Finite generation of canonical rings

Let $X\subset \mathbb{P}_{\mathbb{C}}^N$ be a smooth algebraic variety i.e. a sub-manifold defined by homogeneous polynomials $P_1,...,P_t\in \mathbb{C}[z_0,...,z_N]$. $\Omega ^1_X=T_X^\vee$ denotes the cotangent bundle to $X$ and $\omega _X=\Lambda ^{\rm dim X}\Omega ^1_X$ is the canonical bundle of $X$. $H^0(\omega _X ^{\otimes m})$ denotes the space of global sections of the $m$-th tensor power of the canonical line bundle so that an element $s\in H^0(\omega _X ^{ \otimes n})$ can be written in local coordinates as $f(x_1,...,x_n)(d_{x_1}\wedge ...\wedge d_{x_n})^{\otimes m}$ for some holomorphic function $f$.

The vector spaces $H^0(\omega _X ^{\otimes m})$ play a fundamental role in understanding the geometry of $X$. If ${\rm dim} X=1$, it is well known that ${\rm dim} H^0(\omega _X )=g$ is just the geometric genus of $X$. If ${\rm dim} X=2$, then the geometry of $X$ was well understood (in terms of the groups $H^0(\omega _X ^{\otimes m})$) by the Italian school of Algebraic Geometry a century ago. For example, by a Theorem of Castelnuovo, if ${\rm dim} H^0(\omega _X ^{\otimes 2} )=0$ and ${\rm dim} H^0(\Omega _X ^1)=0$, then $X$ is birational (i.e. isomorphic outside of a measure zero set) to $\mathbb{P}_{\mathbb{C}}^2$. More recently, Mori was awarded the fields medal for completing the ${\rm dim} X=2$ picture to the case of dimension $3$.

In this talk I will discuss recent joint work with Birkar, Cascini and McKernan towards understanding the geometry of algebraic varieties of arbitrary dimension. In particular I will discuss the following:

Theorem 0.1   Let $X$ be a smooth projective algebraic variety. Then the canonical ring

$$R(X)=\bigoplus _{m\geq 0}H^0(\omega _X^{\otimes m})$$

is finitely generated.

Note that this Theorem was independently proven by Siu.