Pluricanonical systems of projective varieties of general type. I.

*(English)*Zbl 1142.14012For any complex projective manifold \(X\) of general type with canonical divisor \(K_X\) the rational map \(\varphi_m\) defined by the pluricanonical system \(| mK_X| \) is birational for \(m\geq m_0\). It was proved by the author in a subsequent paper [Osaka J. Math. 44, No. 3, 723–764 (2007; Zbl 1186.14043)] and independently by other authors, that \(m_0\) only depends on \(\dim X\): For every \(n\in\mathbb N\) there exists a number \(\nu_n\in\mathbb N\) such that \(\varphi_m\) is birational for every \(m\geq \nu_n\), \(n=\dim X\).

In the present paper the proof of this statement is still based on the additional assumption that every projective variety of general type has a minimal model, namely a projective variety \(X_m\) which is birationally equivalent to \(X\), with only \(\mathbb Q\)-factorial terminal singularities and with \(K_{X_m}\) a nef \(\mathbb Q\)-Cartier divisor. The subadjunction theorem of Kawamata is also essential for the proof. It is shown that the theorem is equivalent to the following fact: For every \(n\in\mathbb N\) there exists a positive number \(C_n\) such that \[ n!\cdot\overline{\lim\limits_{m\rightarrow\infty}}m^{-n}\dim H^0(X,{\mathcal O}_X(mK_X))\geq C_n, \] for every complex projective manifold \(X\) of general type with \(\dim X=n\).

In the present paper the proof of this statement is still based on the additional assumption that every projective variety of general type has a minimal model, namely a projective variety \(X_m\) which is birationally equivalent to \(X\), with only \(\mathbb Q\)-factorial terminal singularities and with \(K_{X_m}\) a nef \(\mathbb Q\)-Cartier divisor. The subadjunction theorem of Kawamata is also essential for the proof. It is shown that the theorem is equivalent to the following fact: For every \(n\in\mathbb N\) there exists a positive number \(C_n\) such that \[ n!\cdot\overline{\lim\limits_{m\rightarrow\infty}}m^{-n}\dim H^0(X,{\mathcal O}_X(mK_X))\geq C_n, \] for every complex projective manifold \(X\) of general type with \(\dim X=n\).

Reviewer: Eberhard Oeljeklaus (Bremen)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |

14E05 | Rational and birational maps |

14J40 | \(n\)-folds (\(n>4\)) |

32U05 | Plurisubharmonic functions and generalizations |

14E25 | Embeddings in algebraic geometry |

##### Keywords:

multiplier ideal sheaves; minimal model program; subadjunction theorem; pluricanonical system
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