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The Mathematical Work of Caucher Birkar The Mathematical Work of Caucher Birkar Christopher Hacon University of Utah August, 2018 Christopher Hacon The Mathematical Work of Caucher Birkar Introduction It is a great pleasure to be able celebrate some of the mathematical achievements of Caucher Birkar. Caucher Birkar is being recognized for his outstanding work in Birational Algebraic Geometry of Complex Projective Varieties and especially for: Christopher Hacon The Mathematical Work of Caucher Birkar Introduction 1 His contributions to the Minimal Model Program including the finite generation of the canonical ring and the existence of flips. 2 His proof of the BAB conjecture concerning the boundedness of Fano varieties with mild singularities. Christopher Hacon The Mathematical Work of Caucher Birkar Algebraic Geometry Algebraic Geometry is the study of solutions of polynomial equations. We consider projective varieties N X = V (P1;:::; Pr ) ⊂ PC where Pi 2 C[x0;:::; xn] are homogeneous polynomial equations and N = ( N+1 n 0)¯ = ∗ is the N-dimensional PC C C projective space. N ⊃ N is a natural compactification obtained by adding the PC C hyperplane at infinity H = N n N =∼ N−1. PC C PC Typically we will assume that X is irreducible and smooth, hence a complex manifold of dimension d = dim X . Christopher Hacon The Mathematical Work of Caucher Birkar Birational equivalence Two varieties are birational if they have isomorphic open subsets (in the Zariski topology, closed subsets are zeroes of polynomial equations). It is easy to see that two varieties are birational if they have ∼ the same field of rational functions C(X ) = C(Y ). Recall that by Hironaka's theorem on the resolution of singularities (1964), for any variety X , there is a finite sequence of blow ups along smooth subvarieties 0 X = Xn ! Xn−1 ! ::: X1 ! X such that X 0 is smooth. If Z ⊂ X are smooth varieties, then the blow up blZ (X ) ! X of X along Z replaces the subset Z ⊂ X by the codimension 1 subvariety E = P(NZ X ). We say that E is an exceptional divisor. Christopher Hacon The Mathematical Work of Caucher Birkar Christopher Hacon The Mathematical Work of Caucher Birkar Canonical ring The geometry of varieties X ⊂ N is typically studied in terms PC dim X _ of the canonical line bundle !X = ^ TX . 0 ⊗n A section s 2 H (!X ) can be written in local coordinates as ⊗m f (z1;:::; zn)(dz1 ^ ::: ^ dzn) : Of particular importance is the canonical ring M 0 ⊗n R(!X ) = H (!X ) n≥0 a birational invariant of smooth projective varieties. The Kodaira dimension of X is defined by κ(X ) := tr:deg:CR(!X ) − 1 2 {−1; 0; 1;:::; d = dim X g: Note that complex projective manifolds have no non-constant global holomorphic functions, so it is natural to consider global sections of line bundles. There is only one natural choice: the canonical line bundle! Christopher Hacon The Mathematical Work of Caucher Birkar Canonical ring of hypersurfaces N For example, if X = P then !X = O N (−N − 1). C PC If X ⊂ N is a smooth hypersurface of degree k, then k PC !Xk = O N (k − N − 1)jXk . PC Here O N (l) is the line bundle corresponding to homogeneous PC polynomials of degree l and O N (l)jXk is the line bundle PC obtained by restriction to X ⊂ N . k PC ∼ It is easy to see that if k ≤ N then R(!Xk ) = C and so κ(Xk ) = −1, ∼ if k = N + 1, then R(!Xk ) = C[t] and so κ(Xk ) = 0, and if k ≥ N + 2, then κ(Xk ) = dim Xk . Christopher Hacon The Mathematical Work of Caucher Birkar Canonical ring of curves When d = dim X = 1 we say that X is a curve and we have 3 cases: ∼ 1 κ(X ) = −1: Then X = P is a rational curve. Note that ∼ C ∼ !P1 = OP1 (−2) and so R(!X ) = C. ∼ κ(X ) = 0: Then !X = OX and X is an elliptic curve. There is a one parameter family of these given by the equations x2 = y(y − 1)(y − s): ∼ In this case R(!X ) = C[t]. Christopher Hacon The Mathematical Work of Caucher Birkar Canonical ring of curves If κ(X ) = 1, then we say that X is a curve of general type. These are Riemann surfaces of genus g ≥ 2. For any g ≥ 2 they belong to a 3g − 3 dimensional algebraic family. We have deg(!X ) = 2g − 2 > 0. ⊗m By Riemann Roch, it is easy to see that !X is very ample for m ≥ 3. This means that if s0;:::; sN are a basis of 0 ⊗m H (!X ), then N φm : X ! P ; x ! [s0(x): s1(x): ::: : sN (x)] is an embedding. ⊗m ∼ ∗ Thus !X = φmOPN (1) and in particular R(!X ) is finitely generated. Christopher Hacon The Mathematical Work of Caucher Birkar Christopher Hacon The Mathematical Work of Caucher Birkar Birational equivalence One would like to prove similar results in higher dimensions. If dim X ≥ 2, the birational equivalence relation