The Mathematical Work of

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 of Complex Projective Varieties and especially for:

Christopher Hacon The Mathematical Work of Caucher Birkar Introduction

1 His contributions to the including the finite generation of the 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 ∈ C[x0,..., xn] are homogeneous polynomial equations and N = ( N+1 \ 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−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 ωX = ∧ TX . 0 ⊗n A section s ∈ H (ωX ) can be written in local coordinates as ⊗m f (z1,..., zn)(dz1 ∧ ... ∧ dzn) . Of particular importance is the canonical ring M 0 ⊗n R(ωX ) = H (ωX ) n≥0 a birational invariant of smooth projective varieties. The of X is defined by

κ(X ) := tr.deg.CR(ωX ) − 1 ∈ {−1, 0, 1,..., d = dim X }. 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 ωX = O N (−N − 1). C PC If X ⊂ N is a smooth hypersurface of degree k, then k PC ωXk = O N (k − N − 1)|Xk . PC Here O N (l) is the line bundle corresponding to homogeneous PC polynomials of degree l and O N (l)|Xk is the line bundle PC obtained by restriction to X ⊂ N . k PC ∼ It is easy to see that if k ≤ N then R(ωXk ) = C and so κ(Xk ) = −1, ∼ if k = N + 1, then R(ωXk ) = 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 ∼ ωP1 = OP1 (−2) and so R(ωX ) = C. ∼ κ(X ) = 0: Then ωX = 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(ωX ) = 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(ωX ) = 2g − 2 > 0. ⊗m By Riemann Roch, it is easy to see that ωX is very ample for m ≥ 3. This means that if s0,..., sN are a basis of 0 ⊗m H (ωX ), then

N φm : X → P , x → [s0(x): s1(x): ... : sN (x)]

is an embedding. ⊗m ∼ ∗ Thus ωX = φmOPN (1) and in particular R(ωX ) 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(ωX ) = R(ωX 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 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 , then the canonical ring R(ωX ) 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(ωX )) 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(ωX ).

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 ω⊗n is a line bundle for some n > 0. Xcan ⊗m ω is ample so that ω = O N (1)| 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(ωXcan ) 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(ωXmin ) · 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 ωX negative curves and replace them with points or with ωX 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(ω⊗m) = 0 for all m > 0, it follows easily that if X is P1 0 ⊗m uniruled, then H (ωX ) = 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 (ωX ) 6= 0 for some m ∈ N. This is closely related to the famous . 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. Then there is a finite sequence of flips 0 0 and divisorial contractions X 99K X and a morphism f : X → Z 0 such that: dim X > dim Z, ρ(X /Z) = 1, c1(ωX 0 ) · C < 0 for any curve C contained in a fiber of f .

f : X 0 → Z is a Mori fiber space. The fibers F of f are Fano varieties with terminal −1 singularities so that ωF is an ample Q-line bundle. They are well understood (for example π1(F ) = 0). Thus we think of Fano varieties as the building blocks for uniruled varieties. They play an important role in algebraic geometry and many related subjects.

Christopher Hacon The Mathematical Work of Caucher Birkar Christopher Hacon The Mathematical Work of Caucher Birkar Fano varieties

The most important question related to Fano varieties is: Are Fano varieties with mild singularities (terminal or even -log-terminal singularities) bounded? I.e. do they belong to finitely many algebraic families? Several versions of this question have appeared prominently in the litterature and are known to have many important consequences (existence of K¨ahler-Einsteinmetrics, applications to Cremona groups, ....)

Christopher Hacon The Mathematical Work of Caucher Birkar Boundedness of Fanos

In dimension 2, varieties with terminal singularities are smooth and it is known that there are 10 possibilities (algebraic families). In dimension 3, there are 105 families of smooth Fano’s (Iskovskih 1989 and Mori-Mukai 1981) and many more families of terminal 3-folds. The boundedness of terminal Fano 3-folds was shown by Kawamata (1992) (and by Koll´ar-Mori-Miyaoka-Takagi, 2000 in the canonical case). The boundedness of smooth Fano varieties in any dimension was shown by Nadel, Campana, Koll´ar,Mori and Miyaoka in the early 1990’s.

Christopher Hacon The Mathematical Work of Caucher Birkar BAB conjecture

The boundedness of (-log-) terminal toric Fanos was shown by A. Borisov and L. Borisov in 1993 and for -log terminal surfaces by Alexeev in 1994. The BAB conjecture claims that for any  > 0 Fano varieties of fixed dimension with -log terminal singularities are bounded. In recent spectacular progress, Caucher Birkar was able to prove that this conjecture is true in full generality.

Theorem (Birkar (2016)) The BAB conjecture holds, in particular the set of all terminal Fano varieties in any fixed dimension is bounded.

Christopher Hacon The Mathematical Work of Caucher Birkar Conclusion

In conclusion, I hope that I have conveyed that Birational Geometry is a classical subject which has recently undergone a dramatic transformation due to important sustained contribuitions to the minimal model program by many distinguished researchers. Caucher has played a central role in this process and has been involved in many of the most important breakthroughs. Please join me in congratulating Caucher on his spectacular achievements!

Christopher Hacon The Mathematical Work of Caucher Birkar