Kawamata rationality theorem

A theorem stating that there is a strong restriction for the canonical divisor of an algebraic variety to be negative while the positivity is arbitrary. It is closely related to the structure of the cone of curves and the existence of rational curves.

Definitions and terminology.

Let $X$ be a normal algebraic variety (cf. Algebraic variety). A $\mathbf{Q}$-divisor $B = \sum _ { j = 1 } ^ { t } b _ { j } B _ { j }$ on $X$ is a formal linear combination of a finite number of prime divisors $B _ { j }$ of $X$ with rational number coefficients $b _ { j }$ (cf. also Divisor). The canonical divisor $K _ { X }$ is a Weil divisor on $X$ corresponding to a non-zero rational differential $n$-form for $n = \operatorname { dim } X$ (cf. also Differential form). The pair $( X , B )$ is said to be weakly log-terminal if the following conditions are satisfied:

The coefficients of $B$ satisfy $0 \leq b _ { j } \leq 1$.

There exists a positive integer $r$ such that $r ( K _ { X } + B )$ is a Cartier divisor (cf. Divisor).

There exists a projective birational morphism $\mu : Y \rightarrow X$ from a smooth variety such that the union

\begin{equation*} \sum _ { j = 1 } ^ { t } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } D _ { k } \end{equation*}

is a normal crossing divisor (cf. Divisor), where $\mu_* ^ {- 1 } B _ { j }$ is the strict transform of $B _ { j }$ and $\cup _ { k = 1 } ^ { s } D _ { k }$ coincides with the smallest closed subset $E$ of $Y$ such that $\mu | _ { Y \backslash E } : Y \backslash E \rightarrow X \backslash \mu ( E )$ is an isomorphism.

One can write

\begin{equation*} \mu ^ { * } ( K _ { X } + B ) = K _ { Y } + \sum _ { j = 1 } ^ { t } b _ { j } \mu _ { * } ^ { - 1 } B _ { j } + \sum _ { k = 1 } ^ { s } d _ { k } D _ { k } \end{equation*}

such that $d _ { k } < 1$ for all $k$.

There exist positive integers $e_k$ such that the divisor $- \sum _ { k = 1 } ^ { s } e _ { k } D _ { k }$ is $\mu$-ample (cf. also Ample vector bundle).

For example, the pair $( X , B )$ is weak log-terminal if $X$ is smooth and $B = \sum _ { j = 1 } ^ { t } B _ { j }$ is a normal crossing divisor, or if $X$ has only quotient singularities and $B = 0$.

Rationality theorem.

Let $X$ be a normal algebraic variety defined over an algebraically closed field of characteristic $0$, and let $B$ be a $\mathbf{Q}$-divisor on $X$ such that the pair $( X , B )$ is weakly log-terminal. Let $f : X \rightarrow S$ be a projective morphism (cf. Projective scheme) to another algebraic variety $S$, and let $H$ be an $f$-ample Cartier divisor on $X$. Then (the rationality theorem, [a1])

\begin{equation*} \lambda = \operatorname { sup } \{ t \in \mathbf{Q} : H + t ( K _ { X } + B ) \text { is } f\square \text{ ample} \} \end{equation*}

is either $+ \infty$ or a rational number. In the latter case, let $r$ be the smallest positive integer such that $r ( K _ { X } + B )$ is a Cartier divisor, and let $d$ be the maximum of the dimensions of geometric fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive integers $p$ and $q$. Then $q \leq r ( d + 1 )$.

For example, equality is attained when $X = {\bf P} ^ { d }$, $B = 0$, $S$ is a point, and $H$ is a hyperplane section.

The following theorem asserts the existence of a rational curve, a birational image of the projective line $\mathbf{P}^{1}$, and provides a more geometric picture. However, the estimate of the denominator $q \leq 2 d r$ obtained is weaker: In the situation of the above rationality theorem, if $\lambda \neq + \infty$, then there exists a morphism $g : \mathbf{P} ^ { 1 } \rightarrow X$ such that $f \circ g ( \mathbf{P} ^ { 1 } )$ is a point and $0 < - ( K _ { X } + B ) , g ( \mathbf{P} ^ { 1 } ) \leq 2 d$ [a2].

The two theorems are related in the following way: If $\lambda \neq + \infty$, then $H + \lambda ( K _ { X } + B )$ is no longer $f$-ample. However, there exists a positive integer $m_0$ such that the natural homomorphism

\begin{equation*} f ^ { * } f_{*} \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \rightarrow \end{equation*}

\begin{equation*} \rightarrow \mathcal{O} _ { X } ( m q ( H + \lambda ( K _ { X } + B ) ) ) \end{equation*}

is surjective for any positive integer $m \geq m _ { 0 }$ (the base-point-free theorem, [a1]). Let $\phi : X \rightarrow Y$ be the associated morphism over the base space $S$. Then any positive dimensional fibre of $\phi$ is covered by a family of rational curves as given in the second theorem [a2].

References

[a1]	Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , Algebraic Geometry (Sendai 1985) , Adv. Stud. Pure Math. , 10 , Kinokuniya& North-Holland (1987) pp. 283–360 MR0946243 Zbl 0672.14006
[a2]	Y. Kawamata, "On the length of an extremal rational curve" Invent. Math. , 105 (1991) pp. 609–611 MR1117153 Zbl 0751.14007