moduli functor for log canonically polarized varieties, in his ground breaking papers Birkar [2016a,b], Caucher Birkar proves the BAB Conjecture in full generality. Theorem 5 (Birkar). Fix d N and  > 0. The set of -log terminal projective 2 Fano varieties of dimension d is bounded. In particular d-dimensional terminal Fano varieties are bounded. The proof of this result is a technical tour de force that also sheds light on many im- portant techniques in birational geometry. In particular Birkar proves important results on the ˛-invariants of Fano varieties (answering a question of Tian with implications to K-stability and the existence of Kähler–Einstein metrics) as well as optimal results on the existence of complements that vastly generalize to arbitrary dimensions, results ob- tained by V. Shokurov in the case of surfaces. It is clear that Birkar [2016a] and Birkar [2016b] will have a major impact in birational geometry and related areas. In conclusion, I hope to have conveyed that Birational Geometry is a classical subject which has undergone a dramatic transformation in recent years due to important and sus- tained contributions to the minimal model program by many distinguished researchers. Caucher Birkar has been at the forefront of this progress and has been a key player in several of the most important breakthroughs. It is a pleasure to congratulate him on his spectacular achievements.

Acknowledgements. The author was partially supported by NSF research grants no: DMS-1300750, DMS-1801851, DMS-1265285 and by a grant from the Simons Founda- tion; Award Number: 256202. He would also like to thank the Mathematics Department and the Research Institute for Mathematical Sciences, located in Kyoto University. 68 CHRISTOPHER D. HACON

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Received 2018-06-26.

C D. H D M U U S L C, UT 84112, USA [email protected]