P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (61–70) CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC GEOMETRY C D. H Abstract On Wednesday, August 1st 2018, Caucher Birkar was awarded the Fields Medal for his contributions to the minimal model program and his proof of the bounded- ness of -log canonical Fano varieties. In this note I will discuss some of Birkar’s main achievements. Algebraic geometry studies the solution sets of systems of polynomial equations. A N complex projective variety 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 projective space. 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 ring. 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 line bundle 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 canonical ring 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 birational geometry 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 Kodaira dimension 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.