5.1. Log terminal singularities [02EM]

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5.1. Log terminal singularities

Since this material may not be familiar to complex analysts or differential geometers, we briefly recall some basic facts on some of the singularities encountered in the Minimal Model Program (MMP for short). See [KM] for a detailled account in the algebraic case, the analytic theory being also surveyed there in less detail.

The sheaf of holomorphic functions $\mathcal{O}_{V}$ is the subsheaf of the sheaf of continuous functions on $V$ consisting of the functions whose restriction to $V^{reg}$ is holomorphic. Actually, by Hartogs’ theorem, any holomorphic function on $V^{reg}$ extends to $V$ , which means that $j_{*}\mathcal{O}_{V^{reg}}=\mathcal{O}_{V}$ .

Every meromorphic n-form $\alpha$ on $V^{reg}$ extends to $V$ , i.e. let $\pi:X\to V$ be a resolution of singularities of $V$ , then the meromorphic $n$ -form $\pi^{*}\alpha$ defined on $\pi^{-1}V^{reg}$ extends to a meromorphic $n$ -form on $X$ . Let $\omega_{V^{reg}}$ be the canonical sheaf of the smooth variety $V^{reg}$ . The sheaf $\omega_{V}=j_{*}\omega_{V^{reg}}$ is a coherent analytic sheaf on $V$ .

More generally every meromorphic pluricanonical form on $V^{reg}$ extends to $V$ and $\omega^{[q]}_{V}=j_{*}\omega_{V^{reg}}^{q}$ , $q>0$ is a coherent analytic sheaf on $V$ .

Definition 5.1.

Say $V$ is 1-Gorenstein iff one of the following equivalent conditions holds:

(1)

Every $x\in V$ has an open neighborhood $U$ such that $U^{reg}$ carries an holomorphic $n$ -form with an empty zero divisor.
(2)

$\omega_{V}$ is a rank one locally free sheaf.
(3)

Every $x\in V$ has an open neighborhood $U$ such that $\omega_{U^{reg}}$ is isomorphic to $\mathcal{O}_{V^{reg}}|_{U}$ .

A local section of $\omega_{V}$ defining an holomorphic $n$ -form without zeroes on $V^{reg}$ will be called a local generator of $\omega_{V}$ . If furthermore $V$ is Cohen-Macaulay, $V$ is said to be Gorenstein.

Say $V$ is $\mathbb{Q}$ -Gorenstein iff one of the following equivalent conditions is satisfied:

(1)

Every $x\in V$ has an open neighborhood $U$ such that $U^{reg}$ carries an holomorphic pluricanonical form with an empty zero divisor.
(2)

For every $x\in V$ , there exists $N_{x}\in\mathbb{N}$ and an open neighborhood $U$ of $x$ such that $\omega^{[N_{x}]}_{U}$ is a rank one locally free sheaf.
(3)

For every $x\in V$ there is $N_{x}\in\mathbb{N}$ and an open neighborhood $U$ of $x$ such that $\omega^{N_{x}}_{U^{reg}}$ is isomorphic to $\mathcal{O}_{V^{reg}}|_{U}$ .

A local section of $\omega^{[N]}_{V}$ defining an holomorphic pluricanonical form without zeroes on $V^{reg}$ will be called a local generator of $\omega^{[N]}_{V}$ .

For every $x\in V$ , the smallest $N_{x}$ fulfilling condition 3 near $x$ is called the local index of $V$ at $x$ . The l.c.m. of all local indices, if finite, is called the index of $V$ .

Definition 5.2.

Say $V$ has only canonical singularities iff $V$ is $\mathbb{Q}$ -Gorenstein, of finite index $N$ and one of the following equivalent conditions is fulfilled:

(1)

Let $\pi:X\to V$ be a resolution. Let $\alpha$ be a local generator of $\omega_{V}^{[N]}$ . The meromorphic pluricanonical form $\pi^{*}\alpha$ is holomorphic.
(2)

Let $\pi:X\to V$ be a resolution. For every $m\in\mathbb{N},\ \pi_{*}\omega_{X}^{[Nm]}=\omega_{V}^{[Nm]}$ .
(3)

(Assuming $V$ is an algebraic variety) Let $\pi:X\to V$ be a resolution. Then $K_{X}\cong\pi^{*}K_{V}+\displaystyle\sum a_{E}E$ with $a_{E}\geq 0$ where $\cong$ means numerical equivalence of $\mathbb{Q}$ -Cartier divisors and the sum runs over the exceptional divisors of $\pi$ .

Observe that it is enough to check the first two conditions for some resolution. In the third condition $Na_{E}$ is the order of vanishing of $\pi^{*}\alpha$ along the divisor $E$ .

Definition 5.3.

Say $V$ has only log-terminal singularities iff $V$ is $\mathbb{Q}$ -Gorenstein, of finite index $N$ and the following holds: let $\pi:X\to V$ be a log-resolution and let $\alpha$ be a local generator of $\omega_{V}^{[N]}$ : then the pole along any component $E$ of $exc(\pi)$ of the meromorphic $N$ -canonical form $\pi^{*}\alpha$ on $X$ is of order $\leq N-1$ .

When $V$ is algebraic an equivalent formulation is: let $\pi:X\to V$ be a log-resolution. Then $K_{X}\cong\pi^{*}K_{V}+\displaystyle\sum_{E}a_{E}E$ with $a_{E}>-1$ .

The importance of the class of canonical singularities comes from a theorem due to M. Reid [R 1] (see also [Deb], p. 174):

Theorem 5.4.

Let $X$ be a projective algebraic manifold of general type whose canonical ring $R={\displaystyle\oplus_{n\in\mathbb{N}}}H^{0}(X,\omega_{X}^{n})$ is of finite type. Then the canonical model of $X$ , $X_{can}:=Proj(R)$ has only canonical singularities. If $N=Index(X_{can})$ then $\omega^{[N]}_{X_{can}}$ is ample.

The finiteness of the canonical ring for varieties of general type is known in dimension 3 [Ka]. In higher dimension, Y.Kawamata has proved that it is a consequence of the existence of minimal models. $X_{can}$ is a uniquely defined singular birational model of $X$ . The minimal models of $X$ in the sense of the MMP are crepant terminalizations of $X_{can}$ and do not enjoy the above unicity since they may be related by non trivial flops.

Examples 5.5.

Let $S$ be a normal algebraic surface. The following are equivalent:

(1)

$S$ has only canonical singularities.
(2)

$S$ is locally analytically isomorphic to $X=\mathbb{C}^{2}/G$ , $G\subset SL_{2}(\mathbb{C})$ a finite subgroup.
(3)

The exceptional divisors of the minimal resolution $\pi_{min}$ of $S$ , have simple normal crossings, their components are (-2) smooth rational curves, their incidence graphs are of type A-D-E (Du Val singularities).

The log terminal surface singularities are precisely the singularities of the form $X=\mathbb{C}^{2}/G$ , $G\subset GL_{2}(\mathbb{C})$ a finite subgroup.

Examples 5.6.

In higher dimension, quotient singularities are still log terminal. Fix $n>0$ and let $H\subset\mathbb{C}{\mathbb{P}}^{n+1}$ be a smooth degree $d$ hypersurface. The affine cone over $H$ has only canonical singularities iff $d\leq n+1$ .

In particular, the ordinary double point $x^{2}+y^{2}+z^{2}+t^{2}=0$ has only canonical singularities but it is not a quotient singularity.

The hypersurface singularities of type $A-D-E$ are canonical.