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Pluricanonical systems of projective varieties of general type. I. (English) Zbl 1142.14012
For 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 \(\dim X=n\).

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
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