A canonical bundle formula.

*(English)*Zbl 1032.14014From the paper: If \(f:X\to C\) is a minimal elliptic surface over \(\mathbb{C}\), then the relative canonical divisor \(K_{X/C}\) is expressed as
\[
K_{X/C} =f^* L+\sum_P {m_P-1\over m_P}f^*(P), \tag{1}
\]
where \(L\) is a nef divisor on \(C\) and \(P\) runs over the set of points such that \(f^*(P)\) is a multiple fiber with multiplicity \(m_P>1\). It is the key in the estimates on the plurigenera \(P_n(X)\) that the coefficients \((m_P-1)/m_P\) are ‘close’ to 1. Furthermore \(12L\) is expressed as
\[
12K_{X/C}= f^*j^*{\mathcal O}_\mathbb{P}(1)+12 \sum_P{m_p-1\over m_P}f^* (P)+ \sum \sigma_Qf^*(Q), \tag{2}
\]
where \(\sigma_Q\) is an integer \(\in [0,12)\) and \(j:C\to \mathbb{P}^1\) is the \(j\)-function. The computation of these coefficients is based on the explicit classification of the singular fibers of \(f\), which made the generalization difficult.

The higher dimensional analogue of the formula (2) is treated in section 2 and the log version in section 4. The estimates of the coefficients are treated in 2.8, 3.1 and 4.5. We note that the “coefficients” in the formula (2) are of the form \(1-1/m\) except for the finite number of exceptions \(1/12,\dots, 11/12\).

The following are some of the applications.

1. (Corollary 5.3) If \((X,\Delta)\) is a \(klt\) pair (that is: \(X\) is a normal variety and \(\Delta\) an effective divisor) with \(\kappa (X,K_X+ \Delta)\leq 3\), then its log-canonical ring is finitely generated.

2. (Corollary 6.2) There exists an effectively computable natural number \(M\) such that \(|MK_X |\) induces the Iitaka fibering for every algebraic threefold \(X\) with Kodaira dimension \(\kappa(X)=1\).

To get the analogue for an \((m+1)\)-dimensional \(X\) \((m\geq 3)\) with \(\kappa(X) =1\), it remains to show that an arbitrary \(m\)-fold \(F\) with \(\kappa(F)=0\) and \(p_g(F)=1\) is birational to a smooth projective model with effectively bounded \(m\)-th Betti number.

The higher dimensional analogue of the formula (2) is treated in section 2 and the log version in section 4. The estimates of the coefficients are treated in 2.8, 3.1 and 4.5. We note that the “coefficients” in the formula (2) are of the form \(1-1/m\) except for the finite number of exceptions \(1/12,\dots, 11/12\).

The following are some of the applications.

1. (Corollary 5.3) If \((X,\Delta)\) is a \(klt\) pair (that is: \(X\) is a normal variety and \(\Delta\) an effective divisor) with \(\kappa (X,K_X+ \Delta)\leq 3\), then its log-canonical ring is finitely generated.

2. (Corollary 6.2) There exists an effectively computable natural number \(M\) such that \(|MK_X |\) induces the Iitaka fibering for every algebraic threefold \(X\) with Kodaira dimension \(\kappa(X)=1\).

To get the analogue for an \((m+1)\)-dimensional \(X\) \((m\geq 3)\) with \(\kappa(X) =1\), it remains to show that an arbitrary \(m\)-fold \(F\) with \(\kappa(F)=0\) and \(p_g(F)=1\) is birational to a smooth projective model with effectively bounded \(m\)-th Betti number.