is non-trivial. In dimension 2, any two smooth birational surfaces become isomorphic after finitely many blow ups of smooth points (Zariski, 1931). In dimension ≥ 3 the situation is much more complicated, however it is known by work of Wlodarczyk (1999) and Abramovich-Karu-Matsuki-Wlodarczyk (2002), that the birational equivalence relation is generated by blow ups along smooth centers. It is easy to see that if X ; X 0 are birational smooth varieties, ∼ 0 ∼ then π1(X ) = π1(X ) and R(!X ) = R(!X 0 ). However, typically, they have different Betti numbers eg. 0 b2(X ) 6= b2(X ). Christopher Hacon The Mathematical Work of Caucher Birkar Christopher Hacon The Mathematical Work of Caucher Birkar Higher dimensions One of the main goals of birational geometry is to produce a distinguished representative with good geometric properties in any birational equivalence class. In dimension 2 this was achieved by the Italian school of algebraic geometry at the beginning of the 20-th century. In dimension 3, this was finally proven by Mori and others in the 1980's. Around the same time, the minimal model program (MMP), was established by work of Kawamata, Koll´ar,Mori, Reid, Shokhurov and others. The MMP is a general framework for the birational classification of higher dimensional projective varieties. In arbitrary dimension the program has not yet been completed but there has been some spectacular progress. Christopher Hacon The Mathematical Work of Caucher Birkar Finite generation Theorem (Birkar-Cascini-Hacon-McKernan, Siu 2010) If X is a smooth complex projective variety, then the canonical ring R(!X ) is finitely generated. Corollary (Birkar-Cascini-Hacon-McKernan) If X is of general type (κ(X ) = dim X ), then X has a canonical model Xcan and a minimal model Xmin. Christopher Hacon The Mathematical Work of Caucher Birkar Canonical models The canonical model Xcan := Proj(R(!X )) is a distinguished "canonical" (unique) representative of the birational equivalence class of X which is defined by the generators and relations in the finitely generated ring R(!X ). Xcan may be singular, but its singularities are mild (canonical). In particular they are cohomologically insignificant so that e.g. i ∼ i H (OX ) = H (OXcan ) for 0 ≤ i ≤ dim X . The "canonical line bundle" is now a Q-line bundle which means that !⊗n is a line bundle for some n > 0. Xcan ⊗m ! is ample so that ! = O N (1)j for some m > 0. Xcan Xcan P Xcan This result has lead to the proof of the existence of projective moduli spaces for canonically polarized (stable) varieties. In particular, for any fixed dimension d = dim X and volume d v = c1(!Xcan ) there is an algebraic variety parametrizing all canonical models of dimension d and volume v. Christopher Hacon The Mathematical Work of Caucher Birkar Varieties of intermediate Kodaira dimension When 0 ≤ κ(X ) ≤ dim X − 1 it is still expected that X has a minimal model Xmin which has mild (terminal) singularities, is not unique, and such that c1(!Xmin ) · C ≥ 0 for any curve C ⊂ Xmin. It is conjectured that minimal models X 99K Xmin are obtained by a finite sequence of well understood elementary birational transformations (known as flips and divisorial contractions) that remove certain !X negative curves and replace them with points or with !X positive curves. The existence of flips is another result of fundamental importance proved by Birkar-Cascini-Hacon-McKernan (2010). In order to prove the existence of minimal models it then suffices to prove the termination of flips i.e. that any sequence of flips (and divisorial contractions) is finite. This is an important conjecture and Birkar has proven some of the strongest results towards this conjecture. Christopher Hacon The Mathematical Work of Caucher Birkar Christopher Hacon The Mathematical Work of Caucher Birkar Termination Theorem (Birkar) Assume the non-vanishing conjecture. Then certain sequences of flips (known as flips with scaling) terminate. In particular if X is not uniruled then it has a minimal model X 99K Xmin. Uniruled varieties are varieties covered by rational curves. Since H0(!⊗m) = 0 for all m > 0, it follows easily that if X is P1 0 ⊗m uniruled, then H (!X ) = 0 for all m > 0. The non-vanishing conjecture asserts that if X is a smooth 0 ⊗m projective variety which is not uniruled then H (!X ) 6= 0 for some m 2 N. This is closely related to the famous abundance conjecture. Actually, there is a more precise structure theorem for uniruled varieties. Christopher Hacon The Mathematical Work of Caucher Birkar Mori Fiber Spaces Theorem (Birkar-Cascini-Hacon-McKernan) Let X be a uniruled variety.
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