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Pluricanonical systems on algebraic varieties of general type. (English) Zbl 1108.14031
Let $$X$$ be a smooth complex projective variety of dimension $$n$$. $$X$$ is of general type if $$\text{ vol}(K_X)>0$$. Recall that the $$\text{ vol}(K_X)$$ is the limit of $$n! h^0( mK_X)/m^n$$ as $$m\to \infty$$. Equivalently, $$X$$ is of general type if $$|mK_X|$$ defines a birational map for some $$m>0$$. It is a fundamental problem in algebraic geometry to understand the behaviour of the pluricanonical maps (i.e. the rational maps defined by $$|mK_X|$$ for $$m>0$$). It is well known that if $$X$$ is of general type and $$\text{ dim} X=1$$ (i.e. $$X$$ is a curve of genus $$g\geq 2$$), then $$|mK_X|$$ is very ample (and hence birational) for all $$m\geq 3$$. If $$X$$ is a surface of general type, then by a result of Bombieri, $$|mK_X|$$ is birational for all $$m\geq 5$$. In this paper, the author proves the following:
Theorem. For any integer $$n>0$$, there exists a positive integer $$m_n>0$$ depending only on $$n$$ such that for any smooth complex projective variety of general type and dimension $$n$$, and for all integers $$m>m_n$$ the pluricanonical system $$|mK_X|$$ defines a birational map. Moreover, there exists a constant $$\nu _n>0$$ such that $$\text{ vol}(K_X)>\nu _n$$.
The theorem is not effective in the sense that the constants $$m_n$$ and $$\nu _n$$ can not be computed with the methods of this paper.
The proof relies on many of the key techniques of modern higher dimensional algebraic geometry such as multiplier ideals and log canonical centers. The main new ingredient is a technique to extend pluricanonical divisors from a log canonical center to the ambient variety following ideas of Siu and Tsuji. While the ideas used are very technical, the presentation of this beautiful paper is very clear and precise.
It should be pointed out that the proofs of these results very closely follow the ideas of H. Tsuji given in the preprint [arXiv:math.AG/9909021; cf. Osaka J. Math. 43, No. 4, 967–995 (2006; Zbl 1142.14012); ibid. 44, No. 3, 723–764 (2007; Zbl 1186.14043)] where the above theorem was first announced. Another proof of these results (also following the ideas of Tsuji) due to C. Hacon and J. McKernan can be found in [Invent. Math. 166, No. 1, 1–25 (2006; Zbl 1121.14011)].

##### MSC:
 14J40 $$n$$-folds ($$n>4$$) 14E05 Rational and birational maps
##### Keywords:
pluricanonical maps; varieties of general type
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##### References:
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